q-Laguerre polynomials and related q-partial differential equations |
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Authors: | Da-Wei Niu Long Li |
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Institution: | Department of Mathematics, East China Normal University, Shanghai, P.R. China. |
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Abstract: | In this paper, we define two homogenous q-Laguerre polynomials, by introducing a modified q-differential operator, we prove that an analytic function can be expanded in terms of the q-Laguerre polynomials if and only if the function satisfies certain q-partial differential equations. Using this main result, we derive the generating functions, bilinear generating functions and mixed generating functions for the q-Laguerre polynomials and generalized q-Hahn polynomials. Cigler’s polynomials and its generating functions discussed in J. Cao, D.-W. Niu, A note on q -difference equations for Cigler’s polynomials, J. Difference Equ. Appl. 22 (2016), 1880–1892.] are generalized. At last, we obtain an q-integral identity involving q-Laguerre polynomials. These applications indicate that the q-partial differential equation is an effective tool in studying q-Laguerre polynomials. |
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Keywords: | q-series q-differential operator q-partial differential equation q-Laguerre polynomial little q-Laguerre polynomial q-Hahn polynomial generating function |
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