Monotonicity, convexity, and complete monotonicity of two functions related to the gamma function

Monotonicity and convexity involving generalized elliptic integral of the first kind

“…. is the Pochhammer symbol and (x) = ∞ 0 t x−1 e −t dt [12,17,18,29] is the classical Euler gamma function. In the case that c = a + b + k (with k ∈ N 0 =: N ∪ {0}), the resulting function F(a, b; a + b + k; x) is said to be k-balanced.…”