In this paper, we present some limits for k− Gamma function and k− Beta function by using properties asymptotic of k− Gamma function.

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In the paper, the authors prove that the function x α ln px x+p+1 − ψ p (x) is completely monotonic on (0, ∞) if and only if α ≤ 1, where p ∈ N and ψ p (x) is the p-analogue of the classical psi function ψ(x).

We introduce the -Bernstein functions, for , and give sufficient and necessary conditions for a function to belong to the class of -Bernstein functions. For some classes of functions we give results concerning -completely monotonic and -Bernstein functions. The obtained results are -analogue of known results.

It is defined Γ p,q function, a generalize of Γ function. Also, we defined ψ p,q -analogue of the psi function as the log derivative of Γ p,q . For the Γ p,q -function, are given some properties related to convexity, log-convexity and completely monotonic function. Also, some properties of ψ p,q analog of the ψ function have been established. As an application, when p → ∞, q → 1, we obtain all result of [12] and [21].