Applications of the Mellin transform in quantum calculus

Abstract

In this paper, we study in quantum calculus the correspondence between poles of the q-Mellin transform (see [A. Fitouhi, N. Bettaibi, K. Brahim, The Mellin transform in Quantum Calculus, Constr. Approx. 23 (3) (2006) 305-323]) and the asymptotic behaviour of the original function at 0 and ∞. As applications, we give a new technique (in q-analysis) to derive the asymptotic expansion of some functions defined by q-integrals or by q-harmonic sums. Finally, a q-analogue of the Mellin-Perron formula is given.

Some results for Laplace-type integral operator in quantum calculus

“…On the other hand, some impressive integral transforms also have the corresponding qanalogues in the concept of q-calculus; they include the q-Laplace transforms [10], the qSumudu transforms [9,[11][12][13], the q-Wavelet transform [14], the q-Mellin transform [15], q-E 2,1 -transform [16], q-Mangontarum transforms [17,18], q-natural transforms [19], and so on. Recently, a number of authors have studied various image formulas for these qintegral transforms, associated with a variety of special functions.…”

ZERO-FREE REGIONS OF A q-ANALOGUE OF THE COMPLETE RIEMANN ZETA FUNCTION

“…We can solve this problem if s ∈ R as in the proof of Proposition 3.3 (2). Furthermore, considering Fig.…”

“…The q-Mellin transform (Jackson-Mellin's transform) of suitable functions f : (0, ∞) → C is given by the Jackson integral (see, e.g., [2])…”

Certain results related to the N-transform of a certain class of functions and differential operators

“…Thereafter, some remarkable integral transforms were associated with different analogues in a q-calculus context. Among those examined integrals, we recall the q-Laplace integral operator [2][3][4][5], the q-Sumudu integral operator [6][7][8][9], the Weyl fractional q-integral operator [10], the q-wavelet integral operator [11], the q-Mellin type integral operator [12], Mangontarum transform [13], Widder potential transform [14], E 2;1 -transform [15] and a few others. In this paper, we give some q-analogues of some recently investigated integral transform, named the natural transform, and obtain some new desired q-properties.…”