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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

AbstractWe 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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01 Jan 1988
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
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
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


Journal ArticleDOI
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. ...

4 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) ....

    [...]

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

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  • ...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) ....

    [...]


References
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Abstract: Higher Transcendental Functions Based, in part, on notes left by the late Prof. Harry Bateman, and compiled by the Staff of the Bateman Project. Vol. 1. Pp. xxvi + 302. 52s. Vol. 2. Pp. xvii + 396. 60s. (London: McGraw-Hill Publishing Company, Ltd., 1953.)

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Posted Content
Abstract: We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

1,426 citations