A note on q-difference equations for Cigler’s polynomials

Abstract

Viewing from polynomials solutions of q-difference equations makes certain generating functions calculated easily. In this short paper, we build the relations between Cigler’s polynomials and q-difference equations. In addition, we deduce generating functions and bilinear generating functions for Cigler’s polynomials. More over, we gain multilinear and mixed generating functions for Cigler’s polynomials. At last, we gain $U(n+1)$ type generating functions for Cigler’s polynomials.

Keywords:

• q-difference equation
• Cigler’s polynomial
• generating function
• multilinear generating function
• $U(n+1)$

AMS Subject Classifications:

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

Acknowledgements

The author 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.

Dedicated to Professor Johann Cigler.

Additional information

Funding

This work was supported by the Zhejiang Provincial Natural Science Foundation of China [grant number LQ13A010021]; the National Natural Science Foundation of China [grant number 11501155]; the China Scholarship Council [grant number 201508330101].