q-Difference equations for the generalized Cigler’s polynomials

Abstract

In this paper, we show how to prove several types of generating functions for Cigler’s polynomials involving four variables by the method of homogeneous q-difference equations. We utilize some methods and results of Liu’s [Ramanujan Math. Soc. 20 (2013), pp. 213–250.] and Fang’s [Appl. Math. Comput. 248 (2014), pp. 550–561.] continue to study Cigler’s results [J. Differ. Equ. Appl. 22 (2016), pp. 1880–1892.]. In addition, we deduce $U(n+1)$ type generation functions for Cigler’s polynomials. More over, we build relations between transformation formulas and homogeneous q-difference equations. And then, we gain multilinear and multiple generating functions for Cigler’s polynomials as application. Finally, we generalize Andrews–Askey integrals, moment integrals and Askey–Roy integrals by the method of homogeneous q-difference equations.

Keywords:

• q-Difference equation
• generating functions
• Cigler’s polynomial
• Andrews–Askey integral
• moment integral
• Askey–Roy integral

AMS Subject Classifications:

• 05A30
• 11B65
• 33D15
• 33D45
• 33D60
• 39A13
• 39B32

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 11501155].