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 Ufimsk. Mat. Zh., 2018, Volume 10, Issue 2, Pages 118–126 (Mi ufa426)

Certain generating functions of Hermite–Bernoulli–Legendre polynomials

N. U. Khan, T. Usman

Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India

Abstract: The special polynomials of more than one variable provide new means of analysis for the solutions of a wide class of partial differential equations often encountered in physical problems. Most of the special function of mathematical physics and their generalization have been suggested by physical problems. It turns out very often that the solution of a given problem in physics or applied mathematics requires the evaluation of an infinite sum involving special functions. Problems of this type arise, e.g., in the computation of the higher-order moments of a distribution or while calculating transition matrix elements in quantum mechanics. Motivated by their importance and potential for applications in a variety of research fields, recently, numerous polynomials and their extensions have been introduced and studied. In this paper, we introduce a new class of generating functions for Hermite-Bernoulli-Legendre polynomials and study certain implicit summation formulas by using different analytical means and applying generating function. We also introduce bilateral series associated with a newly-introduced generating function by appropriately specializing a number of known or new partly unilateral and partly bilateral generating functions. The results presented here, being very general, are pointed out to be specialized to yield a number of known and new identities involving relatively simpler and familiar polynomials.

Keywords: 2-variable Hermite polynomials, Generalized Bernoulli numbers and polynomials, 2-variable Legendre polynomials, 3-variable Hermite-Bernoulli-Legendre polynomials, summation formulae, generating functions.

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English version:
Ufa Mathematical Journal, 2018, 10:2, 118–126 (PDF, 311 kB); https://doi.org/10.13108/2018-10-2-118

Bibliographic databases:

UDC: 517.9
MSC: 33B10, 33C45, 33C47, 33C90
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Citation: N. U. Khan, T. Usman, “Certain generating functions of Hermite–Bernoulli–Legendre polynomials”, Ufimsk. Mat. Zh., 10:2 (2018), 118–126; Ufa Math. J., 10:2 (2018), 118–126

Citation in format AMSBIB
\Bibitem{KhaUsm18} \by N.~U.~Khan, T.~Usman \paper Certain generating functions of Hermite--Bernoulli--Legendre polynomials \jour Ufimsk. Mat. Zh. \yr 2018 \vol 10 \issue 2 \pages 118--126 \mathnet{http://mi.mathnet.ru/ufa426} \transl \jour Ufa Math. J. \yr 2018 \vol 10 \issue 2 \pages 118--126 \crossref{https://doi.org/10.13108/2018-10-2-118} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000438890500009} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85048464968} 

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This publication is cited in the following articles:
1. M. Goubi, “A new class of generalized polynomials associated with Hermite-Bernoulli polynomials”, J. Appl. Math. Inform., 38:3-4 (2020), 211–220
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