Generalized Meixner Pollaczek polynomials

R E S E A R C H

Open Access

Generalized Meixner-Pollaczek polynomials

Stanislawa Kanas

1*

_{and Anna Tatarczak}

2

*_{Correspondence:}

[email protected]

1_{Department of Mathematics,}

Rzeszow University of Technology, Rzeszow, Poland

Full list of author information is available at the end of the article

Abstract

We consider the generalized Meixner-Pollaczek (GMP) polynomialsPλ_n(x;

θ

,

ψ

) of a variablex∈Rand parameters

λ

> 0,

θ

∈(0,

π

),

ψ

∈R, definedviathe generating function

Gλ(x;

θ

,

ψ

;z) = 1

(1 –zeiθ₎λ–ix_{(1 –ze}iψ₎λ+ix =

∞

n=0

Pλ_n(x;

θ

,

ψ

)zn, |z|< 1.

We find the three-term recurrence relation, the explicité formula, the hypergeometric representation, the difference equation and the orthogonality relation for GMP polynomialsPλ

n(x;

θ

,

ψ

). Moreover, we study the special case ofPλn(x;

θ

,

ψ

)

corresponding to the choice

ψ

=

π

+

θ

and

ψ

=

π

–

θ

, which leads to some interesting families of polynomials. The limiting case (

λ

→0) of the sequences of polynomialsPλ

n(x;

θ

,

π

+

θ

) is obtained, and the orthogonality relation in the strip

S={z∈C:|(z)|< 1}is shown.

MSC: 33C45; 30C10; 30C45; 39A60

Keywords: Meixner-Pollaczek polynomials; difference equation; generating function; orthogonal polynomials; Fisher information

1 Introduction

The classical Koebe function is a function holomorphic inD={z:|z|< }and given by the formula

k(z) =

z ( –z) =

 

 +z  –z



– 

=z+ z+ z+· · ·, z∈D.

The importance ofkfollows from the extremality for the famous Bieberbach

conjec-ture. The Koebe function is univalent and starlike inDand maps the unit diskDonto the complex plane minus a slit (–∞, –_].

Several generalizations of k appeared in the literature. Robertson [] proved that

k(–α)(z) =_(–z)z(–α) (≤α< ) is the extremal function for the class of functions starlike

of orderα. The function

kα(z) =  α

 +z  –z

α – 

, α∈R\ {},z∈D,

was extensively studied by Pommerenke [], who investigated a universal invariant family Uα.

The definition ofkαwas extended for a nonzero complex numberαby Yamashita []. The classical result of Hille [] ascertains thatkα is univalent inDif and only ifα=  is in the unionAof the closed disks{|z+ | ≤}and{|z– | ≤}. Making use of geometric properties, Yamashita [] described howkαtends to be univalent in the wholeDasαtends

to each boundary point ofAfrom outside.

The properties oflogk_c, where

kc(z) =

 c

 +z  –z

c

– 

, c∈C\ {}, k(z) =

 log

 +z

 –z, z∈D, (.)

were studied in [] by Campbell and Pfaltzgraff. Pommerenke [] examined the special case of (.),i.e.,

kiγ(z) =  iγ

 +z  –z

iγ – 

, γ > ,z∈D,

for which

k_iγ(z) = 

( +z)–iγ_{( –z)}–iγ.

An evident and important extension of (.) was given by the following formulas (θ ∈

R,ψ∈R):

kc(θ,ψ;z) =

 (eiψ_–eiθ_)c

 –zeiθ  –zeiψ

c

– 

, c∈C\ {},eiψ=eiθ,z∈D,

and for the case whenc= ,

k(θ,ψ;z) =

 (eiψ_–eiφ₎log

 –zeiθ  –zeiψ, e

iψ_=eiθ_,z∈D.

We have

k_c(θ,ψ;z) = 

( –zeiθ₎–c_{( –ze}iψ₎+c, c∈C.

Comparing k_iγ(θ,ψ;z) = _(–zeiθ₎–iγ_(–zeψ₎+iγ with the generating function for

Meixner-Pollaczek polynomialsPλ

n(x;θ) [],

Gλ(x;θ, –θ;z) = 

( –zeiθ₎λ–ix_{( –ze}–iθ₎λ+ix =

∞

n=

Pλ_n(x;θ)zn, z∈D,

whereλ> ,θ ∈(,π),x∈R, we were motivated to introduce the generalized

Meixner-Pollaczek polynomials (GMP) []Pλ

n(x;θ,ψ) of a variable x∈Rand parametersλ> ,

θ∈(,π),ψ∈Rviathe generating function

Gλ_{(x;θ,ψ;z) =} 

( –zeiθ₎λ–ix_{( –ze}iψ₎λ+ix=

∞

n=

Pλ

n(x;θ,ψ)zn, z∈D. (.)

Obviously, we havePλ

2 Orthogonal polynomials

LetLdenote themoment functionalthat is a linear mapC[x]→C. A sequence of poly-nomials{Pn(x)}∞n=is anorthogonal polynomials sequence(OPS) with respect toLifPn(x)

has degreen,L[Pm(x)Pn(x)] =  form=nandL[Pn(x)]=  for alln.

In this paper we consider orthogonal polynomial systems defined recursively. Every monic OPSPn(x)∞n=may be described by a recurrence formula of the form

Pn(x) = (x–cn)Pn–(x) –λnPn–(x), n= , , , . . . , (.)

whereP–(x) = ,P(x) = , the numberscnandλnare constants,λn=  forn>  andλ

is arbitrary (see [, Ch. I, Theorem .]). The sequences of orthogonal polynomials are symmetricifPn(x) = (–)nPn(–x) for alln(see [, Ch. I, Theorem .]) or thatcnin (.)

are all zero.

Polynomials with exponential generating functions are among the most often studied polynomials. One of them is the Meixner-Pollaczek polynomials. The Meixner-Pollaczek polynomials were first invented by Meixner []. The same polynomials were also con-sidered independently by Pollaczek []. These polynomials are classified in the Askey-scheme of orthogonal polynomials [, ].

Some of the main properties of these polynomials are presented in Erdélyiet al.[],

Chihara [], Askey and Wilson [] and in the report by Koekoek and Swarttouw []. De-tailed analyses with applications of these polynomials are also made by several authors.

Among others, the works of Rahman [], Atakishiyev and Suslov [], Benderet al.[],

Koornwinder [] and the extensive work of Li and Wong [] may be included.

This paper is mainly concerned about the generalized Meixner-Pollaczek (GMP) poly-nomials. We also study the special cases ofPλ

n(x;θ,ψ), corresponding to the choiceψ=

π+θandψ=π–θ, which lead to some interesting families of polynomials.

For complex numbersa,bandc(c= , –, –, . . .), theGaussian hypergeometric function

F(a,b,c;z) is defined by

F(a,b,c;z) = +∞

k=

(a)k(b)k

(c)k

zk

k!, z∈D, (.)

where (a)nis thePochhammer symboldescribed by

(a)n=a(a+ )(a+ )· · ·(a+n– ), n∈N, (a)= .

Notice thatF(a,b,c;z) is symmetric inaandb, and the series terminates if eitheraorb

is zero or a negative integer. In general, the seriesF(z) is absolutely convergent inD. If

(c–a–b) > , it is also convergent on∂D, and it is known that

F(a,b;c; ) =

(c)(c–a–b)

(c–a)(c–b), Re(c–a–b) > ;c= , –, –, . . . . (.)

3 Generalized Meixner-Pollaczek polynomials

In this section we find the three-term recurrence relation, the explicité formula, the hy-pergeometric representation, the difference equation and the orthogonality relation for

(GMP) polynomialsPλ

Theorem  Let us set Pλ

–= .The polynomials Pλn=Pnλ(x;θ,ψ)have the following

proper-ties:

(a) Pλ

nsatisfy the three-term recurrence relation

Pλ_= ,

nPλ_n= (λ–ix)eiθ+ (λ+ix)eiψ+ (n– )eiθ+eiψPλ_n– – (λ+n– )ei(θ+ψ)Pλ_n–, n≥.

(b) Pλ

nare given by the formula

Pλ_n(x;θ,ψ) =einθ

n

j=

(λ+ix)j(λ–ix)n–j

j!(n–j)! e

ij(ψ–θ)_{, n∈N∪ {}.}

(.)

(c) Pλ

nhave the hypergeometric representation

n!Pλ_n(x;θ,ψ) = (λ)neinθF

–n,λ+ix, λ;  –ei(ψ–θ). (.)

(d) Lety(x) =Pλ

n(x;θ,ψ).The functiony(x)satisfies the following difference equation:

eiθ_{(λ–ix)y(x+i) + ixe}iθ_+eiψ_{– (n+λ)e}iθ_–eiψ_y(x)

–eiψ_{(λ+ix)y(x–i) = . (.)}

Proof

(a) We differentiate the formula(.)with respect toz, and after multiplication by

( –zeiθ_{)( –ze}iψ_{), we compare the leading coefficients ofz}n–_. (b) The Cauchy product of the power series

 –zeiθ–(λ–ix)= ∞

n=

(λ–ix)neinθ

n! z

n

and

 –zeiψ–(λ+ix)= ∞

n=

(λ+ix)neinψ

n! z

n

gives(.).

(c) Applying the formula from [, vol., p.],

( –s)a–c( –s+sz)–a= ∞

n=

(c)n

n! F(–n,a;c;z)s

n_{, |s|< ,s( –z)< ,}

withs=zeiθ_{,a=λ+ix,c= λ,z=  –e}i(ψ–θ)_{, one obtains}

 –zeiθ–(λ–ix) –zeiψ–(λ+ix)= ∞

n=

einθ_(λ)

n

n! F

–n,λ+ix, λ;  –ei(ψ–θ)zn.

(d) Inserting(x+i)and(x–i)instead ofxinto the generating function(.), we find that

y(x+i) =

n–

k=

Pλ_k(x;θ,ψ)ei(n–k)θ–ei[(n–k–)θ+ψ]+Pλ_n,

y(x–i) =

n–

k=

Pλ_k(x;θ,ψ)ei(n–k)ψ–ei[(n–k–)ψ+θ]+Pλ_n,

which implies that

eiθ_{(λ–ix)y(x+i) –e}iψ_{(λ+ix)y(x–i)}

=eiθ–eiψ

n–

k=

P_kλ(x;θ,ψ) (λ–ix)ei(n–k)θ+ (λ+ix)ei(n–k)ψ

+ eiθ(λ–ix) –eiψ(λ+ix)Pλ_n. (.)

Differentiation of the generating function(.)with respect tozand equating the leading coefficient ofzn–yields

nPλ_n(x;θ,ψ) =

n–

k=

Pλ_k(x;θ,ψ)(λ–ix)ei(n–k)θ+ (λ+ix)ei(n–k)ψ,

which together with(.)gives(.).

Theorem  The polynomials Pλ

n(x;θ,ψ)are orthogonal on (–∞, +∞) with the weight

wλ

θ,ψ(x) =πe(

θ–ψ+π)x_|(λ+ix)|_{,forλ> ,θ∈(–}π

,

π

),and

+∞

–∞

wλ θ,ψ(x)P

λ

n(x;θ,ψ)Pmλ(x;θ,ψ)dx=δnm

(n+ λ) n!(cos(θ–ψ+π)/)λ.

Proof LetF(s) andG(s) be the Mellin transforms off(x) andg(x),i.e.,

{Mf}(s) =F(s) =

∞



f(x)xs–dx, {Mg}(s) =G(s) =

∞



g(x)xs–dx.

Then the following formula (Parseval’s identity) holds []:

 πi

c+i∞

c–i∞ F(s)G( –s)ds=

_∞



f(x)g(x)dx (.)

and [] +∞



uα–e–pue–iqudu=(α)p+q–α_e–iαarctan(p/q)_{. (.)}

Forf(x) = x(λ+j)_e–x _{andg(x) = x}(λ+k)–_e–x_{, we haveF(s) =(λ+j+}s

),G(s) =(λ+k+

s–

 ). By the well-known property

×

By the formula (.), the above reduces to



From this and relation (.), it follows that

+∞

Proof Consider the following:

∞

Proposition  The family of generalized Meixner-Pollaczek polynomials Pλ

n(x;θ,ψ)can

be extended to the caseλ= as follows:

P_(x;θ,ψ) = ,

nP_n(x;θ,ψ) =

eiθ_–eiψ i

xP_n–(x;θ,ψ), n≥. (.)

Proof Since

lim λ→P

λ

n(x;θ,ψ) =_λ→lim

(λ)ne

inθ

F

–n,λ+ix, λ;  –e

iψ eiθ

= 

n!e

inθ

 –ei(ψ–θ)(n)(–n)(ix)F

–n+ ,ix+ , ;  –ei(ψ–θ)

= 

n!e

inθ

 –ei(ψ–θ)(n)(–n)(ix)P_n–(x;θ,ψ)n–  ()n–

e–i(n–)θ

=

n

eiθ_–eiψ i

xP_n–(x;θ,ψ),

then (.) is a natural consequence.

4 The case

ψ

=

π

+

θ

Let us consider now the caseψ=π+θ. We observe that such a case leads to the very

interesting family of symmetric polynomials. Some special cases ofPλ

n(x;θ,π+θ;z) are

known in the literature forθ =π_. These are the symmetric Meixner-Pollaczek

polyno-mials, denoted byPλ

n(x/;θ),λ> . For instance, Benderet al.[] and Koornwinder []

have shown that there is a connection between the symmetric Meixner-Pollaczek

polyno-mialsP

 

n(x_,π_) and the Heisenberg algebra. Another example is [], where the symmetric

Meixner-Pollaczek polynomials are considered.

We define the symmetric generalized Meixner-Pollaczek (SGMP) polynomialsSλ

n(x;θ)

by the following generating function:

Gλ(x;θ,π+θ;z) = 

( –zeiθ₎λ–ix_{( +ze}iθ₎λ+ix

=e

–xarctan(zei(θ+π/))

( –z_eiθ₎λ =

∞

n=

Sλ_n(x;θ)zn, z∈D.

This sequence of polynomials has a hypergeometric representation

Sλ

n(x;θ) =einθ

(λ)n

n! F(–n,λ+ix, λ; ), (.)

and an integral representation

Sλ_n(x;θ) =  πi

e–xarctan(zei(θ+π/))

In this section we mainly consider the stripS={z∈C:|z|< }. There are several rea-sons why the strip is of special interest. Letw(x) = 

cosh(πx/). The functionw(x) is a density

function of a probability measure onS. We describe an orthogonal basis for the basis in the Hilbert spaceH(S,P), whereP is the Poisson measure for . The inner product for any two functionsf,g∈H_{(S,P) is given by the formula}

(f,g)_H_(S,P):=

R

f(x+i)g(x+i) +f(x–i)g(x–i)

coshπ_x dx.

Now, we consider the system{σn(x)}given by the recursion relation

σ–= ,σ= , (n+ )σn+(z) +ieiθzσn(z) –eiθ(n– )σn–(z) = . (.)

Theorem  Let the system{σn(x)}∞n=be given by(.),then:

(a) the system satisfies

Gσ(z,s) = ∞

k=

σk(z)sk=e–zarctanse

i(θ+π_{ )}

,

(b) the sequence of polynomials{σn}∞ is an orthogonal basis in the Hilbert space

H_(S,P),

(c) the norm of polynomialsσnis

√

ifk≥andifk= .

Proof

(a) By(.)we have

(k+ )σk+(z) +ieiθzσk(z) –eiθ(k– )σk–(z) = .

Multiplying the above relations bysk_{, summing overkand simplifying, we obtain}

 = ∞

k=

(k+ )σk+(z) +ieiθzσk(z) –eiθ(k– )σk–(z)

sk

= ∞

k=

(k+ )σk+(z)sk+ieiθz

∞

k=

σk(z)sk–eiθ

∞

k=

(k– )σk–(z)sk

=∂Gσ(z,s)

∂s +ie

iθ_{zGσ(z,s) –e}iθ_s∂Gσ(z,s) ∂s .

This implies that

 –eiθs∂Gσ(z,s)

∂s = –ie

iθ

zGσ(z,s),

which in turn implies

∂Gσ(z,s)

∂s = –ieiθ_z

Integrating both sides with respect toswith the conditionGσ(, ) = , we obtain

(b) In order to prove the orthogonality ofσn(x)polynomials and compute their norms,

it suffices to show that

Comparing the coefficients of the powers ofsandt, we obtain the desired result.

Remark  Applying Cauchy’s integral formula to the generating function of the system, one obtains the integral representation

σn(x) =

around a closed contourKabout the origin with radius less than .

Remark  Lety(x) =σn(x). The functiony(x) satisfies the following difference equation:

y(x+i) –y(x–i)

i =

ny(x)

Proposition  The system{σn}satisfies the following relation:

Remark  From Proposition  we get

σn(x) =

We define quasi-symmetric Meixner-Pollaczek (QMP) polynomialsQλ

n(x;θ) by the

gener-(a) The QMP polynomialsQλ

n=Qλn(x;θ)satisfy the three-term recurrence relation

Qλ_–= ,

Qλ_= ,

nQλ_n= i (λ+n– )sinθ–xcosθQλ_n–+ (λ+n– )Qλ_n–, n≥.

(b) The polynomialsQλ

n=Qλn(x;θ)are given by the formula

(c) The polynomialsQλ

n=Qλn(x;θ)have the hypergeometric representation

n(x;θ)satisfy the following difference equation:

(e) The polynomialsQλ

n(x;θ)are orthogonal on(–∞, +∞)with the weight

wλ θ(x) =

 πe

θx_(λ+ix) forλ> andθ∈(–π

,

π

)and

 π

+∞

–∞ e

θx_(λ+ix)_Qλ

n(x;θ)Qλm(x;θ)dx=δmn

(n+ λ)

(cosθ)λ_n!. (.)

The Fisher informationIθ(μ) of a random variableXwith distributionμ(x;θ), whereθ

is a continuous parameter, is defined by []

Iθ(μ) =E

∂ ∂xln(μ)



.

It is named after RA Fisher who invented the concept of maximum likelihood estimator and discovered several of its properties. Over the years, the concept of Fisher information has found many application in physics [], biology [], engineering,etc.In [] Dominici considered a sequencePn(x) of orthogonal polynomials with respect to the weight

func-tionρ(x) satisfying

∞

x=

Pn(x)Pm(x)ρ(x) =hnδnm, n,m= , , . . . .

Introducing the functions

ρn(x) =

[Pn(x)]ρ(x)

hn

, n∈N, (.)

the Fisher information corresponding to the functions (.) may be described as follows:

Iθ(Pn) =

∞

x=

∂ ∂θρn(x)





ρn(x)

, n∈N.

For the familyPn(x) of polynomials defined by

Pn(x) =F –n, –x,c;z(θ)

, n∈N,

in [], it was computed that

∂Pn

∂θ =

n z

∂z

∂θ Pn(x) –Pn–(x)

, n∈N. (.)

In this work we use the ideas of [] to compute the Fisher information of QMP polyno-mials.

Theorem  The Fisher information of QMP polynomials is given by

IθQλ_n=

∞

–∞

∂ ∂θρn(x)





ρn(x)

dx=–[n

_{+ (n+ )λ]}

cos_θ , n∈N,

Proof For GMP we haveρ(x) =wλ

θ(x) =πe

θx_|(λ+ix)|_.

From (.) and (.), we have

∂Qλ

n

∂θ = –ntan(θ)Q

λ

n+i

λ+n– 

cosθ Q

λ

n–,

while (.) and (.) give

ρn(x) =

eθx_|(λ+ix)|_(cosθ)λ_n![Q

n(x)]

π (n+ λ) . (.)

Note that

∞

–∞ρn(x)dx= , n∈N. (.)

Differentiating (.) with respect toθ, we obtain

∂ρn(x)

∂θ =

iρn(x)

cosθQλ

n

(n+ )Qλ_n+– (λ+n– )Qλ_n–.

Therefore

∂ ∂θρn(x)





ρn(x)

= –ρn(x) cos_θ(Qλ

n)

× (n+ )Qλ

n+



– (n+ )(λ+n– )Qλ

n+Qλn–+ (λ+n– )

Qλ

n–



= 

cos_θ

(n+ )(n+ λ)ρn+(x) +n(n+ λ– )ρn–

– (n+ )(λ+n– ) (cosθ)

λ_n!

(n+ λ)ρ(x)Qλ

n+Qλn–

. (.)

Integrating (.) and using the orthogonality relation (.), and (.), we get

IθQλ_n=

_∞

–∞

∂ ∂θρn(x)





ρn(x)

dx= –

cos_θ (n+ )(n+ λ) +n(n+ λ– )

and the result follows.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Rzeszow University of Technology, Rzeszow, Poland.}2_{Department of Mathematics, Maria}

Curie-Sklodowska University in Lublin, Lublin, Poland.

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doi:10.1186/1687-1847-2013-131