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Complete asymptotic expansions for certain multiple q-integrals and q-differentials of Thomae-Jackson type

Masanori Katsurada,
published 2010-07-27

Let q be a complex parameter with q < 1. We shall study in this paper asymptotic aspects when q to 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), for which a satisfactory extension of the domain where our main formulae (2.4) and (2.9) are valid is possible under the restriction that 0 < q < 1 (Theorem 3). Several applications of Theorem 3 are futher given for the generalized Lerch zeta-function defined by (3.2) (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 in the proofs of Theorems 1 and 2; crucial roles are especially played by the Mellin transform formulae (4.8) and (6.7).

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