q-Laguerre polynomials and related q-partial differential equations

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.

Keywords:

• q-series
• q-differential operator
• q-partial differential equation
• q-Laguerre polynomial
• little q-Laguerre polynomial
• q-Hahn polynomial
• generating function

AMS Subject Classifications:

• 05A30
• 11B65
• 33D15
• 33D45
• 39A13

Acknowledgements

The authors would like to thank the referees and editors for their many valuable comments and suggestions.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11571114].