Analytic summability of real and complex functions

Abstract

Gamma-type functions satisfying the functional equation f(x+1) = g(x)f(x) and limit summability of real and complex functions were introduced by Webster (1997) and Hooshmand (2001). However, some important special functions are not limit summable, and so other types of such summability are needed. In this paper, by using Bernoulli numbers and polynomials B_n(z), we define the notions of analytic summability and analytic summand function of complex or real functions, and prove several criteria for analytic summability of holomorphic functions on an open domain D. As consequences of our results, we give some criteria for absolute convergence of the functional series

\(\sum\nolimits_{n = 0}^\infty {{c_n}\sigma \left( {{Z^n}} \right)} ,where\sigma \left( {{Z^n}} \right) = {S_n}\left( z \right) = \frac{{{B_{n + 1}}\left( {z + 1} \right) - {B_{n + 1}}\left( 1 \right)}}{{n + 1}}\)

. Finally, we state some open problems.