The Ramanujan Journal

, Volume 48, Issue 1, pp 63–73 | Cite as

A note on q-difference equations for Ramanujan’s integrals

  • Jian CaoEmail author


This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to obtain the expression of certain q-orthogonal polynomials as solutions of q-difference equations. In addition, we show how to generalize Ramanujan’s integrals by the technique of q-difference equation. More over, we find two generalized q-Chu–Vandermonde formulas from the perspective of the method of q-difference equations.


Solutions of q-difference equation Generating function Al-Salam–Carlitz polynomial Ramanujan’s integral 

Mathematics Subject Classification

05A30 11B65 33D15 33D45 39A13 



The author would like to thank the referees and editors for their many valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 11501155).


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Authors and Affiliations

  1. 1.Department of MathematicsHangzhou Normal UniversityHangzhouPeople’s Republic of China

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