Raşa's logarithmic-convexity conjecture for squared Baskakov functions

For $n in\mathbb{N}$, $c>0$, and

$$\psi_{n,c}(x)=\sum_{k=0}^{\infty}\left(p_{n,k}^{[c]}(x)\right)^2,$$

where

$$p_{n,k}^{[c]}(x)=\binom{-n/c}{k}\left(-cx\right)^k\left(1+cx\right)^{-n/c-k},\qquad x\geq 0,$$

Raşa's conjecture. The function $\psi_{n,c}$ is logarithmically convex. The paper proves complete monotonicity of $\psi_{n,c}$, which implies logarithmic convexity; thus this conjecture is resolved.