Generalization of Mellin derivative and its applications

A combinatorial analysis of higher order generalised geometric polynomials: A generalisation of barred preferential arrangements

“…This is a generalisation of the work done in [6]. (1−x(e t −1)) r , for instance in [7,10,11,12,14,15,16,17,18,21]. Hence, our results in Theorem 3.1 offers a generalised combinatorial interpretation of these geometric polynomials.…”

“…Part II of [9]). These polynomials are well known in the literature, and it is also well known that these polynomials arise from variations of the generating function e rt (1−x(e t −1)) r , for instance in [7,10,11,12,14,15,16,17,18,21]. In this study our combinatorial interpretation of the integer sequences arising from the generating function…”

Harmonic number identities via polynomials with r-Lah coefficients

“…In particular n k 0 = n k is Lah numbers or rarely called Stirling numbers of the third kind [45]. The Mellin derivative (xD) = x d d x has been used for many different purposes, such as evaluating some power series, integrals [7,13,23,33,36] and also introducing some new families of polynomials [7,22,23,33,34]. When it is applied to a n-times differentiable function f we have [7] (xD…”

Identities and congruences involving the geometric polynomials

“…The geometric polynomials are defined by w n .x/ D P n kD0˚n k « kŠx k and satisfy the recurrence relation .x C 1/w n .x/ D x P n j D0 n j w j .x/; n 1; [9], where˚n k « is the .n; k/-th Stirling number of the second kind [2,26]. These polynomials have attracted attention from many researchers, see for instance [9,10,[15][16][17]. For x D 1 we obtain the geometric numbers w n WD w n .1/ D P n kD0˚n k « kŠ, for more information about these numbers, see [6][7][8]11,12,14,28,29].…”