GAUSS SUMMATION FORMULA FOR LIMIT SUMMAND FUNCTIONS AND RELATED RESULTS

Abstract

‎The concept of limit summability of real functions was introduced and studied by the second author in 2001‎. ‎Concerning this‎, ‎the limit summand function $f_{\sigma}$ corresponds to a real or complex function $f$ with domain $D_f\supseteq\mathbb{N^*}$‎, ‎which satisfies the difference functional equation $f_{\sigma}(x) = f(x)‎ + ‎f_{\sigma}(x-1)$‎. ‎By considering the special case of $g:\mathbb{R^+} \rightarrow \mathbb{R^+}$ and by putting $f=\log g$‎, ‎he showed that $\Gamma$-type functions can be considered as a special case of limit summability and the basic relation $g^*(x+1) = e^{f_{\sigma}(x)}$ holds (where $g^*$ is its $\Gamma$-type function)‎. ‎As a result‎, ‎he improved one of the main theorems for $\Gamma$-type functions due to Webster in 1997‎. ‎Regarding the Gauss multiplication formula for $\Gamma$-type functions‎, ‎we introduce its dual for limit summand functions‎, ‎that is, Gauss \textit{summation} formula‎. ‎Also‎, ‎we show that not only the Gauss multiplication for $\Gamma$-type functions is a simple result of this formula‎, ‎but also provide an improvement of it with several consequences and applications‎. ‎Finally‎,‎‎ ‎we point out that in two theorems of Webster some conditions are superfluous‎.