q-Hypergeometric Functions

Throughout this chapter, \(q\in \left ] 0,1\right [ \) is a real number. We present here a short introduction to q-hypergeometric functions, which yield the usual hypergeometric functions as a limit case when \( q\rightarrow 1.\) Hence, we define, in Sect. 10.1, q-analogues of Pochhammer’s symbol, binomial coefficients, differentiation and integration, and exponential and logarithm functions. In Sect. 10.2, we introduce the most general q-hypergeometric functions and study in some detail the q-analogues of the binomial function and of the Gauss hypergeometric function. Finally, Sect. 10.3 is devoted to the proof of some summation formulas.