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Journal ArticleDOI

Generalized Meixner-Pollaczek polynomials

07 May 2013-Advances in Difference Equations (Springer International Publishing)-Vol. 2013, Iss: 1, pp 131
TL;DR: In this paper, the generalized Meixner-Pollaczek polynomials P λ (x; θ, ψ )o f a variable x ∈ R and parameters λ > 0,θ ∈ (0, π ), ψ ∈ r,d ef ined via the generating function
Abstract: We consider the generalized Meixner-Pollaczek (GMP) polynomials P λ (x; θ , ψ )o f a variable x ∈ R and parameters λ >0 ,θ ∈ (0, π ), ψ ∈ R ,d ef inedvia the generating function

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Citations
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01 Jan 1988
TL;DR: In this paper, an alternative proof is given for the connection between a system of continuous Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56, 2445 (1986); J. Math. 28, 509 (1987)].
Abstract: An alternative proof is given for the connection between a system of continuous Hahn polynomials and identities for symmetric elements in the Heisenberg algebra, which was first observed by Bender, Mead, and Pinsky [Phys. Rev. Lett. 56, 2445 (1986); J. Math. Phys. 28, 509 (1987)]. The continuous Hahn polynomials turn out to be Meixner–Pollaczek polynomials. Use is made of the connection between Laguerre polynomials and Meixner–Pollaczek polynomials, the Rodrigues formula for Laguerre polynomials, an operational formula involving Meixner–Pollaczek polynomials, and the Schrodinger model for the irreducible unitary representations of the three‐dimensional Heisenberg group.

34 citations

Journal ArticleDOI
TL;DR: In this article, the main purpose of a paper is a solution of coefficients problems in. Problem related to the well-known Zalcman conjecture is presented, where the generalized Chebyshev polynomials of the second kind are defined as follows.
Abstract: For let denote the class of generalized typically real functions i.e. the class of functions of a formwhere , and is the unique probability measure on the interval . For the same range of parameters, let the generalized Chebyshev polynomials of the second kind be defined as followsWe see thatThe main purpose of a paper is a solution of coefficients problems in . Problem related to the well-known Zalcman conjecture is presented.

6 citations

Journal ArticleDOI
TL;DR: In this paper, a new proof of Wimp's formula for the associated Pollaczek polynomials Pnλz;a,b,c,d,e,f,c was presented.
Abstract: This paper is mainly devoted to generating functions of Pollaczek and other related polynomials. We first present a new proof of Wimp's formula for the associated Pollaczek polynomials Pnλz;a,b,c. ...

6 citations


Additional excerpts

  • ...1: Kanas and Tatarczak [28] studied the generalized Meixner-Pollaczek polynomials Qn(x; θ ,ψ) generated by ∞ ∑ n=0 Qλn (x; θ ,ψ) t n = (1 − t eiθ )−λ+ix (1 − t eiψ)−λ−ix (|t| < 1; λ > 0, θ ∈ (0,π) , ψ ∈ R) ....

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  • ...[28] Kanas S, Tatarczak A. Generalized Meixner-Pollaczek polynomials....

    [...]

  • ...Remark 5.1: Kanas and Tatarczak [28] studied the generalized Meixner-Pollaczek polynomials Qλn(x; θ ,ψ) generated by ∞∑ n=0 Qλn (x; θ ,ψ) t n = (1 − t eiθ )−λ+ix (1 − t eiψ)−λ−ix (|t| < 1; λ > 0, θ ∈ (0,π) , ψ ∈ R) ....

    [...]

Journal ArticleDOI
TL;DR: In this article, the first associated Meixner-Pollaczek polynomials arising from nonlinear coherent states with anti-holomorphic coefficients were identified as orthogonal polynomial arising from coherent states.
Abstract: While considering nonlinear coherent states with anti-holomorphic coefficients z¯n/xn!, we identify as first-associated Meixner–Pollaczek polynomials the orthogonal polynomials arising from...

4 citations

References
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Book
01 Oct 1998
TL;DR: This book discusses Fourier Series, as well as special functions Asymptotic Expansions, Signals and Their Representation, and Recursive Filters Satisfying Prescribed Specifications.
Abstract: Signal processing is a broad and timeless area. The term "signal" includes audio, video, speech, image, communication, geophysical, sonar, radar, medical, and more. Signal processing applies to the theory and application of filtering, coding, transmitting, estimating, detecting, analyzing, recognizing, synthesizing, recording, and reproducing signals. Handbook of Formulas and Tables for Signal Processing a must-have reference for all engineering professionals involved in signal and image processing. Collecting the most useful formulas and tables - such as integral tables, formulas of algebra, formulas of trigonometry - the text includes:Material for the deterministic and statistical signal processing areasExamples explaining the use of the given formulaNumerous definitionsMany figures that have been added to special chaptersHandbook of Formulas and Tables for Signal Processing brings together - in one textbook - all the equations necessary for signal and image processing for professionals transforming anything from a physical to a manipulated form, creating a new standard for any person starting a future in the broad, extensive area of research.

295 citations

Journal ArticleDOI
TL;DR: In this article, the Hahn and Meixner polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter, leading to the origination of two systems of real polynomial systems orthogonal with respect to a continuous measure.
Abstract: The Hahn and Meixner polynomials belonging to the classical orthogonal polynomials of a discrete variable are analytically continued in the complex plane both in variable and parameter. This leads to the origination of two systems of real polynomials orthogonal with respect to a continuous measure. The Meixner polynomials of an imaginary argument obtained in this manner turned out to be known in the literature as the Pollaczek polynomials. The orthogonality relation for the Hahn polynomials with respect to a continuous measure is apparently new. A close connection between the Hahn polynomials of an imaginary argument and representations of the Lorentz group SO(3,1) is considered.

100 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that 2 is the best possible constant in (2) in the following sense: for every C>2 there exists a function f (z) such that f or \\z\\ < 1, we have (i) f(z) is holoniorphic, (ii) f{z) takes on the value one infinitely often, and (iii) | {f(z), z) | ^ C[\\ − | z\\ 2 ] ~ 2 with equality f or real values of z.
Abstract: then f {z) is univalent f or \\z\\ < 1 . The object of the present note is to show that 2 is the best possible constant in (2) in the following sense: For every C>2 there exists a function f (z) such that f or \\z\\ < 1 we have (i) f(z) is holoniorphic, (ii) f{z) takes on the value one infinitely often, and (iii) | {f(z), z) | ^ C[\\ — | z\\ 2 ] ~ 2 with equality f or real values of z. An explicit example of such a function is given by

75 citations