- Open Access
Generalized Meixner-Pollaczek polynomials
© Kanas and Tatarczak; licensee Springer 2013
Received: 4 February 2013
Accepted: 22 April 2013
Published: 7 May 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 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 .
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.
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.
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.
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 .
- (a)satisfy the three-term recurrence relation
- (b)are given by the formula(3.1)
- (c)have the hypergeometric representation(3.2)
- (d)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 .
- (b)The Cauchy product of the power series
- (c)Applying the formula from [, vol.1, p.82],
- (d)Inserting and instead of x into the generating function (1.2), we find that
which together with (3.4) gives (3.3).
Comparing both sides of the above, we get the equality (3.9). □
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.
- (a)the system satisfies
the sequence of polynomials is an orthogonal basis in the Hilbert space ,
the norm of polynomials is if and 1 if .
- (a)By (4.2) we have
- (b)In order to prove the orthogonality of polynomials and compute their norms, it suffices to show that(4.3)
- (c)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. □
around a closed contour K about the origin with radius less than 1.
5 The case
- (a)The QMP polynomials satisfy the three-term recurrence relation
- (b)The polynomials are given by the formula
- (c)The polynomials have the hypergeometric representation(5.1)
- (d)The polynomials satisfy the following difference equation:
- (e)The polynomials are orthogonal on with the weight
In this work we use the ideas of  to compute the Fisher information of QMP polynomials.
with defined as in (5.3).
Proof For GMP we have .
and the result follows. □
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