Multiplication formula for the Nörlund sum operator and their applications

Abstract The first aim of this paper is to present fundamental properties of the Nörlund sum (or the Nörlund sum operator) with their applications. The second purpose of this paper is to find not only some new formulas involving some certain classes of polynomials and finite sums with aid of the Nörlund sum, but also novel multiplication formula for the Nörlund sum. Moreover, by applying this multiplication formula to analytic functions and meromorphic functions, we derive the Bernoulli polynomials and a multiplication formula for the Hurwitz zeta function. We show that special case of this multiplication formula gives the Raabe’s multiplication formula for the Bernoulli polynomials. Finally, we give relations among the Nörlund sum, indefinite sum, and the Volkenborn integral.