GAUSS SUMMATION FORMULA FOR LIMIT SUMMAND FUNCTIONS AND RELATED RESULTS
DOI:
https://doi.org/10.30495/jme.v15i0.1287Abstract
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.
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