• 제목, 요약, 키워드: q-Laguerre polynomials

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q-SOBOLEV ORTHOGONALITY OF THE q-LAGUERRE POLYNOMIALS {Ln(-N)(·q)}n=0 FOR POSITIVE INTEGERS N

  • Moreno, Samuel G.;Garcia-Caballe, Esther M.
    • 대한수학회지
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    • v.48 no.5
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    • pp.913-926
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    • 2011
  • The family of q-Laguerre polynomials $\{L_n^{(\alpha)}({\cdot};q)\}_{n=0}^{\infty}$ is usually defined for 0 < q < 1 and ${\alpha}$ > -1. We extend this family to a new one in which arbitrary complex values of the parameter ${\alpha}$ are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter ${\alpha}$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving q-derivatives with respect to which the generalized q-Laguerre polynomials $\{L_n^{(-N)}({\cdot};q)\}_{n=0}^{\infty}$, for positive integers N, become orthogonal.

(p, q)-EXTENSION OF THE WHITTAKER FUNCTION AND ITS CERTAIN PROPERTIES

  • Dar, Showkat Ahmad;Shadab, Mohd
    • 대한수학회논문집
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    • v.33 no.2
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    • pp.619-630
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    • 2018
  • In this paper, we obtain a (p, q)-extension of the Whittaker function $M_{k,{\mu}}(z)$ together with its integral representations, by using the extended confluent hypergeometric function of the first kind ${\Phi}_{p,q}(b;c;z)$ [recently extended by J. Choi]. Also, we give some of its main properties, namely the summation formula, a transformation formula, a Mellin transform, a differential formula and inequalities. In addition, our extension on Whittaker function finds interesting connection with the Laguerre polynomials.

The Incomplete Lauricella Functions of Several Variables and Associated Properties and Formulas

  • Choi, Junesang;Parmar, Rakesh K.;Srivastava, H.M.
    • Kyungpook Mathematical Journal
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    • v.58 no.1
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    • pp.19-35
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    • 2018
  • Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [30] and the second Appell function [6], we introduce here the incomplete Lauricella functions ${\gamma}^{(n)}_A$ and ${\Gamma}^{(n)}_A$ of n variables. We then systematically investigate several properties of each of these incomplete Lauricella functions including, for example, their various integral representations, finite summation formulas, transformation and derivative formulas, and so on. We provide relevant connections of some of the special cases of the main results presented here with known identities. Several potential areas of application of the incomplete hypergeometric functions in one and more variables are also pointed out.