q-Difference equations of moment integrals for transformational identities and generating functions

Abstract

In this paper, we show how to deduce several moment integrals by the method of $q$-difference equation. In addition, we obtain several generalizations of transformational identities by the technique of moment integrals. Meanwhile, we obtain some special cases of bilateral–unilateral series and the Rogers–Ramanujan identities as by-products. Moreover, we give two mixed generating functions for generalized Rogers–Szegö polynomials by moment integrals and derive two transformational identities by symmetry. At last, a trilinear generating function for generalized Rogers–Szegö polynomials is given by moment integrals.

Keywords::

• moment integral
• generating function
• q-difference equation
• Rogers–Szegö polynomial
• Rogers–Ramanujan identity

MSC (2010) Classification::

• 05A30
• 11B65
• 33C45
• 33D15
• 33D45
• 39A13

Acknowledgements

The author would like to thank the referee and editor for many valuable comments and suggestions.

Notes

^1. Email: [email protected]

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11351002); China Postdoctoral Science Foundation [grant number 2012M521155]; Zhejiang Projects for Postdoctoral Research Preferred Funds [grant number Bsh1201021]; and Zhejiang Provincial Natural Science Foundation of China [grant number LQ13A010021].