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 vertical bar q vertical bar < 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).