Relations for Barnes' type multiple Bernoulli polynomials and generalized Stirling numbers via laplace transform and generating functions


Hacer Ozden Ayna, Yilmaz Simsek




The aim of this paper is not only to survey some properties of Barnes' multiple Changi $q$-Bernoulli polynomials with Barnes' zeta function, but also to derive some relations for Barnes' multiple Changi $q$-Bernoulli polynomials and Barnes' multiple Changi $q$-zeta functions using the $q$-integral method and modified generating functions. By applying the Laplace transform to these generating functions, we obtain infinite series and integral representation for these polynomials. Using the power series of the function $\sinh x$ and these generating functions, we obtain some new formulas for generalized Stirling numbers of the second kind. Finally, we give some special values of our general results.