Asymptotic expansions for certain multiple q‐integrals and q‐differentials of Thomae‐Jackson type

This is a summarized version of the forthcoming paper [11]. Let q be a complex parameter with |q|<1. We shall study in this paper asymptotic aspects when q→1 of certain general classes of q‐integral and q‐differential operations given in (1.5) and (1.6) below respectively; this leads us to establish complete asymptotic expansions for their iterated extensions (Theorems 1 and 2) under fairly generic situations (Theorem 3). Several applications of of our main formulae (2.4) and (2.9) are further given for the generalized Lerch zeta‐function defined by (3.3) (Theorems 4–6 and Corollary 6.1), the q‐factorials (Corollary 4.1), q‐analogues of the exponential (Corollary 4.2), the binomial (Corollary 4.3), and the poly‐logarithmic functions (Corollaries 4.4 and 5.1).