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. □
- Robertson MS: On the theory of univalent functions. Ann. Math. 1936, 37: 374–408. 10.2307/1968451View ArticleGoogle Scholar
- Pommerenke C: Linear-invariant Familien analytischer Funktionen. Math. Ann. 1964, 155: 108–154. 10.1007/BF01344077MathSciNetView ArticleGoogle Scholar
- Yamashita S: Nonunivalent generalized Koebe function. Proc. Jpn. Acad., Ser. A, Math. Sci. 2003, 79(1):9–10. 10.3792/pjaa.79.9View ArticleGoogle Scholar
- Hille E: Remarks on a paper by Zeev Nehari. Bull. Am. Math. Soc. 1949, 55: 552–553. 10.1090/S0002-9904-1949-09243-1MathSciNetView ArticleGoogle Scholar
- Campbell DM, Pfaltzgraff JA:Mapping properties of . Colloq. Math. 1974, 32: 267–276.MathSciNetGoogle Scholar
- Koekoek, R, Swarttouw, RF: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Report 98–17, Delft University of Technology (1998)Google Scholar
- Naraniecka I, Szynal J, Tatarczak A: The generalized Koebe function. Tr. Petrozavodsk. Gos. Univ. Ser. Mat. 2010, 17: 62–66.MathSciNetGoogle Scholar
- Chihara TS: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York; 1978.Google Scholar
- Meixner J: Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion. J. Lond. Math. Soc. 1934, 9: 6–13.MathSciNetView ArticleGoogle Scholar
- Pollaczak F: Sur une famille de polynomes orthogonaux qui contient les polynomes d’Hermite et de Laguerre comme cas limites. C. R. Acad. Sci. Paris 1950, 230: 1563–1565.MathSciNetGoogle Scholar
- Askey R, Wilson J: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 1985., 54: Article ID 319Google Scholar
- Erdélyi A, et al. I. In Higher Transcendental Functions. McGraw-Hill, New York; 1953. Bateman Manuscript Project.Google Scholar
- Rahman M: A generalization of Gasper’s kernel for Hahn polynomials: application to Pollaczek polynomials. Can. J. Math. 1978, 30(1):133–146. 10.4153/CJM-1978-011-7View ArticleGoogle Scholar
- Atakishiyev NM, Suslov SK: The Hahn and Meixner polynomials of an imaginary argument and some of their applications. J. Phys. A, Math. Gen. 1985, 18: 1583–1596. 10.1088/0305-4470/18/10/014MathSciNetView ArticleGoogle Scholar
- Bender CM, Mead LR, Pinsky S: Continuous Hahn polynomials and the Heisenberg algebra. J. Math. Phys. 1987, 28(3):509–513. 10.1063/1.527635MathSciNetView ArticleGoogle Scholar
- Koornwinder TH: Meixner-Pollaczek polynomials and the Heisenberg algebra. J. Math. Phys. 1989, 30(4):767–769. 10.1063/1.528394MathSciNetView ArticleGoogle Scholar
- Li X, Wong R: On the asymptotics of the Meixner-Pollaczek polynomials and their zeros. Constr. Approx. 2001, 17: 59–90.MathSciNetView ArticleGoogle Scholar
- Poularikas AD: The Handbook of Formulas and Tables for Signal Processing. CRC Press, Boca Raton; 1999.Google Scholar
- Araaya, TK: The symmetric Meixner-Pollaczek polynomials. Uppsala Dissertations in Mathematics, Department of Mathematics, Uppsala University (2003)Google Scholar
- Fisher RA: Statistical Methods and Scientific Inference. Hafner Press, New York; 1973.Google Scholar
- Friden BR: Science from Fisher Information. Cambridge University Press, Cambridge; 2004.View ArticleGoogle Scholar
- Zheng G, Gastwirth JL: Fisher information in randomly sampled sib pairs and extremely discordant sib pairs in genetic analysis for a quantitative trait locus. J. Stat. Plan. Inference 2005, 130(1–2):299–315. 10.1016/j.jspi.2003.08.019MathSciNetView ArticleGoogle Scholar
- Dominici D: Fisher information of orthogonal polynomials. J. Comput. Appl. Math. 2010, 233(6):1511–1518. 10.1016/j.cam.2009.02.066MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.