Trabajo de Investigacion Tutelada

Programa de Doctorado en Fısica yMatematicas(FISYMAT)

Propiedades teorico-informacionales delos polinomios de Jacobi

Estudiante: Angel Guerrero Martınez

Tutores: Pablo Sanchez-Moreno yJesus Sanchez-Dehesa Moreno-Cid

Instituto Carlos I de Fısica Teorica y ComputacionalDpto. de Fısica Atomica, Molecular y Nuclear

Dpto. de Matematica AplicadaUniversidad de Granada

Granada, Septiembre 2010

Indice general

Resumen y estructura del trabajo 5

Artıculo: Information-theoretic lengths of Jacobi polynomials(publicado en J. Physics A: Math. Theor. 43, 305203, 2010) 7

Apendices 11A.1. Polinomios ortogonales clasicos . . . . . . . . . . . . . . . . . 11

A.1.1. Ortogonalidad de los P.O.C. . . . . . . . . . . . . . . . 13A.1.2. Relacion de recurrencia a tres terminos . . . . . . . . 14A.1.3. Densidad de Rakhmanov . . . . . . . . . . . . . . . . 15

A.2. Desarrollo de los P.O.C. en serie de potencias . . . . . . . . . 16A.3. Desarrollo de potencias en serie de P.O.C. . . . . . . . . . . . 19A.4. Valores esperados . . . . . . . . . . . . . . . . . . . . . . . . . 20

A.4.1. Relacion de recurrencia a tres terminos para obtener〈xk〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

A.4.2. Metodo general para calcular 〈xk〉 . . . . . . . . . . . 23

3

4 INDICE GENERAL

Resumen y estructura deltrabajo

Los polinomios de Jacobi P α,βn (x), (α, β > 1) son polinomios hiperge-

ometricos reales ortogonales con respecto a la funcion peso (1 − x)α(1 +x)β en el intervalo [−1,+1]. Estas funciones especiales aparecen frecuente-mente en numerosos problemas de Matematica Aplicada, Fısica Matematicay Fısica Cuantica. En este trabajo se investiga la distribucion de la densi-dad de Rakmanov de estos polinomios por medio de las longitudes teorico-informacionales de varios tipos (Renyi, Shannon, Fisher) recientemente usa-dos en el estudio del esparcimiento de otras funciones especiales en su dominiode definicion [6, 7]. Estas magnitudes permiten obtener informacion local yglobal de los polinomios de Jacobi de caracter complemetario a la aportadapor la desviacion estandar.

En primer lugar se obtienen las expresiones explıcitas de la desviacionestandar y de la informacion de Fisher. Las longitudes de Renyi se expre-san en terminos de los polinomios multivariados de Bell, que dependen delgrado n y de los parametros α y β. La longitud de Shannon, que no puedecalcularse analıticamente de forma exacta debido a la naturaleza logarıtmicade su forma funcional, es acotada variacionalmente y su asintotica es tambi-en estudiada por medio de una tecnica teorico-informacional. Finalmente, selleva a cabo un analisis computacional de todas estas magnitudes.

Este trabajo ha dado lugar a un artıculo de coordenadas “Journal ofPhysics A: Matematics and Theoretical 43, 305203 (2010)”, que esta estruc-turado de la siguiente forma. En la Seccion 1 se describen los polinomios deJacobi y sus aplicaciones en varios campos cientıficos y tecnologicos. En laSeccion 2, se introducen las longitudes teorico-informacionales de una den-sidad de probabilidad y se pone de manifiesto su significado matematicorelacionado con el esparcimiento polinomico. A continuacion, en la Seccion3, se calculan explıcitamente los momentos de la densidad de Rakhmanov delos polinomios de Jacobi ası como su desviacion estandar y su informacion deFisher. En la Seccion 4 se describe un metodo de calculo de las longitudes de

5

6 Resumen y estructura del trabajo

Renyi de los polinomios de Jacobi que hace uso del desarrollo de una poten-cia entera de un polinomio de Jacobi en serie de potencias, cuyos coeficientesestan controlados por polinomios multivariados de Bell. En la Seccion 5 seutiliza el metodo variacional para acotar superiormente la entropıa de Shan-non de densidades de probabilidades generales con intervalo soporte [−1,+1]en terminos de varios valores esperados de tipos potencia y logarıtmico. En laSeccion 6, se analiza la asintotica de la longitud teorico-informacional de lospolinomios de Jacobi y se obtienen varias cotas superiores variacionales de es-ta magnitud. En la Seccion 7 se estudian numericamente las cotas obtenidasen la seccion previa. Ademas, se realiza un estudio exhaustivo de los val-ores maximos de la longitud de Shannon en funcion de n, α y β. Despues,se ponen de manifiesto algunos problemas abiertos y se mencionan algunasconclusiones.

Finalmente, se detallan algunas propiedades generales de los polinomiosortogonales clasicos, se describen los desarrollos de estos polinomios en seriede potencias y de su problema inverso (haciendo hincapie en los polinomios deJacobi), y se muestran dos metodos de determinacion de los valores esperadosde potencias enteras de la variable en los Apendices A.1, A.2, A.3 y A.4,respectivamente.

Artıculo: Information-theoreticlengths of Jacobi polynomials(publicado en J. Physics A:Math. Theor. 43, 305203, 2010)

En este capıtulo se incluye el artıculo “Information-theoretic lengths ofJacobi polynomials”, publicado en Journal of Physics A: Mathematical andTheoretical, volumen 43 (2010), artıculo 305203., cuyos autores son AngelGuerrero, Pablo Sanchez-Moreno y Jesus S. Dehesa. En el se estudian lospolinomios de Jacobi desde un punto de vista teorico-informacional.

Haciendo uso de las numerosas propiedades algebraicas de los polinomiosde Jacobi [2, 5] , se investigan sus propiedades teorico-informacionales. Estaspropiedades permiten un conocimiento mucho mas completo de la distribu-cion de tales polinomios en el intervalo de ortogonalidad que la conocidadesviacion estandar. Se introducen las medidas de longitud informacionalesde Fisher, Renyi y Shannon [8, 6], que comparten un conjunto de propiedadescomunes con la desviacion estandar: tienen las mismas unidades que la vari-able independiente, son invariantes bajo reflexiones (y traslaciones bajo cier-tas circunstancias), y convergen a cero cuando la densidad de probabilidadtiende a una distribucion delta de Dirac.

Tras la introduccion inicial, la estructura del artıculo, seccion a seccion,es la siguiente:

Seccion 2: Se presentan las definiciones de las medidas de informa-cion que aparecen el artıculo: entropıas de Renyi, Tsallis y Shannon,informacion de Fisher, y desviacion estandar. Tambien aparecen lasdefiniciones de las correspondientes longitudes de Renyi, Shannon yFisher.

Seccion 3: Se procede a la obtencion de la desviacion estandar a partirde los momentos ordinarios, o valores esperados, 〈xk〉. El calculo com-

7

8 Artıculo publicado

pleto de estas magnitudes se puede encontrar en el Apendice A.4 deesta memoria. Ademas, se presenta el valor de la informacion de Fisher

F [ρ] :=

⟨[

d

dxln ρ(x)

]⟩

=

∫ 1

−1

[ρ′(x)]2

ρ(x)dx,

y de la longitud asociada.

δx :=1

√

F [ρ]

Seccion 4: Se introducen los momentos entropicos Wq[ρn,α,β] de la den-sidad de Rakhmanov asociada a los polinomos de Jacobi:

Wq[ρn,α,β] =

∫ 1

−1

[ρn,α,β(x)]qdx,

que son evaluados de forma exacta para valores de q tales que 2q ∈ N,usando para ello los polinomios de Bell. A partir de estos momentosentropicos se obtiene la longitud de Renyi, definida como

LRq [ρn,α,β] = {Wq[ρn,α,β]}−

1

q−1 .

Seccion 5: Se estudia la entropıa de Shannon

S[ρn,α,β] = −∫ 1

−1

ρn,α,β(x) ln (ρn,α,β(x)) dx,

cuyo valor para los polinomios de Jacobi no puede se determinado deforma exacta. Por este motivo se estudia el comportamiento asintoticode esta magnitud cuando el gradpo n tiende a infinito. Ademas, me-diante un calculo variacional se obtienen diversas cotas superiores ala entropıa de Shannon. Dichas cotas se expresan en terminos de losvalores esperados 〈ln(x2)〉, 〈ln(1± x)〉, 〈ln(1− x2)〉, 〈x〉 y 〈x2〉.

Seccion 6: Se aplican los resultados de la seccion anterior a la longitudde Shannon, definida como

N [ρn,α,β ] = exp (S[ρn,α,β]) .

Seccion 7: Se abordan y resuelven los tres problemas computacionalessiguientes. Primero, la identificacion de los pares (α, β) para los que lalongitud de Shannon de un polinomio de Jacobi de grado n es maxi-ma. Segundo, la determinacion de la precision numerica de las cotasobtenidas previamente mencionadas. Tercero, la comparacion mutuade las cuatro magnitudes isodimensionales (desviacion estandar, lastres longitudes teorico-informacionales) de los polinomios de Jacobi.

Artıculo publicado 9

Finalmente se senalan algunos problemas abiertos y se dan las conclu-siones

10 Artıculo publicado

IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL

J. Phys. A: Math. Theor. 43 (2010) 305203 (19pp) doi:10.1088/1751-8113/43/30/305203

Information-theoretic lengths of Jacobi polynomials

A Guerrero1,2, P Sanchez-Moreno2,3 and J S Dehesa1,2

1 Departamento de Fısica Atomica, Molecular y Nuclear, Universidad de Granada, Granada,Spain2 Instituto “Carlos I” de Fısica Teorica y Computacional, Universidad de Granada, Granada, Spain3 Departamento de Matematica Aplicada, Universidad de Granada, Granada, Spain

E-mail: agmartinez@ugr.es, pablos@ugr.es and dehesa@ugr.es

Received 15 March 2010, in final form 22 May 2010Published 28 June 2010Online at stacks.iop.org/JPhysA/43/305203

AbstractThe information-theoretic lengths of the Jacobi polynomials P

(α,β)n (x), which

are information-theoretic measures (Renyi, Shannon and Fisher) of theirassociated Rakhmanov probability density, are investigated. They quantifythe spreading of the polynomials along the orthogonality interval [−1, 1] in acomplementary but different way as the root-mean-square or standard deviationbecause, contrary to this measure, they do not refer to any specific point ofthe interval. The explicit expressions of the Fisher length are given. TheRenyi lengths are found by the use of the combinatorial multivariable Bellpolynomials in terms of the polynomial degree n and the parameters (α, β). TheShannon length, which cannot be exactly calculated because of its logarithmicfunctional form, is bounded from below by using sharp upper bounds to generaldensities on [−1, +1] given in terms of various expectation values; moreover, itsasymptotics is also pointed out. Finally, several computational issues relativeto these three quantities are carefully analyzed.

PACS number: 89.70.CfMathematics Subject Classification: 33C45, 94A17, 62B10, 65C60

1. Introduction

The Jacobi polynomials P(α,β)n (x), (α, β > −1), are the real hypergeometric orthogonal

polynomials with respect to the weight function

ωα,β(x) = (1 − x)α(1 + x)β

in the interval [−1, 1]. These polynomials (or some subclasses of them, such as the Chebyshevand Gegenbauer polynomials) play a fundamental role not only in applied mathematics[1–4] and mathematical physics [5–10] but also in numerous scientific and technologicalfields such as e.g. atomic physics [11–14], classical [15] and quantum [16] optics, information

1751-8113/10/305203+19$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA 1

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

theory [17–21], molecular physics [22–24], non-relativistic [6, 25, 26] and relativistic[27–33] quantum physics, many-body spin physics [34, 35], physics of the Calogero–Moserand Ruijsenaars–Schneieder systems [36], polymer physics [37], QCD analysis of the polarizedquark distribution [38], quantum algebras and groups [39–43], quantum walk [44], Rydbergphysics [45, 46] and supersymmetric quantum mechanics [22, 47–49]. Let us highlight thatthe Jacobi polynomials control the angular part of the position wavefunctions of the quantumstates of any spherically symmetric quantum-mechanical potential in arbitrary dimensions; itis well known, for example, that central potentials are able to explain a great deal of naturalphenomena such as periodicity in the periodic table of elements and numerous macroscopic andspectroscopic properties of atomic and molecular systems. Moreover, the Jacobi polynomialsalso control the radial part of the momentum wavefunctions of the hydrogenic systems [50],the explicit expression of the representation functions of certain quantum groups [39–41] andsome coupling coefficients of angular momenta [42].

Although many algebraic properties (orthogonality, three-term recurrence relation, ladderrelations, second-order differential equation, etc) of these polynomials are well known [1–3,5, 6], this is not the case for their spreading measures beyond the root-mean-square or thestandard deviation. These quantities measure the distribution of the Rakhmanov probabilitydensity of the Jacobi polynomials (and so, of the polynomials themselves) defined by [51]

ρn,α,β(x) = [P (α,β)

n (x)]2

ωα,β(x) (1)

along the orthogonality interval [−1, 1] in various complementary ways, where P(α,β)n (x) are

the orthonormal Jacobi polynomials. Physically, this function characterizes the quantum-mechanical density of the stationary states of numerous physical systems [24–27, 47, 52]. Inthis work we investigate the direct spreading measures of the Rakhmanov density (1) withemphasis on those of information-theoretic origin. The direct spreading measures [53] ofa probability density defined in a finite interval share the same properties as the standarddeviation: same units as the variable x, reflection invariance and vanishing when the densitytends to a Dirac delta. To this family of measures belongs not only the standard deviationbut also the information-theoretic lengths of Fisher, Renyi and Shannon types [53, 54], whosedefinition and meaning for the Rakhmanov density (1), together with the basic properties ofthe Jacobi polynomials, will be given in section 2. It is interesting to remark that, contraryto the standard deviation which measures the concentration of the probability cloud around aparticular point (the centroid) of the orthogonality interval, the information-theoretic lengthsdo not refer to any specific point of that interval.

This is a reason why the information-theoretic lengths are physically much moreappropriate uncertainty measures than the standard deviation in quantum systems. Moreover,they have finite values for probability distributions where the standard deviation does not exist[53]. These lengths are much more convenient measures of the probability distribution than(more conventional) quantifiers such as the associated Shannon entropy, Fisher informationand Renyi entropy because, contrarily to these quantities, all the information-theoretic lengthsmentioned above have the same dimensions as the variable x, so, allowing a natural comparisonamong them and with the standard deviation. For explicit physical applications to variousquantum mechanical potentials, see [55, 56].

The structure of the paper is as follows. In section 2, the notion and meaning of thespreading measures of a probability density are briefly described for the sake of clarity andto fix notations. In section 3 we give the explicit values of the moments with respect tothe origin 〈xk〉, k ∈ Z, the standard deviation �x and the Fisher length of the polynomials.Then, in section 4, we compute the Renyi lengths in terms of the polynomial degree n andthe parameters (α, β) by the use of an error-free approach based on the multivariable Bell

2

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

polynomials of Combinatorics [57]. The Shannon length of the Jacobi polynomials cannot becalculated in an exact manner because of its logarithmic-functional form; so, it seems naturalto try to obtain sharp bounds and to study its asymptotics. Keeping in mind this purpose, wefirst obtain in section 5 some upper bounds to the Shannon entropy for general probabilitydensities with support [−1, +1] in terms of various power and logarithmic expectation valuesand then, in section 6, we determine the asymptotics and some upper bounds for the Shannonlength of the Jacobi polynomials. In section 7 three computational issues are tackled: (i) theidentification of the pairs (α, β) for which the Shannon length is maximal for a given degreen, (ii) the determination of the numerical accuracy of the upper bounds previously mentionedand (iii) the mutual comparison of the isodimensional or direct spreading quantities (standarddeviation and information theoretic lengths) of the Jacobi polynomials. Finally, some openproblems are posed and conclusions are given.

2. Basics on spreading measures

In this section we gather the notion and meaning of the spreading measures of a probabilitydensity. For a probability density ρ(x), x ∈ [−1, 1], corresponding to some random variableX, there exist two main classes of measures which quantify the distribution or the spread of Xalong its interval of definition [−1, 1]: the moments with respect to the origin 〈xk〉, k ∈ Z, andthe frequency or entropic moments 〈[ρ(x)]k〉 (see e.g. [58]). Heretoforth, the symbol 〈f (x)〉denotes the expectation value of f (x) with respect to the density ρ(x) as

〈f (x)〉 :=∫ 1

−1f (x)ρ(x) dx. (2)

The complete knowledge of each set of moments determines the probability density ρ(x) undercertain conditions as the ordinary [59] and the entropic [60] Hausdorff moment problems state.

Often, it is more convenient to use instead some related quantities such as the centralmoments (or moments around the centroid 〈x〉) given by the expectation values 〈(x − 〈x〉)k〉,the Renyi entropies [61] defined by

Rq[ρ] := 1

1 − qln〈[ρ(x)]q−1〉; q > 0, q �= 1, (3)

and the Tsallis entropies [62] given by

Tq[ρ] := 1

q − 1[1 − 〈[ρ(x)]q−1〉]; q > 0, q �= 1,

together with their q → 1 limit, the Shannon entropy

S[ρ] := −∫ 1

−1ρ(x) ln(ρ(x)) dx. (4)

These quantities, however, are not direct measures of spreading [53] of ρn(x) in the sensethat they do not have the same units as x. The most well-known direct measure of spreadingare the root-mean-square or standard deviation

�x :=√

〈(x − 〈x〉)2〉 =√

〈x2〉 − 〈x〉2 (5)

and, in a less sense, the Onicescu–Heller length [63, 64] (also called inverse participation ratio[63, 64], disequilibrium [65], linear entropy [66] and Brukner–Zeilinger entropy [67] in othercontexts)

L[ρ] := 1

〈ρ(x)〉 .

3

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

Recently, other direct measures of spreading related to the entropic moments have beenintroduced: the Renyi and Shannon lengths [53] defined by

LRq [ρ] := exp(Rq[ρ]), (6)

and

N [ρ] := limq→1

LRq [ρ] = exp(S[ρ]), (7)

respectively. Remark that LR2 [ρ] = L[ρ]. It is worth noting that

• the standard deviation and the Renyi and Shannon lengths are global spreading measuresbecause they are powerlike (standard deviation and Renyi lengths) or logarithmic(Shannon length) functionals of the density,

• the standard deviation is a measure of separation of the probability cloud from a particularpoint of the support interval of the density (namely the mean value or centroid) and

• the Renyi and Shannon lengths are measures of the extent to which the density is in factconcentrated.

In addition, there is another direct measure of spreading which is qualitatively differentfrom the previous one. It is the Fisher length [68], defined by

δx := 1√F [ρ]

, (8)

where F [ρ(x)] denotes the Fisher information of the density ρ(x) given by

F [ρ] :=⟨[

d

dxln ρ(x)

]⟩=

∫ 1

−1

[ρ ′(x)]2

ρ(x)dx. (9)

The Fisher length, then, is a functional of the derivative of the density; so, it is very sensitive tothe fluctuations of the density in contrast to the global direct measures of spreading mentionedpreviously. This local quantity measures the pointwise probability concentration of the densityalong its support interval.

The four direct spreading measures (standard deviation and the Renyi, Shannon and Fisherlengths) defined in a finite interval, besides the properties mentioned in the previous section,enjoy an uncertainty property; see [53] for the standard deviation, Renyi and Shannon cases,and [69, 70] for the Fisher case.

3. Moments, standard deviation and Fisher length

In this section we show the values of the ordinary moments 〈xk〉n,α,β , the standard deviation(�x)n,α,β and the Fisher length (δx)n,α,β of the Rakhmanov probability density ρn,α,β(x) ofthe Jacobi polynomials given by equation (1). The Jacobi polynomials P

(α,β)n (x) are known

to satisfy the orthonormality relation∫ 1

−1P (α,β)

n (x)P (α,β)m (x)ωα,β(x) dx = δnm, (10)

for α, β > −1, so that P(α,β)n (x) = dn,α,β P

(α,β)n (x) gives the relation between the orthonormal

and orthogonal polynomials, where the constant

d2n,α,β = 2α+β+1�(n + α + 1)�(n + β + 1)

n!(2n + α + β + 1)�(n + α + β + 1).

4

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

The moments around the origin 〈xk〉n,α,β of the Jacobi polynomials, according to equation (2),are defined by

〈xk〉n,α,β :=∫ +1

−1xkρn,α,β(x) dx =

∫ +1

−1xk

[P (α,β)

n (x)]2

ωα,β(x) dx.

The use of the expansion of xkP(α,β)n (x) in series of orthogonal polynomials [71] together with

the orthogonality relation (10) has allowed us to obtain that

〈xk〉n,α,β =n∑

m=0

a(α,β)m,n

n∑l=n−k

(m

l

)(−1)m−lb

(α,β)

k+l,n, (11)

where the coefficients a(α,β)m,n and b

(α,β)

k+l,n are given by

a(α,β)m,n =

(n

m

)�(α + n + 1)�(α + β + n + m + 1)

n!2m�(α + β + n + 1)�(α + m + 1), (12)

and

b(α,β)

k+l,n = (k + l)!(−1)k+l−n2n�(n + α + β + 1)

(k + l − n)!�(2n + α + β + 1)2F1

(n − k − l, n + β + 1

2n + α + β + 2; 2

), (13)

respectively. For k = 1 and k = 2, equations (11)–(13) provide the following values

〈x〉n,α,β = β2 − α2

(2n + α + β)(2n + α + β + 2), (14)

and

〈x2〉n,α,β = 4(n + 1)(n + α + 1)(n + β + 1)(n + α + β + 1)

(2n + α + β + 1)(2n + α + β + 2)2(2n + α + β + 3)

+4n(n + α)(n + β)(n + α + β)

(2n + α + β − 1)(2n + α + β)2(2n + α + β + 1)+ 〈x〉2, (15)

for the first and second moments. Then, the standard deviation of the Jacobi polynomials is,according to equations (1) and (5), the expression

(�x)n,α,β := (〈x2〉n,α,β − 〈x〉2n,α,β

) 12 ,

which together with equations (14) and (15) provides the value

(�x)n,α,β =[

4(n + 1)(n + α + 1)(n + β + 1)(n + α + β + 1)

(2n + α + β + 1)(2n + α + β + 2)2(2n + α + β + 3)

+4n(n + α)(n + β)(n + α + β)

(2n + α + β − 1)(2n + α + β)2(2n + α + β + 1)

]1/2

,

in agreement with the expression recently obtained by other means [72]. Remark that

(�x)n,α,β ≈ 1√2, n → +∞. (16)

Moreover, the Fisher length can be obtained by the use of equations (1), (8) and (9) as

(δx)n,α,β = 1√F [ρn,α,β ]

,

where the Fisher information F [ρn,α,β ] has the expression [73]

F [ρn,α,β ] =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

2n+α+β+14(n+α+β−1)

[n(n + α + β − 1)

(n+αβ+1 + 2 + n+β

α+1

)+ (n + 1)(n + α + β)

(n+αβ−1 + 2 + n+β

α−1

)], α, β > 1,

2n+β+14

[n2

β+1 + n + (4n + 1)(n + β + 1) + (n+1)2

β−1

], α = 0, β > 1,

2n(n + 1)(2n + 1), α, β = 0,

∞, otherwise

5

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

for the Jacobi polynomials P(α,β)n (x) with α, β > −1 and n = 0, 1, 2, . . . ; keep in

mind, additionally, that F [ρn,α,β(x)] = F [ρn,β,α(x)] due to the (α, β)-symmetry of Jacobipolynomials. Let us highlight, in particular, that the Cramer–Rao inequality δx � �x,valid for continuous densities on the whole real line, is satisfied by the Rakhmanov densitiesρn,α,β(x), with support on [−1, +1], when α, β > 1. This is because these densities have zerosat x = ±1 and they can be considered as continuous densities on the (−∞, +∞) interval.

4. Renyi’s lengths

In this section we compute the Renyi lengths LRq [ρn,α,β(x)] of the Jacobi polynomials by

means of an algebraic methodology which uses the explicit expression of the polynomials andthe multivariable Bell polynomials of Combinatorics [57]. According to equations (1), (3)and (6), these quantities are given by

LRq [ρn,α,β ] = {Wq[ρn,α,β ]}− 1

q−1 ; q > 0, q �= 1,

where

Wq[ρn,α,β ] := 〈[ρn,α,β(x)]q−1〉 =∫ +1

−1[ρn,α,β(x)]q dx

=∫ +1

−1

[P (α,β)

n (x)]2q

[ωαβ(x)]q dx (17)

are the frequency or entropic moments of the Rakhmanov density (1) of the Jacobi polynomials.To calculate this functional we begin with the explicit expression of the Jacobi polynomials,which can be formulated as the following power series:

P (α,β)n (x) =

n∑k=0

ckxk, (18)

where the expansion coefficients have the values

ck =√

�(α + n + 1)(2n + α + β + 1)

n!2(α + β + 1)�(α + β + n + 1)�(n + β + 1

×n∑

i=k

(−1)i−k

(n

i

)(i

k

)�(α + β + n + i + 1)

2i�(α + i + 1). (19)

Recently it has been found (see the appendix of [54]) that a finite power of a polynomial canbe expressed by means of the multivariable Bell polynomials of Combinatorics [57]. Thisresult applied to Jacobi polynomials (18) gives[P (α,β)

n (x)]p =

[n∑

k=0

ckxk

]p

=np∑

k=0

p!

(k + p)!Bk+p,p(c0, 2!c1, . . . , (k + 1)!ck)x

k, (20)

with ci = 0 for i > n, and the remaining coefficients are given by equation (19). Moreoverthe Bell polynomials are given by

Bm,l(c1, . . . , cm−l+1) =∑

π(m,l)

m!

j1!j2! · · · jm−l+1!

(c1

1!

)j1(c2

2!

)j2 · · ·(

cm−l+1

(m − l + 1)!

)jm−l+1

,

where the sum runs over all partitions π(m, l) such that

j1 + j2 + · · · + jm−l+1 = l, and j1 + 2j2 + · · · + (m − l + 1)jm−l+1 = m.

6

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

The substitution of expression (20) with p = 2q into equation (17) yields the value

Wq[ρn,α,β ] =2nq∑k=0

(2q)!

(k + 2q)!Bk+2q,2q(c0, 2!c1, . . . , (k + 1)!ck)I(k, q, α, β), (21)

where

I(k, q, α, β) =∫ +1

−1xk(1 − x)αq(1 + x)βqdx

= (−1)k21+αq+βq�(αq + 1)�(βq + 1)

�(αq + βq + 2)2F1

(−k, 1 + βq

2 + (α + β)q; 2

).

For the sake of checking we first note that for q = 1

W1[ρn,α,β ] =∫ +1

−1ρn,α,β(x) dx = 1,

and for q = 2,

W2[ρn,α,β ] =∫ +1

−1[ρn,α,β(x)]2 dx

=4n∑

k=0

4!

(k + 4)!Bk+4,4(c0, 2!c1, . . . , (k + 1)!ck)I(k, 2, α, β),

which immediately produces the Onicescu–Heller length L[ρn,α,β ] of the Jacobi polynomials,

since L[ρn,α,β ] = 〈ρn,α,β(x)〉−1 = {W2[ρn,α,β ]

}−1. The Renyi lengths of these polynomials

can be obtained from (6) and (21), obtaining the expression

LRq [ρn,α,β ] = {Wq[ρn,α,β ]}− 1

q−1

=(

2nq∑k=0

(2q)!

(k + 2q)!Bk+2q,2q(c0, 2!c1, . . . , (k + 1)!ck)I(k, q, α, β)

)− 1q−1

,

with q = 2, 3, . . .. For the two Jacobi polynomials with lowest degrees one has, then, thevalues

LRq [ρ0,α,β ] =

[(�(α + β + 2)

2α+β+1�(α + 1)�(β + 1)

)2 21+αq+βq�(αq + 1)�(βq + 1)

�(αq + βq + 2)

]− 1q−1

,

LRq [ρ1,α,β ] =

[2q∑

k=0

(2q

n

)c

2q−k

0 ck1I(k, q, α, β)

]− 1q−1

,

for the Renyi lengths, respectively, and the values

L[ρ0,α,β ] =[(

�(α + β + 2)

2α+β+1�(α + 1)�(β + 1)

)q 21+2α+2β�(2α + 1)�(2β + 1)

�(2α + 2β + 2)

]−1

,

L[ρ1,α,β ] =[

4∑k=0

(4

n

)c4−k

0 ck1I(k, 2, α, β)

]−1

,

for the Onicescu–Heller length of the polynomials P(α,β)n (x) with n = 0, 1 respectively.

7

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

5. Upper bounds to the Shannon entropy for general densities on [−1, +1]

Before starting the study of the Shannon length of the Jacobi polynomials, in the next sectionwe find here several general upper bounds to the Shannon entropy S[ρ] and the Shannon lengthN [ρ] for any density ρ(x) in the [−1, +1] interval, with different constraints in the form ofexpectation values.

First, we find the well-known maximum value of this information measure for anynormalized density by considering the following Lagrangian:

L = −∫ 1

−1ρ(x) ln ρ(x) dx − λ

(∫ 1

−1ρ(x) dx − 1

).

The associated Euler–Lagrange equation

d

dx

∂L∂ρ ′ − ∂L

∂ρ= 0,

yields

ln ρmax(x) + 1 + λ = 0.

So ρmax(x) is a constant distribution that, once normalized to unity and obtained the value ofλ, becomes ρmax(x) = 1/2. The maximum Shannon entropy is S[ρmax] = ln 2, so we havethe well-known general upper bound

S[ρ] � ln 2,

and taking into account equation (7), we obtain

N [ρ] � 2. (22)

Now let us find the variational bounds to the Shannon entropy of a general probabilitydensity subject not only to the normalization to unity but also to the known expectation value〈f (x)〉 of an arbitrary (but given) function f (x). Working similarly as before, we start withthe Lagrangian associated with this problem:

L = −∫ 1

−1ρ(x) ln ρ(x) dx − λ1

(∫ 1

−1ρ(x) dx − 1

)− λ2

(∫ 1

−1f (x)ρ(x) dx − 〈f 〉

).

Then the corresponding Euler–Lagrange equation yields

ρmax(x) = e−1−λ1 e−λ2f (x).

The Lagrange multipliers λ1 and λ2 are obtained by imposing the two conditions∫ 1

−1ρmax(x) dx = 1, (23)

and ∫ 1

−1f (x)ρmax(x) dx = 〈f 〉. (24)

The analytical determination of the upper bound S [ρmax] to the Shannon entropy S[ρ]will depend on the explicit form of f (x). In the following we give the upper bounds associatedwith different choices of f (x).

• Bound in terms of the expectation value 〈ln(x2)〉. In this case we have

ρmax(x) = e−1−λ1 e−λ2 ln(x2) = e−1−λ1 |x|−2λ2 .

8

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

Then, condition (23) yields∫ 1

−1ρmax(x) dx = e−1−λ1

∫ 1

−1|x|−2λ2 dx = e−1−λ1

2

1 − 2λ2= 1, if λ2 <

1

2.

so, λ1 can be expressed in terms of λ2 and ρmax becomes

ρmax(x) = 1 − 2λ2

2|x|−2λ2 .

The second condition (24) yields∫ 1

−1ln(x2)ρmax(x) dx = 1 − 2λ2

2

∫ 1

−1ln(x2)|x|−2λ2 dx

= 2

2λ2 − 1= 〈ln(x2)〉, if λ2 <

1

2.

Thus, λ2 is expressed in terms of 〈ln(x2)〉 as λ2 = 1/〈ln(x2)〉 + 1/2, so the conditionλ2 < 1/2 is satisfied; and ρmax is finally written as

ρmax(x) = − 1

〈ln(x2)〉 |x|− 2〈ln(x2)〉 −1

.

The corresponding value of the Shannon entropy for this density is

S[ρmax] = −∫ 1

−1ρmax(x) ln ρmax(x)dx = 1 +

〈ln(x2)〉2

+ ln(−〈ln(x2)〉).

Then, the variational upper bound to the Shannon entropy, with the constraint 〈ln(x2)〉, isgiven by

S[ρ] � 1 +〈ln(x2)〉

2+ ln(−〈ln(x2)〉). (25)

• Bound in terms of the expectation value 〈ln(1 − x)〉. Working similarly, we have that

ρmax(x) = 2−1

ln 2−〈ln(1−x)〉

ln 2 − 〈ln(1 − x)〉 (1 − x)1

ln 2−〈ln(1−x)〉 −1,

and the corresponding upper bound to the Shannon entropy of any distribution withsupport [−1, +1] and constraint 〈ln(1 − x)〉 is given by

S[ρ] � 1 + 〈ln(1 − x)〉 + ln(ln 2 − 〈ln(1 − x)〉). (26)

• Bound in terms of the expectation value 〈ln(1 + x)〉. In this case we obtain that

ρmax(x) = 2−1

ln 2−〈ln(1+x)〉

ln 2 − 〈ln(1 + x)〉 (1 + x)1

ln 2−〈ln(1+x)〉 −1,

and the upper bound

S[ρ] � 1 + 〈ln(1 + x)〉 + ln(ln 2 − 〈ln(1 + x)〉). (27)

• Bound in terms of the expectation value 〈ln(1 − x2)〉. In this case the expression of〈ln(1 − x2)〉 in terms of the Lagrange multiplier λ2 is not invertible, so that we obtain

ρmax(x) = �(

32 − λ2

)√

π�(1 − λ2)(1 − x2)−λ2 ,

and the upper bound

S[ρ] � − ln�

(32 − λ2

)√

π�(1 − λ2)+ λ2

(ψ(1 − λ2) − ψ

(3

2− λ2

)), (28)

9

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

where ψ(x) is the digamma function, with the implicit relation

ψ(1 − λ2) − ψ

(3

2− λ2

)= 〈ln(1 − x2)〉 (29)

that must be solved numerically for every value of 〈ln(1 − x2)〉.• Bound in terms of the expectation value 〈x〉. As in the previous case, we have

ρmax(x) = λ2

2 sinh(λ2)e−λ2x,

and the upper bound

S[ρ] � − lnλ2

2 sinh(λ2)− λ2 coth(λ2) − 1, (30)

with the implicit relation

1

λ2− coth(λ2) = 〈x〉. (31)

• Bound in terms of the expectation value 〈x2〉. As in the two previous cases, we have

ρmax(x) =√

λ2

π

1

erf(√

λ2)e−λ2x

2,

and the upper bound

S[ρ] � 1

2− ln

(√λ2

π

1

erf(√

λ2)

)−

√λ2

π

e−λ2

erf(√

λ2), (32)

where erf(x) is the error function, with the implicit relation

1

λ2− e−λ2

√λ2πerf(

√λ2)

= 〈x2〉. (33)

Finally, let us point out that the combination of equation (7) with each of the bounds(25)–(30) and (32) provides the corresponding upper bounds to the Shannon length of anyprobability density with the support [−1, +1].

6. The Shannon length

The purpose of this section is twofold. First, to fix the asymptotics of the Shannon lengthN [ρn,α,β ] of the Jacobi polynomials P

(α,β)n and its relation to the standard deviation. Second,

to determine sharp upper bounds to N [ρn,α,β ] by using the results of the previous section.With respect to the first purpose let us start writing that, according to equations (4) and (7),

N [ρn,α,β(x)] = exp{S[ρn,α,β(x)]}, (34)

with

S[ρn,α,β ] = −∫ +1

−1ωαβ(x)

[P (α,β)

n (x)]2

ln{ωαβ(x)

[P (α,β)

n (x)]2}

dx

= E[P (α,β)

n

]+ I

[P (α,β)

n

],

with the entropic functionals [70, 74]

E[P (α,β)

n

] = −∫ +1

−1ωα,β(x)

[P (α,β)

n (x)]2

ln[P (α,β)

n (x)]2

dx

= ln π − 1 − (α + β) ln 2 + o(1), (35)

10

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

and

I[P (α,β)

n

] = −∫ +1

−1ωα,β(x)

[P (α,β)

n (x)]2

ln ωα,β(x)

= −αψ(n + α + 1) − βψ(n + β + 1) + (α + β)

×[− ln 2 +

1

2n + α + β + 1+ 2ψ(2n + α + β + 1) − ψ(n + α + β + 1)

].

The logarithmic functional E[P

(α,β)n

]cannot be exactly calculated, but its asymptotics has

been fully determined by means of the Lp-method of Aptekarev et al [75]. Taking into accountthis result (given by equation (35)) together with the fact that limn→∞ I

[P

(α,β)n

] = (α +β) ln 2one has that

S[ρn,α,β ] = ln π − 1 + o(1); n → +∞,

for the Shannon entropy, and with equation (34)

N [ρn,α,β ] ≈ π

e; n → ∞, (36)

for the Shannon length of the Jacobi polynomials. Moreover, from equations (16) and (36)we find the asymptotical relation

N [ρn,α,β ] ≈ π√

2

e(�x)n,α,β ≈ 1.6389 (�x)n,α,β; n → ∞ (37)

between the Shannon length and the standard deviation of the Jacobi polynomials. It isinteresting to highlight that this asymptotical linear correlation between these two quantitieshas exactly the same form as for Hermite [54, 76] and Laguerre polynomials [76, 77].

In order to obtain upper bounds for the Shannon length, we can use the variational boundsfound in the previous section for the Shannon entropy. Then, from equation (22) we wouldobtain the general upper bound

N [ρn,α,β ] � 2.

Imposing now the different constraints based on the expectation values considered before, weobtain the following upper bounds in a straightforward manner:

• Bound with the constraint 〈ln(x2)〉:

N [ρn,α,β ] � −〈ln(x2)〉 exp

(1 +

〈ln(x2)〉2

). (38)

• Bound with the constraint 〈ln(1 ± x)〉:N [ρn,α,β ] � (ln 2 − 〈ln(1 ± x)〉) exp(1 + 〈ln(1 ± x)〉). (39)

• Bound with the constraint 〈ln(1 − x2)〉:

N [ρn,α,β ] �√

π�(1 − λ2)

�(

32 − λ2

) exp

[λ2

(ψ(1 − λ2) − ψ

(3

2− λ2

))], (40)

where λ2 is related to 〈ln(1 − x2)〉 through equation (29).• Bound with the constraint 〈x〉:

N [ρn,α,β ] � 2 sinh(λ2)

λ2exp(−1 − λ2 coth(λ2)), (41)

where λ2 is given in terms of 〈x〉 by equation (31), and 〈x〉 is given explicitly byequation (14) in terms of (n, α, β).

11

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

0

2

4

6

8

10

0 2 4 6 8 10

β

α

n=

0

n=

1n

=2

n=

3n

=4

n=

5n

=6

n=

7n

=8

n=

9n

=10

Figure 1. Numerical study of the maximal Shannon length of the Jacobi polynomials P(α,β)n (x).

Regions in the plane (α, β) where the Jacobi polynomials P(α,β)n (x) have maximal Shannon lengths.

Note that the degree n varies from a region to the following.

• Bound with the constraint 〈x2〉:

N [ρn,α,β ] �√

π

λ2erf(

√λ2) exp

(1

2−

√λ2

π

e−λ2

erf(√

λ2)

), (42)

where λ2 is given in term of 〈x2〉 by equation (33), and 〈x2〉 is given by equation (15) interms of (n, α, β).

The accuracy of these bounds is numerically studied in the next section.

7. Some computational issues

In this numerical section, we first investigate the regions in the plane (α, β) where the Shannonspreading length of the Jacobi polynomials is maximal for various degrees n. Then, we analyzethe accuracy of the upper bounds to the Shannon length obtained in the previous section andfinally we study the mutual comparison of the four direct spreading lengths considered in thispaper, namely the standard deviation and the Fisher, Renyi and Shannon lengths.

In figure 1, we show the regions where the maximal Shannon length of the Jacobipolynomial P

(α,β)n (x) in the plane (α, β) is obtained for various degrees n going from 0 to 10.

Only half of the plane is shown because the Shannon length is invariant under the exchangeof α and β. We observe a rich structure of successive black and gray regions, which is anopen analytical problem. All points lying down in a certain region correspond to obtainingthe maximum value of the Shannon length for the same degree n. Note the region for n = 0,indicating that the maximum Shannon length is N [ρ0,α,β ]. This region is surrounded by otherregions where the maximum is sequentially achieved for polynomials with increasing degree,as is shown in the figure.

In figures 2–4 we study numerically the variational bounds (38)–(42) to the Shannonlength of the Jacobi polynomial P

(α,β)n (x) obtained in the previous section with respect to

its exact value N [ρn,α,β ] for n going from 0 to 20 and the parameters (α, β) are equal to

12

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 5 10 15 20

N[ρ

n,0

,0]an

dupper

bou

nds

n

Figure 2. Numerical study of the Shannon length N [ρn,α,β ] (�) and its upper bounds in terms ofthe expectation values 〈ln(x2)〉 (×), 〈ln(1 − x)〉 (�), 〈ln(1 + x)〉 (�), 〈ln(1 − x2)〉 (•), 〈x〉 (◦), 〈x2〉(+), as a function of n for (α, β) = (0, 0).

(0, 0),(− 1

3 ,− 13

)and (2, 5), respectively. The bounds are controlled by the expectation values

〈ln(x2)〉, 〈ln(1 − x)〉, 〈ln(1 + x)〉 , 〈ln(1 − x2)〉, 〈x〉, 〈x2〉, respectively. Several observationscan be done. First, all the bounds lie above, or equal in the case n = α = β = 0, to theexact value; moreover all of them improve the trivial bound 2, except the one controlled by〈x〉 in the cases when α = β, where the density is even and 〈x〉 = 0. Second, expression(40) controlled by the expectation value 〈ln(1 − x2)〉 provides the best upper bound in bothfigures 2 and 3, except for n = 1, but that is not true in figure 4. Third, there is still plenty ofspace for the improvement of the upper bounds in the three figures.

In figures 5 and 6 we make a similar study of the previous bounds to the Shannon lengthof the Jacobi polynomial P

(α,β)n (x) as a function of the parameter α when the pair of values

(n, β) is equal to (0, 2) and (1, 0), respectively. It is found that, at times, some upperbounds are equal or very close to the exact value of the Shannon length; this occurs, e.g.,for the bounds depending on the expectation values 〈ln(1 − x)〉 and 〈ln(1 − x2)〉 in figure 5,and for the bound depending on 〈ln(x2)〉 in figure 6. Nevertheless, none of the expectationvalues chosen as constraint supply a sufficiently good upper bound for the whole intervalconsidered.

Finally, in figure 7 we compare the four direct spreading measures of the Jacobi polynomialP

(α,β)n (x) among themselves for (α, β) = (2, 2) and n going from 0 to 80: the Shannon length

N [ρn,α,β ], the Renyi lengths Rq[ρn,α,β ] with q = 2, 5 and 10, the Fisher length (δx)n,α,β

and the standard deviation (�x)n,α,β . This is possible because all of them have the samedimensions. It is observed that (i) the Fisher length is smaller than the remaining spreadingmeasures, mainly because of its locality property, and it tends toward zero for large values ofthe polynomial degree n, (ii) the Shannon length is bigger than the rest of spreading measuresand, together with the standard deviation, tends asymptotically (i.e. for large n) to constantvalues, which are mutually related by equation (37) and (iii) the three Renyi lengths consideredin the figure, contrary to the two previous global quantities, asymptotically vanish because thepower functionals of the densities decrease along the support interval when the polynomialdegree is increasing.

13

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20

N[ρ

n,−

1 3,−

1 3]an

dupper

bou

nds

n

Figure 3. Numerical study of the Shannon length N [ρn,α,β ] (�) and its upper bounds in terms ofthe expectation values 〈ln(x2)〉 (×), 〈ln(1 − x)〉 (�), 〈ln(1 + x)〉 (�), 〈ln(1 − x2)〉 (•), 〈x〉 (◦), 〈x2〉(+), as a function of n for (α, β) = (−1/3, −1/3).

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 5 10 15 20

N[ρ

n,2

,5]an

dupper

bou

nds

n

Figure 4. Numerical study of the Shannon length N [ρn,α,β ] (�) and its upper bounds in terms ofthe expectation values 〈ln(x2)〉 (×), 〈ln(1 − x)〉 (�), 〈ln(1 + x)〉 (�), 〈ln(1 − x2)〉 (•), 〈x〉 (◦), 〈x2〉(+), as a function of n for (α, β) = (2, 5).

8. Open problems and conclusions

In this work we first give the moments-around-the-origin 〈xk〉, the standard deviation �x andthe Fisher length δx of the Jacobi polynomials P

(α,β)n (x) in terms of the degree n and the

parameters (α, β). Second, we calculate the Renyi lengths of these mathematical objects bymeans of the Bell polynomials, so much used in Combinatorics.

14

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1 0 1 2 3 4 5

N[ρ

0,α,2]an

dupper

bou

nds

α

Figure 5. Numerical study of the Shannon length N [ρn,α,β ] (�) and its upper bounds in termsof the expectation values 〈ln(x2)〉 (long dashed line), 〈ln(1 − x)〉 (dash-dotted line), 〈ln(1 + x)〉(dash-dot-dotted line), 〈ln(1−x2)〉 (short dashed line), 〈x〉 (dotted line), 〈x2〉 (dash-dot-dot-dottedline), as a function of α for (n, β) = (0, 2).

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-1 0 1 2 3 4 5

N[ ρ

1,α,0]an

dupper

bou

nds

α

Figure 6. Numerical study of the Shannon length N [ρn,α,β ] (�) and its upper bounds in termsof the expectation values 〈ln(x2)〉 (long dashed line), 〈ln(1 − x)〉 (dash-dotted line), 〈ln(1 + x)〉(dash-dot-dotted line), 〈ln(1−x2)〉 (short dashed line), 〈x〉 (dotted line), 〈x2〉 (dash-dot-dot-dottedline), as a function of α for (n, β) = (1, 0).

Then, since the analytical determination of the Shannon length N is a formidable task, wecalculate upper bounds of variational type to the Shannon entropy S[ρ] of general densitiesρ(x), x ∈ [−1, +1] . These results are subsequently used to yield upper bounds to the Shannonlength of the Jacobi polynomials, N [ρn,α,β ], via some power and logarithmic expectation valueswhich can be, at times, analytically calculated. Moreover, the accuracy of these bounds isnumerically computed and discussed for various polynomial degrees and pairs of parameters(α, β).

15

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

0.01

0.1

1

1 2 5 10 20 50

Spre

adin

gle

ngt

hs

n

Figure 7. N [ρn,α,β ] (solid line), R2[ρn,α,β ] (long dashed line), R5[ρn,α,β ] (dash-dotted line),R10[ρn,α,β ] (dash-dotted line), (�x)n,α,β (short dashed line) and (δx)n,α,β (dotted line), as afunction of n, for (α, β) = (2, 2).

Finally, the mutual comparison of the global (standard deviation, Renyi andShannon lengths) and local (Fisher length) spreading measures of a Jacobi polynomial iscomputationally performed since all of them have the same dimensions as the variable x. Theygrasp various aspects of the polynomial, quantifying its distribution all over the support indifferent ways.

There are a number of open information-theoretic issues which deserve to be consideredfor a deeper understanding of the spread of Jacobi polynomials along its support interval.Beyond the exact calculation of the Shannon entropy of the Jacobi polynomials (which is aformidable open task because of its logarithmic functional form), it is interesting from boththeoretical and applied points of view to find the asymptotics of the Renyi lengths LR

q [ρn,α,β ]and the entropic moments Wq[ρn,α,β ] in the following three following cases: (i) n → +∞ and(q, α, β) fixed, (ii) q → +∞ and (n, α, β) fixed and (iii) α and β large but (n, q) fixed. Afurther open problem is the determination of the analytical boundaries of the regions of themaximal Shannon length in the plane (α, β).

Acknowledgments

We are very grateful to Junta de Andalucıa for the grants FQM-2445 and FQM-4643, and theMinisterio de Ciencia e Innovacion for the grant FIS2008-02380. We belong to the researchgroup FQM-207.

References

[1] Chihara Th S 1978 An Introduction to Orthogonal Polynomials (New York: Gordon and Breach)[2] Andrews G E, Askey R and Roy R 1999 Special functions Encyclopedia for Mathematics and Its Applications

(Cambridge: Cambridge University Press)

16

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

[3] Ismail M E H 2005 Classical and quantum orthogonal polynomials in one variable Encyclopedia for Mathematicsand Its Applications (Cambridge: Cambridge University Press)

[4] Brushi M, Calogero F and Droghei R 2007 Tridiagonal matrices, orthogonal polynomials and diophantinerelations: I J. Phys. A: Math. Theor. 40 9793

[5] Temme N M 1996 Special Functions: An Introduction to the Classical Functions of Mathematical Physics(New York: Wiley Interscience)

[6] Nikiforov A F and Uvarov V B 1988 Special Functions in Mathematical Physics (Basel: Birkauser)[7] Basor E, Chen Y and Erhardt T 2010 Painleve V and time-dependent Jacobi polynomials J. Phys. A: Math.

Theor. 43 015204[8] Chen Y and Griffin J 2002 Krall-type polynomials via the Heine formula J. Phys. A: Math. Gen. 35 637[9] Doha E H 2002 On the coefficients of differentiated expansions and derivatives of Jacobi polynomials J. Phys.

A: Math. Gen. 35 3467[10] Doha E H 2004 On the construction of recurrence relations for the expansion and connection coefficients in

series of Jacobi polynomials J. Phys. A: Math. Gen. 37 657[11] Chuluunbaatar O, Puzynin I V and Vinitsky S I 2001 Uncoupled correlated calculations of helium isoelectronic

bound states J. Phys. B: At. Mol. Opt. Phys. 34 L425[12] Sorevik T, Madsen L B and Hansen J P 2005 A spectral method for integration of the time-dependent Schrodinger

equation in hyperspherical coordinates J. Phys. A: Math. Gen. 38 6977[13] Dodonov V V, Man’ko O V, Man’ko V I, Polynkin P G and Rosa L 1995 Delta-kicked Landau levels J. Phys.

A: Math. Gen. 28 197[14] Izen S H 1988 A series inversion for the x-ray transform in n dimensions Inverse Problems 4 725[15] Scheel S and Audenaert K M R 2005 Scaling of success probabilities for linear optics gates New J. Phys. 7 149[16] Xiang-Guo M, Ji-Suo W and Bao-Long L 2008 Two-mode excited squeezed vacuum state and its properties

Commun. Theor. Phys. 49 1299[17] Yanez R J, Sanchez-Moreno P, Zarzo A and Dehesa J S 2008 Fisher information of special functions and

second-order differential equations J. Math. Phys. 49 082104[18] Dehesa J S, Lopez-Rosa S, Olmos B and Yanez R J 2006 Fisher information of D-dimensional hydrogenic

systems in position and momentum spaces J. Math. Phys. 47 052104[19] Dehesa J S, Lopez-Rosa S and Yanez R J 2007 Information-theoretic measures of hyperspherical harmonics

J. Math. Phys. 48 043503[20] Dehesa J S, Martınez-Finkelshtein A and Sanchez-Ruiz J 2001 Quantum information entropies and orthogonal

polynomials J. Comput. Appl. Math. 133 23–46[21] Bhattacharya A, Talukdar B, Roy U and Ghosh A 1998 Quantal information entropies for atoms Int. J. Theor.

Phys. 37 1667[22] Compean C B and Kirschbach M 2006 The trigonometric Rosen–Morse potential in the supersymmetric quantum

mechanics and its exact solutions J. Phys. A: Math. Gen. 39 547[23] Carvajal M 2000 Analytical methods to treat anharmonic potentials in molecular physics PhD Thesis Universidad

de Sevilla, Spain[24] Atre R, Kumar A, Kumar N and Panigrahi P K 2004 Quantum-information entropies of the eigenstates and the

coherent state of the Poschl–Teller potential Phys. Rev. A 69 052107[25] Flugge S 1971 Practical Quantum Mechanics (Berlin: Springer)[26] de Lange O L and Raab R E 1991 Operator Methods in Quantum Mechanics (Oxford: Clarendon)[27] Bagrov V G and Gitman D M 1990 Exact Solutions of Relativistic Wavefunctions (Dordrecht: Kluwer Academic)[28] Aldaya V, Bisquert J and Navarro-Salas J 1991 The quantum relativistic harmonic oscillator: generalized

Hermite polynomials Phys. Lett. A 156 381[29] Vignat C 2009 Old and new results about relativistic Hermite polynomials arXiv:0911.5513v1 [math-ph][30] Ismail M E H 1996 Relativistic orthogonal polynomials are Jacobi polynomials J. Phys. A: Math. Gen. 29 3199[31] Berkdemir C and Chen Y F 2009 On the exact solutions of the Dirac equation with a novel angle-dependent

potential Phys. Scr. 79 035003[32] Alhaidari A D 2004 Solution of the relativistic Dirac–Hulthen problem J. Phys. A: Math. Gen. 37 5805[33] Midya B and Roy B 2009 A generalized quantum nonlinear oscillator J. Phys. A: Math. Theor. 42 285301[34] Enciso A, Finkel F, Gonzalez-Lopez A and Rodrıguez M A 2007 Exchange operator formalism for N -body

spin models with near-neighbours interactions J. Phys. A: Math. Theor. 40 1857[35] Winternitz P and Yurdusen I 2009 Integrable and superintegrable systems with spin in three-dimensional

Euclidean space J. Phys. A: Math. Theor. 42 385203[36] Odake S and Sasaki R 2004 Resonance and web structure in discrete soliton systems: the two-dimensional Toda

lattice and its fully discrete and ultra-discrete analogues J. Phys. A: Math. Gen. 37 11841[37] Dongen P G J van and Ernst M H 1984 Size distribution in the polymerisation model Af RBg J. Phys. A: Math.

Gen. 17 2281

17

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

[38] Tehrani S A and Khorramian A N 2007 The Jacobi polynomials QCD analysis for the polarized structurefunction J. High Energy Phys. JHEP07(2007)048

[39] Aizawa A 2000 Representation functions for Jordanian quantum group SLh(2) and Jacobi polynomials J. Phys.A: Math. Gen. 33 3735

[40] Alisauskas A 2004 Coupling coefficients of SO(n) and integrals involving Jacobi and Gegenbauer polynomialsJ. Phys. A: Math. Gen. 37 1093

[41] Aizawa A, Chakravarti R, Naina-Mohammed S S and Segar J 2007 Basic hypergeometric functions and covariantspaces for even-dimensional representations of Uq [osp(1/2)] J. Phys. A: Math. Theor. 40 14985

[42] Alisauskas A 2002 Coupling coefficients of SO(n) and integrals involving Jacobi and Gegenbauer polynomialsJ. Phys. A: Math. Gen. 35 7323

[43] Macfarlane A J 2003 Solution of the Schrodinger equation of the complex manifold CPn J. Phys. A: Math.Gen. 36 9689

[44] Carteret H A, Ismail M E H and Richmond B 2003 Three routes to the exact asymptotics for the one-dimensionalquantum walk J. Phys. A: Math. Gen. 36 8775

[45] Dewangan D P and Neerja Basuchoudhury K 2005 An accurate quantum expression for radiative transitionsbetween the Stark levels of nearby Rydberg states J. Phys. B: At. Mol. Opt. Phys. 38 1209

[46] Dewangan D P 2008 An accurate quantum expression of the z-dipole matrix element between nearby Rydbergparabolic states and the correspondence principle J. Phys. B: At. Mol. Opt. Phys. 41 015002

[47] Cooper F, Khare A and Sukhature U 2001 Supersymmetry in Quantum Mechanics (London: World Scientific)[48] Levai G 1991 A class of exactly solvable potentials related to the Jacobi polynomials J. Phys. A: Math.

Gen. 24 131[49] Quesne C and Tkachuk V M 2003 Harmonic oscillator with nonzero minimal uncertainties in both position and

momentum in a SUSYQM framwork J. Phys. A: Math. Gen. 36 10373[50] Dehesa J S, Lopez-Rosa S, Martınez-Finkelshtein A and Yanez R J 2010 Information theory of D-dimensional

hydrogenic systems: Application to circular and Rydberg states Int. J. Quantum Chem. 110 1529–48[51] Rakhmanov E A 1977 On the asymptotics of the ratio of orthogonal polynomials Math. USSR Sb. 32 199–213[52] Bagchi B, Gorain P, Quesne C and Roychoudhury R 2005 New approach to (quasi-)exactly solvable Schrodinger

equations with a position-dependent effective mass Europhys. Lett. 72 155[53] Hall M J W 1999 Universal geometric approach to uncertainty, entropy and information Phys. Rev.

A 59 2602–15[54] Sanchez-Moreno P, Dehesa J S, Manzano D and Yanez R J 2010 Spreading lengths of Hermite polynomials

J. Comput. Appl. Math. 233 2136–48[55] Omiste J J, Yanez R J and Dehesa J S 2009 Information-theoretic properties of the half-line Coulomb potential

J. Math. Chem. 47 911–28[56] Sanchez-Moreno P, Omiste J J and Dehesa J S 2010 Entropic functionals of Laguerre polynomials and

complexity properties of the half-line Coulomb potential Int. J. Quantum Chem. (DOI:10.1002/qua.22552)submitted

[57] Riordan J 1980 An Introduction to Combinatorial Analysis (Princeton, NJ: Princeton University Press)[58] Kendall M G and Stuart A 1969 The Advanced Theory of Statistics vol 1 (London: Charles Griffin)[59] Shohat J A and Tamarkin J D 1943 The Problem of Moments (Mathematical Surveys vol 1) (New York: American

Mathematical Society)[60] Romera E, Angulo J C and Dehesa J S 2001 The Hausdorff entropic moment problem J. Math. Phys. 42 2309–14[61] Renyi A 1970 Probability Theory (Amsterdam: North-Holland)[62] Tsallis C 1998 Possible generalization of Boltzmann–Gibbs statistics J. Stat. Phys. 52 479–87[63] Onicescu O 1966 Theorie de l’information. Energie informationelle C. R. Acad. Sci., Paris A 263 25[64] Heller E 1987 Quantum localization and the rate of exploration of phase space Phys. Rev. A 35 1360–70[65] Anteneodo C and Plastino A R 1996 Some features of the Lopez–Ruiz–Mancini–Calbet (LMC) statistical

measure of complexity Phys. Lett. A 223 348–54[66] Zurek W H, Habid S and Paz J P 1993 Coherent states via decoherence Phys. Rev. Lett. 70 1187–90[67] Brukner C and Zeilinger A 2001 Conceptual inadequacy of the Shannon information in quantum mechanics

Phys. Rev. A 63 022113[68] Hall M J W 2001 Exact uncertainty relations Phys. Rev. A 64 052103[69] Dehesa J S, Gonzalez-Ferez R and Sanchez-Moreno P 2007 The Fisher-information-based uncertainty relation,

Cramer–Rao inequality and kinetic energy for the D-dimensional central problem J. Phys. A: Math.Theor. 40 1845–56

[70] Dehesa J S, Martınez-Finkelshtein A and Sorokin V N 2006 Information-theoretic measures for Morse andPoschl–Teller potentials Mol. Phys. 104 613–22

[71] Sanchez-Ruiz J and Dehesa J S 1997 Expansions in series of orthogonal hypergeometric polynomials J. Comput.Appl. Math. 89 155–70

18

J. Phys. A: Math. Theor. 43 (2010) 305203 A Guerrero et al

[72] Dehesa J S, Sanchez-Moreno P and Yanez R J 2006 Cramer–Rao information plane of orthogonalhypergeometric polynomials J. Comput. Appl. Math. 186 523–41

[73] Sanchez-Ruiz J and Dehesa J S 2005 Fisher information of orthogonal hypergeometric polynomials J. Comput.Appl. Math. 182 150–64

[74] Sanchez-Ruiz J and Dehesa J S 2000 Entropic integrals of orthogonal hypergeometric polynomials with generalsupports J. Comput. Appl. Math. 118 311–22

[75] Aptekarev A I, Buyarov V S and Dehesa J S 1995 Asymptotic behavior of the Lp -norms and the entropy forgeneral orthogonal polynomials Russian Acad. Sci. Sb. Math. 82 373–95

[76] de Vicente J I, Sanchez-Ruiz J and Dehesa J S 2004 Information entropy and standard deviation of probabilitydistributions involving orthogonal polynomials Communication to IWOP 2004 (Madrid)

[77] submittedSanchez-Moreno P, Manzano D and Dehesa J S 2010 Direct spreading measures of Laguerrepolynomials J. Comput. Appl. Math. submitted

19

Apendices

A.1. Polinomios ortogonales clasicos: clasifi-

cacion y propiedades

Los polinomios ortogonales clasicos son funciones especiales de la matematicaaplicada y de la fısica matematica, cuyo estudio comienza en el siglo XVIII enrelacion con la resolucion de problemas relacionados con ciertos fenomenos dela mecanica celeste [1]. Estas funciones satisfacen la ecuacion hipergeometrica

σ(x) y′′(x) + τ(x) y′(x) + λ y(x) = 0, (A.1)

donde σ(x) es un polinomio de grado menor o igual a 2, τ(x) es un polinomiode grado menor o igual a 1 y λ es una constante. La solucion depende dela forma que tomen los polinomios σ(x) y τ(x) y la constante λ. A nosotrosnos interesa el estudio de las soluciones polinomicas y para ello se debe decumplir la siguiente condicion para la constante λ [2]:

λ = λn = −n τ ′(x)− n(n− 1)

2σ′′(x), (A.2)

donde n = 0, 1, 2, . . . es un numero natural. En este caso especial obtendremossoluciones polinomicas y(x) = yn(x) de grado n para la ecuacion de tipohipergeometrico (A.1).

Las soluciones polinomicas yn(x) satisfacen tambien la formula de Ro-drigues [2]

yn(x) =Bn

ω(x)

dn

dxn[σn(x)ω(x)] , (A.3)

donde Bn es una constante y ω(x) es la funcion peso que verifica la ecuacionde Pearson

[σ(x)ω(x)]′

= τ(x)ω(x). (A.4)

11

12 Apendices

Las ecuaciones (A.1)-(A.2) tienen tres familias de soluciones polinomi-cas reales canonicas: los polinomios de Hermite, Laguerre y Jacobi, cuyaspropiedades se detallan a continuacion

Hermite Hn(x)

σ(x) = 1, ω(x) = e−x2 ⇒ τ(x) = −2x ⇒ λn = 2n

La ecuacion diferencial correspondiente es

H′′

n(x)− 2xH′

n(x) + 2nHn(x) = 0, (A.5)

y la formula de Rodrigues con Bn = (−1)n

Hn(x) = (−1)ndn

dxn

[

e−x2]

. (A.6)

Laguerre L(α)n (x)

σ(x) = x, ω(x) = xα e−x ⇒ τ(x) = −x+ α + 1 ⇒ λn = n

La ecuacion diferencial correspondiente es

x[

L(α)n

(

x)]′′

+ (α + 1− x)[

L(α)n (x)

]′

+ n[

L(α)n (x)

]

= 0, (A.7)

y la formula de Rodrigues con Bn = 1n!

L(α)n (x) =

1

n!ex x−α dn

dxn

[

xα+n e−x]

. (A.8)

Jacobi P(α,β)n (x)

σ(x) = 1 − x2, ω(x) = (1 − x)α(1 + x)β ⇒ τ(x) = −(α + β + 2)x +β − α ⇒λn = n(n+ α + β + 1)

La formula de Rodrigues con Bn = 12n n!

es

P (α,β)n (x) =

1

2n n!(1− x)−α(1 + x)−β dn

dxn

[

(1− x)n+α(1 + x)n+β]

.

(A.9)

A.1. POLINOMIOS ORTOGONALES CLASICOS 13

Hermite Hn(x) Laguerre L(α)n (x) Jacobi P

(α,β)n (x)

σ(x) 1 x 1− x2

τ(x) −2x 1 + α− x −(α + β + 2)x+ β − αλn 2n n n(n+ α + β + 1)Bn (−1)n 1/n! (−1)n/(2n n!)

Tabla A.1: Parametros de los polinomios ortogonales clasicos

A.1.1. Ortogonalidad de los P.O.C.

Los polinomios yn(x) presentados cumplen la condicion de ortogonalidadsiguiente:

∫ b

a

yn(x) ym(x)ω(x) dx = d2n δnm, (A.10)

donde ω(x) es la funcion peso, dn es la constante de normalizacion, δnm esla delta de Kronecker y [a, b] es el intervalo de ortogonalidad. Para cadauno de los P.O.C., con sus respectivos pesos tendremos distintos valores dela constante d2n (tambien presentamos su valor para los polinomios monicos,aquellos que tienen como coeficiente principal 1) y del intervalo de ortogonal-idad. Todos estos datos los reflejamos a continuacion de forma esquematica(nos podemos fijar en la tabla 2, donde aparecen resumidos los valores de d2n):

Hermite Hn(x)

ω(x) = e−x2

d2n = 2n n!√π y para monicos d2n = n!

√π

2n

(a, b) = (−∞,+∞)

Laguerre L(α)n (x)

ω(x) = xα e−x

d2n = Γ(n+α+1)n!

y para monicos d2n = n! Γ(n+ α + 1)

(a, b) = (0,+∞)

Jacobi P(α,β)n (x)

ω(x) = (1− x)α(1 + x)β

d2n = 2α+β+1 Γ(n+α+1) Γ(n+β+1)n!(2n+α+β+1)Γ(n+α+β+1)

14 Apendices

Hermite Hn(x) Laguerre L(α)n (x) Jacobi P

(α,β)n (x)

d2n 2n n!√π Γ(n+α+1)

n!2α+β+1 Γ(n+α+1) Γ(n+β+1)n!(2n+α+β+1)Γ(n+α+β+1)

d2n monicos n!√π

2nn! Γ(n+ α + 1) 2α+β+2n+1 n! Γ(n+α+1) Γ(n+β+1)

(2n+α+β+1) Γ(n+α+β+1) (n+α+β+1)2n

Tabla A.2: Valores de dn para polinomios normales y monicos. Tenemos encuenta que (a)k =

Γ(a+k)Γ(a)

y para monicos d2n = 2α+β+2n+1 n! Γ(n+α+1) Γ(n+β+1)(2n+α+β+1) Γ(n+α+β+1) (n+α+β+1)2n

(a, b) = (−1,+1)

A.1.2. Relacion de recurrencia a tres terminos

Una de las principales caracterısticas de los polinomios ortogonales clasicoses que satisfacen una relacion de recurrencia a tres terminos que conecta trespolinomios de grados consecutivos yn−1(x), yn(x), yn+1(x). La expresion dedicha relacion viene dada por

x yn(x) = αn yn+1(x) + βn yn(x) + γn yn−1(x) (A.11)

donde αn, βn y γn son constantes (se suele tomar y−1(x) = 0 y y0(x) = 1).Para los P.O.C. dichas constantes son bien conocidas; las presentamos acontinuacion

Hermite Hn(x)

αn = 12

βn = 0 γn = n

Laguerre L(α)n (x)

αn = −(n + 1) βn = 2n+ α + 1 γn = −(n + α)

Jacobi P(α,β)n (x)

αn = 2(n+1)(n+α+β+1)(2n+α+β+1)(2n+α+β+2)

βn = β2−α2

(2n+α+β)(2n+α+β+2)

γn = 2(n+α)(n+β)(2n+α+β)(2n+α+β+1)

En la Seccion A.3 se emplean unos valores distintos de los αn, βn y γnque tenemos arriba para hacer calculos; se utilizan los correspondientes a lospolinomios monicos (polinomios cuyo coeficiente principal es 1). Escribimosaquı tambien dichas constantes a continuacion:

A.1. POLINOMIOS ORTOGONALES CLASICOS 15

Hermite Hn(x) Laguerre L(α)n (x) Jacobi P

(α,β)n (x)

αn 1/2 −(n+ 1) 2(n+1)(n+α+β+1)(2n+α+β+1)(2n+α+β+2)

βn 0 2n+ α+ 1 β2−α2

(2n+α+β)(2n+α+β+2)

γn n −(n+ α) 2(n+α)(n+β)(2n+α+β)(2n+α+β+1)

Tabla A.3: Valores de αn, βn y γn en la relacion de recurrencia a tres terminospara polinomios normales

Hermite Hn(x) Laguerre L(α)n (x) Jacobi P

(α,β)n (x)

αn 1 1 1

βn 0 2n+ α + 1 β2−α2

(2n+α+β)(2n+α+β+2)

γn n/2 n(n + α) 4n(n+α)(n+β)(n+α+β)(n+α+β+1)(2n+α+β−1)(2n+α+β)2(2n+α+β+1)

Tabla A.4: Valores de αn, βn y γn en la relacion de recurrencia a tres terminospara polinomios monicos

Hermite Hn(x)

αn = 1 βn = 0 γn = n2

Laguerre L(α)n (x)

αn = 1 βn = 2n+ α + 1 γn = n(n+ α)

Jacobi P(α,β)n (x)

αn = 1 βn = β2−α2

(2n+α+β)(2n+α+β+2)

γn = 4n(n+α)(n+β)(n+α+β)(n+α+β+1)(2n+α+β−1)(2n+α+β)2(2n+α+β+1)

A.1.3. Densidad de Rakhmanov

Vamos a introducir un concepto que tiene su origen en la propiedad deortogonalidad de los polinomios ortogonales (A.10).

∫ b

a

yn(x) ym(x)ω(x) dx = d2n δnm.

En la expresion (A.10) tomamos m = n, que hace la integral no nula al serd2n 6= 0,

16 Apendices

∫ b

a

y2n(x)ω(x) dx = d2n ⇒∫ b

a

1

d2ny2n(x)ω(x) dx = 1. (A.12)

Podemos ver el integrando de esta expresion como una funcion densidad yaque su integral a lo largo del intervalo (a, b) es igual a 1, es decir

∫ b

aρn(x) dx =

1. La densidad de Rakhmanov [3] se define entonces como

ρn(x) =1

d2ny2n(x)ω(x). (A.13)

Esta funcion juega un papel fundamental para entender la asintotica del co-ciente de dos polinomios con grados consecutivos, tal y como demostro EvgeniRakhmanov [3] en 1977.

A.2. Desarrollo de los P.O.C. en serie de po-

tencias

En la seccion anterior hemos presentado a los P.O.C. con su formula deRodrigues. Esta forma de introducirlos esta bien para entender teoricamentecon que elementos estamos tratando y de donde provienen, pero a veces esinteresante tener al alcance otras formas de expresion matematica que nospermitan trabajar con ellas mas comodamente o por lo menos de manera masintuitiva. Por ejemplo, es bien conocida la expresion de dichos polinomios enfuncion de las funciones hipergeometricas [1] pero todavıa no tienen una for-ma que los caracterice explıcitamente como polinomios. Aun hay otra muchomas intuitiva; nuestro objetivo en esta nueva seccion es exponer los desar-rollos en serie de potencias de los polinomios de Hermite, Laguerre y Jacobi;es decir, si yn(x) es un polinomio de grado n, obtener una expresion como lasiguiente:

yn(x) =

n∑

m=0

anm xm, (A.14)

donde anm son constantes. Como vemos, de esta forma tenemos un autenticodesarrollo de los polinomios que nos servira posteriormente para hacer calcu-los de valores esperados (en la Seccion A.4 veremos esta aplicacion). Puedeser que en otros estudios sea de utilidad este desarrollo pero en este trabajosolo nos limitaremos a la aplicacion ya mencionada y que desarrollaremosmas adelante en la seccion posterior.

A.2. DESARROLLO DE LOS P.O.C. EN SERIE DE POTENCIAS 17

A continuacion presentaremos los distintos desarrollos para los P.O.C. queestudiamos en este trabajo. Para los polinomios de Hermite y de Laguerreestas expresiones se encuentran facilmente en la literatura [4]. Los polinomiosde Jacobi requieren un poco de trabajo adicional.

Hermite Hn(x)

Hn(x) = n!

[n2 ]∑

m=0

(−1)m1

m!(n− 2m)!(2x)n−2m =

[n2 ]∑

m=0

bn,m xn−2m,

(A.15)donde

[

n2

]

indica la parte entera de n2, donde los coeficientes bn,m tienen

la forma

bn,m = (−1)m2n−2m n!

m!(n− 2m)!. (A.16)

Laguerre L(α)n (x)

L(α)n (x) =

n∑

m=0

(−1)m(

n+ α

n−m

)

1

m!xm, (A.17)

Podemos hacer mas compacta esta expresion tal y como lo hemos hechoen el caso de Hermite; introducimos notacion

aα,n,m = (−1)m(

n + α

n−m

)

1

m!⇒ L(α)

n (x) =n∑

m=0

aα,n,m xm. (A.18)

Jacobi P(α,β)n (x)

Partimos de la siguiente expresion que aparece en la literatura [4]

P (α,β)n (x) =

n∑

m=0

(

n

m

)

Γ(α + n+ 1) Γ(α+ β + n+m+ 1)

n! 2m Γ(α + β + n+ 1) Γ(α+m+ 1)(x− 1)m,

(A.19)donde usaremos la notacion

aα,βn,m =

(

n

m

)

Γ(α + n+ 1) Γ(α+ β + n+m+ 1)

n! 2m Γ(α + β + n+ 1) Γ(α+m+ 1), (A.20)

con lo que se obtiene

P (α,β)n (x) =

n∑

m=0

aα,βn,m (x− 1)m. (A.21)

18 Apendices

El siguiente paso es desarrollar (x − 1)m utilizando la formula del bi-nomio de Newton,

(x− 1)m =m∑

l=0

(

m

l

)

(−1)m−l xl. (A.22)

Sustituyendo lo anterior en (A.20) nos queda

P (α,β)n (x) =

n∑

m=0

aα,βn,m

m∑

l=0

(

m

l

)

(−1)m−l xl

=n∑

m=0

m∑

l=0

aα,βn,m

(

m

l

)

(−1)m−l xl. (A.23)

Utilizamos la notacion

bm,l = aα,βn,m

(

m

l

)

(−1)m−l, (A.24)

para simplificar la expresion anterior y poder operar mas comodamente.Por tanto, nos queda

P (α,β)n (x) =

n∑

m=0

m∑

l=0

bm,l xl. (A.25)

Desarrollando esta ultima expresion se obtiene

P (α,β)n (x) = b00 + b10 + b11 x+ b20 + b21 x+ b22 x

2 + . . .+

+ (bn0 + bn1 x+ . . .+ bnn xn)

=

n∑

i=0

bi0 +

n∑

i=1

bi1 x+ . . .+

n∑

i=n

bin xn

=

n∑

k=0

ck xk, (A.26)

donde los coeficiente ck tienen la forma

ck =n∑

i=k

bi,k =n∑

i=k

aα,βn,i

(

i

k

)

(−1)i−k

⇒ ck =n∑

i=k

(

n

i

)

Γ(α + n+ 1) Γ(α+ β + n + i+ 1)

n! 2i Γ(α + β + n+ 1) Γ(α+ i+ 1)

(

i

k

)

(−1)i−k.

(A.27)

A.3. DESARROLLO DE POTENCIAS EN SERIE DE P.O.C. 19

A.3. Desarrollo de potencias en serie de P.O.C.

En este apartado estudiamos el problema de la inversion, que consisteen obtener los desarrollos de xm(m es natural) en serie de polinomios or-togonales clasicos yn(x). La deduccion se realiza a partir de la propiedad deortogonalidad que ya presentamos al principio en la primera seccion. Apartese utilizara otra propiedad de los P.O.C. bien conocida que se escribe de lasiguiente manera matematicamente [2]:

σ(x)ω(x) xk = 0 para x = a, b k = 0, 1, · · · (A.28)

donde (a, b) es el intervalo de ortogonalidad. Tambien podemos presentarmatematicamente lo que significa el desarrollo en serie de P.O.C. de xm:

xm =

m∑

n=0

cm,n yn(x), (A.29)

donde cmn son los coeficientes que se deben de determinar para cerrar laexpresion. Aquı trataremos los casos de Hermite, Laguerre y Jacobi. El interesde estos desarrollos se vera en la Seccion A.4.

Lo primero que debemos de hacer es obtener una formula general paralos coeficientes cmn basandonos en las propiedades que ya hemos citado an-teriormente. El primer paso es multiplicar ambos lados de la ecuacion (A.29)por yr(x)ω(x), obteniendo

xm yr(x)ω(x) =m∑

n=0

cm,n yn(x) yr(x)ω(x)

Integrando entre a y b y utilizando (A.10), se llega a

∫ b

a

xm yr(x)ω(x) dx =m∑

n=0

cm,n

∫ b

a

yn(x) yr(x)ω(x) dx =m∑

n=0

cm,n d2r δn,r.

Esta ultima expresion nos lleva a poder despejar el valor de la constante cmn

pues en la sumatoria solo nos queda el termino con n = r debido a la deltade Kronecker:

cmn =1

d2n

∫ b

a

xm yn(x)ω(x) dx. (A.30)

Por ahora tenemos una expresion general para los coeficientes de la expansion(A.29) pero podemos avanzar un poco mas para obtener cmn en funcion solode los parametros de la ecuacion diferencial hipergeometrica de la cual son

20 Apendices

solucion los polinomios yn(x). Para ello, utilizamos la formula de Rodrigues(A.3) para yn(x) y la introducimos en (A.30):

cm,n =Bn

d2n

∫ b

a

xm dn

dxn[σn(x)ω(x)] dx. (A.31)

El siguiente paso es eliminar la derivada n-esima que aparece en el integrando.Para ello integramos por partes n veces y aplicamos la propiedad (A.28). Elresultado final es por tanto el siguiente:

cm,n =(−1)n m!

(m− n)!

Bn

d2n

∫ b

a

xm−n σn(x)ω(x) dx, (A.32)

que viene dado en funcion de los datos conocidos y tabulados que podemosencontrar en la primera seccion del apendice. Ahora lo unico que tenemos quehacer es aplicar esta formula obtenida a los polinomios de Hermite, Laguerrey Jacobi, obteniendo los valores siguientes:

Hermite Hn(x)

cm,n =

m!2m n! ((m−n)/2)!

para m− n par

0 para m− n impar

(A.33)

Laguerre L(α)n (x)

cm,n =(−1)n m! Γ(m+ α + 1)

(m− n)! Γ(n+ α + 1). (A.34)

Jacobi P(α,β)n (x)

cm,n =m! (−1)m−n 2n Γ(n+ 2α + 1)

(m− n)! Γ(2n+ 2α + 1)2F1(n−m,n+β+1; 2n+α+β+2; 2).

(A.35)

donde 2F1(a, b; c; 2) es la funcion hipergeometrica de Gauss.

A.4. Valores esperados 〈xk〉En esta seccion el objetivo es calcular 〈xk〉 utilizando los conceptos y

propiedades que hemos tratado anteriormente. Sera importante tener en

A.4. VALORES ESPERADOS 21

cuenta el significado probabilistico de la densidad de Rakhmanov (A.13) puesen base a ella realizaremos los calculos pertinentes. De hecho, el concepto devalor esperado de una funcion f(x) viene dado por la siguiente expresion:

〈f(x)〉n =

∫ b

a

f(x) ρn(x) dx, (A.36)

donde (a, b) es el intervalo donde vamos a considerar el calculo y ρn(x) es ladensidad de Rakhmanov.

Primeramente haremos uso de la relacion de recurrencia a tres terminosde los P.O.C. para hacer los calculos de 〈xk〉 con valores pequenos determi-nados de la constante entera k. Una vez alcanzada dicha meta presentare-mos un metodo general que considerara cualquier valor de k. Tanto en unprocedimiento como en otro obtendremos resultados correspondientes a lospolinomios de Hermite, Laguerre y Jacobi.

A.4.1. Relacion de recurrencia a tres terminos para

obtener 〈xk〉Vamos a desarrollar un poco la formula anterior (A.36) para ver cual es

el procedimiento que vamos a seguir en este punto:

〈xk〉 =∫ b

a

xk ρn(x) dx =1

d2n

∫ b

a

xky2n(x)ω(x) dx, (A.37)

Ahora utilizaremos la relacion de recurrencia a tres terminos de los P.O.C.para obtener lo que queremos; recordamos la formula que vamos a utilizar:

x yn(x) = αn yn+1(x) + βn yn(x) + γn yn−1(x). (A.38)

El procedimiento a seguir es sustituir el producto x yn(x) por su expresionequivalente dada en (A.38) y aplicar la propiedad de ortogonalidad (A.10) queya conocemos. Realizaremos el procedimiento para los casos k = 0, 1, 2 comoejemplo; el resto de casos sigue un camino similar pero el unico inconvenientees que las cuentas se hacen muy largas. Veamos los pasos a seguir:

k = 0 ⇒〈x0〉 =

∫ b

a

x0 ρn(x) dx =

∫ b

a

ρn(x) dx = 1. (A.39)

Como vemos, el primer caso no tiene ninguna dificultad; solo hemostenido que aplicar la definicion de densidad de Rakhmanov.

22 Apendices

k = 1 ⇒

〈x〉 =∫ b

a

x ρn(x) dx =1

d2n

∫ b

a

ω(x) [x yn(x)] yn(x) dx

=1

d2n

[

αn

∫ b

a

ω(x) yn+1(x) yn(x) dx+ βn

∫ b

a

ω(x) y2n(x) dx

+γn

∫ b

a

ω(x) yn(x) yn−1(x) dx

]

Ahora utilizando la propiedad de ortogonalidad de los polinomios yn(x),se obtiene

〈x〉 = 1

d2nβn

∫ b

a

ω(x) y2n(x) dx = βn. (A.40)

k = 2 ⇒Ahora tenemos que

〈x2〉 =∫ b

a

x2 ρn(x) dx =

∫ b

a

x2 1

d2ny2n(x)ω(x) dx =

=1

d2n

[∫ b

a

ω(x) y2n+1(x) dx+ β2n

∫ b

a

ω(x) y2n(x) dx

+γ2n

∫ b

a

ω(x) y2n+1(x) dx

]

,

donde se ha sustituido x yn(x) por su correspondiente expresion dadaen (A.38) y se ha operado elevando al cuadrado primeramente dichaformula que aparece en el integrando. Se ha vuelto a usar la propiedadde ortogonalidad para llegar hasta donde hemos mostrado; volvemos autilizar dicha propiedad para obtener el resultado final:

〈x2〉 = d2n+1

d2n+ β2

n + γ2n

d2n−1

d2n. (A.41)

Ya hemos mostrado como se opera en los primeros casos; para los siguientesvalores de k se sigue la misma idea de ir sustituyendo x yn(x) por su cor-respondiente expresion de la relacion de recurrencia a tres terminos. A con-tinuacion mostramos resultados correspondientes a los distintos P.O.C. queestamos estudiando desde el principio. Tenemos en cuenta que se han uti-lizado las constantes correspondientes a los polinomios monicos aunque sihubiesemos utilizado las otras obtendrıamos los mismos resultados.

A.4. VALORES ESPERADOS 23

Hermite Hn(x)

Para los Hermite se obtiene que

〈x0〉 = 1 〈x〉 = 0 〈x2〉 = n+1

2

〈x3〉 = 0 〈x4〉 = 3

2

[

n2 + n+1

2

]

Laguerre L(α)n (x)

Para los Laguerre se obtiene que

〈x0〉 = 1 〈x〉 = 2n+ α + 1

〈x2〉 = 6n2 + 6(α + 1)n+ (α + 1)(α+ 2)

〈x3〉 = 4n4+(2α+7)n3+(10α+13)n2+(4α2+12α+14)n+(α+1)(3α+4)

Jacobi P(α,β)n (x)

Para los Jacobi se obtiene que

〈x0〉 = 1 〈x〉 = β2 − α2

(2n+ α + β)(2n+ α + β + 2)

〈x2〉 = 4(n+ 1)(n + α+ 1)(n+ β + 1)(n+ α + β + 1)

(2n+ α + β + 1)(2n+ α + β + 2)2(2n+ α + β + 3)+ 〈x〉2

+4n(n + α)(n+ β)(n+ α+ β)

(2n+ α+ β − 1)(2n+ α+ β)2(2n + α+ β + 1)

A.4.2. Metodo general para calcular 〈xk〉Ahora nos planteamos el mismo problema que antes pero con la diferencia

de que esta vez consideraremos k un entero cualquiera ya sea pequeno o muygrande. La potencia del resultado final es por tanto mucho mayor y masinteresante que el obtenido anteriormente con la relacion de recurrencia atres terminos. Muchos de los conceptos que hemos tratado en los puntosanteriores apareceran aquı para que con su aplicacion se llegue al objetivoque nos planteamos. Comentamos a continuacion las herramientas necesariaspara ir operando en el camino a seguir:

24 Apendices

Lo primero que tendremos en cuenta es que al utilizar polinomios or-togonales clasicos nos podemos aprovechar de la propiedad de ortogo-nalidad (A.10). Lo que verdaderamente nos interesa es una equivalenciarelacionada con esta propiedad; mostramos dicha equivalencia:

∫ b

a

yn(x) ym(x)ω(x) dx = d2n δnm

⇔∫ b

a

xm yn(x)ω(x) dx = 0, (m < n) (A.42)

Mas adelante veremos el interes de esta propiedad

Tenemos tambien presente el desarrollo de las potencias xm como unacombinacion lineal de polinomios ortogonales dada en (A.29):

xm =

m∑

n=0

cmn yn(x), (A.43)

donde cmn son constantes cuyos valores ya presentamos en la seccionA.3 para cada caso concreto (Hermite, Laguerre o Jacobi).

Se hace uso de la expresion de yn(x) como una serie de potencias cuyoestudio ya hicimos en la seccion A.3. La idea central es la utilizacionde la formula (A.14):

yn(x) =

n∑

m=0

anm xm, (A.44)

donde anm son constantes cuyos valores ya estan contemplados en lasegunda seccion.

Cada uno de los tres puntos presentados son importantes para desarrollarel metodo general que queremos presentar aquı. Por suspuesto tambien uti-lizaremos la ya conocida densidad de Rakhmanov. Comenzamos a trabajarcon la definicion de xk:

〈xk〉 =∫ b

a

xk ρn(x) dx =1

d2n

∫ b

a

xk ω(x) yn(x) yn(x) dx.

Ahora sustituimos uno de los polinomios yn(x) por su correspondiente desar-rollo de potencias dado en (A.44)

A.4. VALORES ESPERADOS 25

〈xk〉 = 1

d2n

∫ b

a

xk ω(x) yn(x)

(

n∑

m=0

anm xm

)

dx

=1

d2n

n∑

m=0

anm

(∫ b

a

xk+m yn(x)ω(x) dx

)

.

Aquı aplicamos (A.42), por lo cual sabemos que las integrales se anulan paravalores de m que cumplan k +m < n ⇒ m < n− k

〈xk〉 = 1

d2n

n∑

m=n−k

anm

(∫ b

a

xk+m yn(x)ω(x) dx

)

.

Segun (A.44) cuando m < 0 tenemos anm = 0. Continuamos nuestro caminoutilizando el desarrollo que aparece en (A.43); en concreto utilizamos quexk+m =

∑k+mp=0 ck+m,p yp(x)

〈xk〉 = 1

d2n

n∑

m=n−k

anm

[

∫ b

a

(

k+m∑

p=0

ck+m,p yp(x)

)

yn(x)ω(x) dx

]

=1

d2n

n∑

m=n−k

anm

[

k+m∑

p=0

ck+m,p

(∫ b

a

yp(x) yn(x)ω(x) dx

)

]

.

Aplicamos que∫ b

ayp(x) ym(x)ω(x) dx = d2n δnp de manera que la sumatoria

que hay dentro del corchete se reduce a un termino. Veamos lo que nos queda:

〈xk〉 = 1

d2n

n∑

m=n−k

anm ck+m,n d2n =

n∑

m=n−k

anm ck+m,n. (A.45)

Ya tenemos el resultado que ibamos buscando. Ahora podemos aplicarlo a loscasos particulares de los polinomios ortogonales clasicos de Hermite, Laguerrey Jacobi; solo hay que seguir los pasos que hemos descrito anteriormente yutilizar los datos concretos que hemos presentado y obtenido en las seccionesA.2 y A.3. A continuacion presentamos cada uno de los casos concretos quehemos citado lıneas atras. Es interesante tener expresiones tan claras y cer-radas como nos aparecen para cada uno de los polinomios

Hermite Hn(x)

26 Apendices

Para los Hermite tenemos

〈xk〉 =

∑

k2

m=0(−1)m (k+n−2m)!2k m! (n−2m)! [(k−2m)/2]!

para k par

0 para k impar

(A.46)

Laguerre L(α)n (x)

Para los Laguerre tenemos

〈xk〉 = 1

Γ(n + α+ 1)

n∑

p=n−k

(−1)p+n

(

n + α

n− p

)

(k + p)! Γ(k + p+ α + 1)

(k + p− n)! p!.

(A.47)

Jacobi P(α,β)n (x)

Para los Jacobi tenemos

〈xk〉 =n∑

i=n−k

(

n∑

t=i

(−1)t−i

(

n

t

)

Γ(α + n+ 1) Γ(α+ β + n + t+ 1)

n! 2t Γ(α + β + n + 1) Γ(α+ t+ 1)

(

t

i

)

)

× (−1)k+i−n 2n (k + i)! Γ(n + α+ β + 1)

(k + i− n)! Γ(2n+ α + β + 1)

× 2F1

(

n− k − i, n+ β + 12n+ α + β + 2

∣

∣

∣

∣

2

)

. (A.48)

Bibliografıa

[1] Renato Alvarez-Nodarse, Polinomios Hipergeometricos y q-polinomios,Universidad de Zaragoza 2003

[2] A. F. Nikiforov and V. B. Uvarov, Special Functions in Mathematical

Physics, Birkauser-Verlag, Basel 1988

[3] E. A. Rakhmanov, On the asymptotics of the ratio of orthogonal poly-

nomials, Math. USSR Sb. 32 (1977) 199–213

[4] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions

with Formulas, Graphs, and Mathematical Tables, National Bureau ofStandars, U.S. Goverment Printing Office, Washington D.C. 1964

[5] Th. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon andBreach, New York 1978

[6] P. Sanchez-Moreno and J. S. Dehesa and D. Manzano and R. J. YanezSpreading lengths of Hermite polynomials, J. Comput. Appl. Math. 233(2010) 2136–2148

[7] P. Sanchez-Moreno and D. Manzano and J. S. Dehesa, Direct spread-ing measures of Laguerre polynomials, J. Comput. Appl. Math. (2010).Disponible online. DOI:10.1016/j.cam.2010.07.022

[8] M. J. W. Hall, Universal geometric approach to uncertainty, entropy and

information, PRA 59 (1999) 2602–2615

27