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.