Abstract
Abstract In this manuscript, we present the complete monotonicity of functions defined in terms of the poly-double gamma function ψ 2 ( n ) ( x ) = ( − 1 ) n + 1 n ! ∑ k = 0 ∞ ( 1 + k ) ( x + k ) n + 1 , x > 0 , n ≥ 2 . $$\begin{aligned} \psi _{2}^{(n)}(x) = (-1)^{n+1} n! \sum _{k=0}^{\infty } \dfrac{(1+k)}{(x+k)^{n+1}}, \quad x > 0, \ n\geq 2. \end{aligned}$$ Consequently, we derive bounds for the ratio involving ψ 2 ( n ) ( x ) $\psi _{2}^{(n)}(x)$ and apply these bounds to establish the convexity, sub-additivity and super-additivity of ψ 2 ( n ) ( x ) $\psi _{2}^{(n)}(x)$ . In the process, recurrence relations and asymptotic expansions for ψ 2 ( n ) ( x ) $\psi _{2}^{(n)}(x)$ are established. Moreover, we obtain integral representations, complete monotonicity, and related inequalities for the poly-multiple gamma function ψ p ( n ) ( x ) $\psi _{p}^{(n)}(x)$ . Graphical illustrations are provided to support the theoretical results.