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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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
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
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....
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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) ....
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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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01 Jan 2003
TL;DR: In this paper, the authors considered the problem of finding orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard's three term recurrence relation condition.
Abstract: The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(λ)(x) instead of the standard notation pn(λ) (x/2, π/2), where λ > 0. The limiting case of these sequences of polynomials pn(0) (x) =limλ→0 pn(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}.From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn(0) (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn(0)(x) is obtained, and is foundto be ℙ = {{pn(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.The polynomials {pn(λ) (x)}∞n=0, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn(λ)(x) is orthogonal. Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.
7 citations
01 Jan 2003
TL;DR: In this article, the authors make use of a geometric quantity to describe how f α continuously tends to be univalent in the whole U as a tends to each boundary point of A from outside.
Abstract: The function f α (z) = ({(1 + z)/(1 - z)} α - 1)/(2α) with a complex constant α ¬= 0 is not univalent in the disk U = {|z| < 1} if and only if a is not in the union A of the closed disks {|z + 1| ≤1} and {|z - 1| ≤1}. By making use of a geometric quantity we can describe how f α continuously tends to be univalent in the whole U as a tends to each boundary point of A from outside.
7 citations
6 citations
TL;DR: In this article, the Fisher information of the Meixner-Pollaczek polynomials has been computed for the first time, and the results are shown to be similar to those of Dehesa and his collaborators.
4 citations
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TL;DR: The Fisher information of the Meixner-Pollaczek, Mexner, Krawtchouk and Charlier polynomials is computed.
Abstract: Following the lead of J. Dehesa and his collaborators, we compute the Fisher information of the Meixner-Pollaczek, Meixner, Krawtchouk and Charlier polynomials.
4 citations