limit summability and gamma type functions of order two

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)σ satisfies (ln f)σ(x) = ln f(x) + (ln f)σ(x − 1) and e(ln f)σ (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 fσ2 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.