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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TL;DR: In this paper, the Meixner-Pollaczek polynomials were shown to recover pair of the known recurrence relations for the generalized Laguerre polynomial.
Abstract: We report on existence of pair of new recurrence relations (or difference equations) for the Meixner-Pollaczek polynomials. Proof of the correctness of these difference equations is also presented. Next, we found that subtraction of the forward shift operator for the Meixner-Pollaczek polynomials from one of these recurrence relations leads to the difference equation for the Meixner-Pollaczek polynomials generated via $\cosh$ difference differentiation operator. Then, we show that, under the limit $\varphi \to 0$, new recurrence relations for the Meixner-Pollaczek polynomials recover pair of the known recurrence relations for the generalized Laguerre polynomials. At the end, we introduced differentiation formula, which expresses Meixner-Pollaczek polynomials with parameters $\lambda>0$ and $0 \lt \varphi \lt \pi$ via generalized Laguerre polynomials.
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TL;DR: In this article, the generalization of typically real functions in the unit disk was studied and the Zalcman conjecture for generalized typically real function was shown to hold for generalized real functions.
Abstract: Abstract In this paper, we are mainly interested to study the generalization of typically real functions in the unit disk. We study some coefficient inequalities concerning this class of functions. In particular, we find the Zalcman conjecture for generalized typically real functions.
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TL;DR: Higher Transcendental Functions Based on notes left by the late Prof. Harry Bateman, and compiled by the Staff of the Bateman Project as discussed by the authors, are presented in Table 1.
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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TL;DR: The Askey-scheme of hypergeometric orthogonal polynomials was introduced in this paper, where the q-analogues of the polynomial classes in the Askey scheme are given.
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.
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