Applications of the Mellin transform in quantum calculus
In this paper the authors study in quantum calculus the correspondence between poles of the \(q\)-Mellin transform and the asymptotic behaviour of the original function at \(0\) and \(\infty\). They prove that each individual term in such asymptotic expansion of a function \(f\) taking the form \(x^{b}(\operatorname{Log} x)^{k}\), \(k \in \mathbb{N}\), is linked to a pole of its \(q\)-Mellin transform at \(s=-b\) with multiplicity \(k+1\). As applications they give a new technique to derive the asymptotic expansion of some functions defined by \(q\)-integrals or by \(q\)-harmonic sums. They also obtain a \(q\)-analogue of the generalized Euler-Maclaurin formula as well as a \(q\)-analogue of the Mellin-Perron formula.
scientific article; zbMATH DE number 5124085