### Limit summability and gamma type functions of order two

#### Abstract

Gamma type functions satisfying the difference functional equations $f(x+1)=g(x)f(x)$ and limit summability of functions were studied and introduced by R.J. Webster and M.H. Hooshmand, respectively. It is shown that the topic of gamma type functions can be considered as a subtopic of limit summability. Indeed, if $\ln f$ is limit summable, then its limit summand function $(\ln f)_\sigma$ satisfies $(\ln f)_\sigma(x)=\ln f(x)+(\ln f)_\sigma(x-1)$ and $e^{(\ln f)_\sigma(x)}$ is gamma type function of $f(x+1)$. In this paper, we introduce and study limit summability of order two, 2-limit summand function $f_{\sigma^2}$ and its results as gamma type functions of order two and also limit summand of multipliers. Finally, as an application of the study, we obtain a criteria for existence of gamma type function of the function ${f(x)}^x$ and give some related examples and corollaries.

#### Keywords

Limit summability, summand function, gamma-type function, difference functional equation, convex function

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