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Generalized Meixner-Pollaczek polynomials
Advances in Difference Equations volume 2013, Article number: 131 (2013)
We consider the generalized Meixner-Pollaczek (GMP) polynomials of a variable and parameters , , , defined via the generating function
We find the three-term recurrence relation, the explicité formula, the hypergeometric representation, the difference equation and the orthogonality relation for GMP polynomials . Moreover, we study the special case of corresponding to the choice and , which leads to some interesting families of polynomials. The limiting case () of the sequences of polynomials is obtained, and the orthogonality relation in the strip is shown.
MSC:33C45, 30C10, 30C45, 39A60.
The classical Koebe function is a function holomorphic in and given by the formula
The importance of follows from the extremality for the famous Bieberbach conjecture. The Koebe function is univalent and starlike in and maps the unit disk onto the complex plane minus a slit .
Several generalizations of appeared in the literature. Robertson  proved that () is the extremal function for the class of functions starlike of order α. The function
was extensively studied by Pommerenke , who investigated a universal invariant family .
The definition of was extended for a nonzero complex number α by Yamashita . The classical result of Hille  ascertains that is univalent in if and only if is in the union A of the closed disks and . Making use of geometric properties, Yamashita  described how tends to be univalent in the whole as α tends to each boundary point of A from outside.
The properties of , where
An evident and important extension of (1.1) was given by the following formulas (, ):
and for the case when ,
Comparing with the generating function for Meixner-Pollaczek polynomials ,
where , , , we were motivated to introduce the generalized Meixner-Pollaczek polynomials (GMP)  of a variable and parameters , , via the generating function
Obviously, we have .
2 Orthogonal polynomials
Let denote the moment functional that is a linear map . A sequence of polynomials is an orthogonal polynomials sequence (OPS) with respect to if has degree n, for and for all n.
In this paper we consider orthogonal polynomial systems defined recursively. Every monic OPS may be described by a recurrence formula of the form
where , , the numbers and are constants, for and is arbitrary (see [, Ch. I, Theorem 4.1]). The sequences of orthogonal polynomials are symmetric if for all n (see [, Ch. I, Theorem 4.3]) or that in (2.1) 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 considered independently by Pollaczek . These polynomials are classified in the Askey-scheme of orthogonal polynomials [6, 11].
Some of the main properties of these polynomials are presented in Erdélyi et al. , Chihara , Askey and Wilson  and in the report by Koekoek and Swarttouw . Detailed analyses with applications of these polynomials are also made by several authors. Among others, the works of Rahman , Atakishiyev and Suslov , Bender et al. , Koornwinder  and the extensive work of Li and Wong  may be included.
This paper is mainly concerned about the generalized Meixner-Pollaczek (GMP) polynomials. We also study the special cases of , corresponding to the choice and , which lead to some interesting families of polynomials.
For complex numbers a, b and c (), the Gaussian hypergeometric function is defined by
where is the Pochhammer symbol described by
Notice that is symmetric in a and b, and the series terminates if either a or b is zero or a negative integer. In general, the series is absolutely convergent in . If , it is also convergent on , and it is known that
3 Generalized Meixner-Pollaczek polynomials
In this section we find the three-term recurrence relation, the explicité formula, the hypergeometric representation, the difference equation and the orthogonality relation for (GMP) polynomials .
Theorem 1 Let us set . The polynomials have the following properties:
satisfy the three-term recurrence relation
are given by the formula(3.1)
have the hypergeometric representation(3.2)
Let . The function satisfies the following difference equation:(3.3)
We differentiate the formula (1.2) with respect to z, and after multiplication by , we compare the leading coefficients of .
The Cauchy product of the power series
Applying the formula from [, vol.1, p.82],
with , , , , one obtains
Comparing the coefficients of the power series, we get (3.2).
Inserting and instead of x into the generating function (1.2), we find that
which implies that
Differentiation of the generating function (1.2) with respect to z and equating the leading coefficient of yields
which together with (3.4) gives (3.3).
Theorem 2 The polynomials are orthogonal on with the weight , for , , and
Proof Let and be the Mellin transforms of and , i.e.,
Then the following formula (Parseval’s identity) holds :
For and , we have , . By the well-known property
Consecutively, applying first the formula () and (3.5), and then setting , , in (3.6), we have
Using (3.7) and (2.2), we obtain
By the formula (2.3), the above reduces to
Since for , then (3.8) is nonzero only for the case . Then
From this and relation (3.7), it follows that
Remark 1 For , , and , the following explicité formula holds:
Proof Consider the following:
Comparing both sides of the above, we get the equality (3.9). □
Proposition 1 The family of generalized Meixner-Pollaczek polynomials can be extended to the case as follows:
then (3.10) is a natural consequence. □
4 The case
Let us consider now the case . We observe that such a case leads to the very interesting family of symmetric polynomials. Some special cases of are known in the literature for . These are the symmetric Meixner-Pollaczek polynomials, denoted by , . For instance, Bender et al.  and Koornwinder  have shown that there is a connection between the symmetric Meixner-Pollaczek polynomials and the Heisenberg algebra. Another example is , where the symmetric Meixner-Pollaczek polynomials are considered.
We define the symmetric generalized Meixner-Pollaczek (SGMP) polynomials by the following generating function:
This sequence of polynomials has a hypergeometric representation
and an integral representation
In this section we mainly consider the strip . There are several reasons why the strip is of special interest. Let . The function is a density function of a probability measure on . We describe an orthogonal basis for the basis in the Hilbert space , where is the Poisson measure for 0. The inner product for any two functions is given by the formula
Now, we consider the system given by the recursion relation
Theorem 3 Let the system be given by (4.2), then:
the system satisfies
the sequence of polynomials is an orthogonal basis in the Hilbert space ,
the norm of polynomials is if and 1 if .
By (4.2) we have
Multiplying the above relations by , summing over k and simplifying, we obtain
This implies that
which in turn implies
Integrating both sides with respect to s with the condition , we obtain
In order to prove the orthogonality of polynomials and compute their norms, it suffices to show that(4.3)
To this end, let take , and the formula . Then
In the light of (a) and equation (4.3), we have
Comparing the coefficients of the powers of s and , we obtain the desired result. □
Remark 2 Applying Cauchy’s integral formula to the generating function of the system, one obtains the integral representation
around a closed contour K about the origin with radius less than 1.
Remark 3 Let . The function satisfies the following difference equation:
Proposition 2 The system satisfies the following relation:
Proof By (4.1) and by the definition of , we have
Remark 4 From Proposition 1 we get
5 The case
We define quasi-symmetric Meixner-Pollaczek (QMP) polynomials by the generating function
The QMP polynomials satisfy the three-term recurrence relation
The polynomials are given by the formula
The polynomials have the hypergeometric representation(5.1)
The polynomials satisfy the following difference equation:
The polynomials are orthogonal on with the weight
for and and
The Fisher information of a random variable X with distribution , where θ is a continuous parameter, is defined by 
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 sequence of orthogonal polynomials with respect to the weight function satisfying
Introducing the functions
the Fisher information corresponding to the functions (5.3) may be described as follows:
For the family of polynomials defined by
in , it was computed that
In this work we use the ideas of  to compute the Fisher information of QMP polynomials.
Theorem 4 The Fisher information of QMP polynomials is given by
with defined as in (5.3).
Proof For GMP we have .
From (5.4) and (5.1), we have
while (5.3) and (5.2) give
Differentiating (5.5) with respect to θ, we obtain
Integrating (5.7) and using the orthogonality relation (5.2), and (5.6), we get
and the result follows. □
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The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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Kanas, S., Tatarczak, A. Generalized Meixner-Pollaczek polynomials. Adv Differ Equ 2013, 131 (2013). https://doi.org/10.1186/1687-1847-2013-131
- Meixner-Pollaczek polynomials
- difference equation
- generating function
- orthogonal polynomials
- Fisher information