Generating functions of Pollaczek polynomials: a revisit
01 Jul 2019-Integral Transforms and Special Functions (Taylor & Francis)-Vol. 30, Iss: 11, pp 893-919
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. ...
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01 Jan 2016
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TL;DR: In this paper, two concise inequalities for the associated Pollaczek polynomials are proved for the orthogonal expansions with positive coefficients, using Askey's theorem and triple integral representation.
Abstract: In this paper, we prove two concise inequalities for the associated Pollaczek polynomials. The first one is obtained by using Askey’s theorem (SIAM J. Math. Anal. 2:340–346, 1971) on orthogonal expansions with positive coefficients. The second one is proved by using a triple integral representation due to the authors (Integral Transforms Spec. Funct. 30:893–919, 2019). In the concluding section, we briefly point out some useful variations and known cases of our inequalities.
2 citations
TL;DR: In this article, a new form of a generating function for the c-associated Meixner-Pollaczek polynomials was derived, which is of particular importance since its several r...
Abstract: In the present paper we first derive a new form of a generating function for the c-associated Meixner-Pollaczek polynomials. This generating function is of particular importance since its several r...
2 citations
15 Mar 2022
TL;DR: In this article , the spherical clothoid is revisited as a nonlinear spline primitive for 3-space and its Cartesian coordinate functions using confluent hypergeometric functions (the Kummer functions) and its stereographic projection onto the complex plane.
Abstract: We revisit a nonlinear spline primitive for 3-space first studied by Even Mehlum. It is the spherical clothoid, the spherical curve with geodesic curvature a linear function of arc length. We present its Cartesian coordinate functions using confluent hypergeometric functions (the Kummer functions) and its stereographic projection onto the complex plane. New Humbert series results are also presented along with generating function formulas related to the associated Meixner-Pollaczek polynomials.
05 Jun 2023
TL;DR: This article gave new explicit representations for the associated Meixner, Charlier, Laguerre, and Krawtchouk polynomials and derived new generating functions and convolution-type formulas of the corresponding classical polynomial formulas.
Abstract: We give new explicit representations as well as new generating functions for the associated Meixner, Charlier, Laguerre, and Krawtchouk polynomials. The obtained results are then used to derive new generating functions and convolution-type formulas of the corresponding classical polynomials. Some consequences of our results are also mentioned.
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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.)
4,428 citations
Book•
01 May 2010
TL;DR: This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators and is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun.
Abstract: Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. These functions appear whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are performed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requires practitioners to have ready access to a reliable collection of their properties. This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators. Printed in full color, it is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun. Included with every copy of the book is a CD with a searchable PDF of each chapter.
3,646 citations
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22 Oct 2010
TL;DR: In this paper, Orthogonal Polynomial Solutions of Differential Equations of Real Difference Equations (DDEs) were used to solve Eigenvalue Problems. But they were not used in the context of orthogonal polynomials.
Abstract: Definitions and Miscellaneous Formulas- Classical orthogonal polynomials- Orthogonal Polynomial Solutions of Differential Equations- Orthogonal Polynomial Solutions of Real Difference Equations- Orthogonal Polynomial Solutions of Complex Difference Equations- Orthogonal Polynomial Solutions in x(x+u) of Real Difference Equations- Orthogonal Polynomial Solutions in z(z+u) of Complex Difference Equations- Hypergeometric Orthogonal Polynomials- Polynomial Solutions of Eigenvalue Problems- Classical q-orthogonal polynomials- Orthogonal Polynomial Solutions of q-Difference Equations- Orthogonal Polynomial Solutions in q?x of q-Difference Equations- Orthogonal Polynomial Solutions in q?x+uqx of Real
1,288 citations
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01 Jan 1985
1,030 citations
"Generating functions of Pollaczek p..." refers background or methods in this paper
...The technique presented in [3] can be used to determine its region of convergence as has been applied above in Section 3....
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...It follows therefore from [3] that the triple hypergeometric series is convergent provided that |t| < 1 and |zt| < 1....
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...For more details about this function, the interested reader may refer to [3,20,21]....
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