Infinite log-convexity conjecture for Apéry, Catalan–Larcombe–French, and related sequences

Let A=a_i_{0\leq i\leq\infty} be a sequence, and define

$$\mathcal{L}(A)=\{b_i\}_{0\leq i\leq\infty},\qquad b_i=a_{i-1}a_{i+1}-a_i^2\quad(i\geq1).$$

A sequence is $k$-log-convex if $\mathcal{L}^j(A)$ is log-convex for $j=0,1,\ldots,k-1$, and infinitely log-convex if $\mathcal{L}^k(A)$ is log-convex for every $k\geq0$. Infinite log-convexity conjecture. The sequences $\{A_n\}_{n=0}^\infty$, $\{B_n\}_{n=0}^\infty$, $\{U_n\}_{n=0}^\infty$ and $\{s_n\}_{n=0}^\infty$ are infinitely log-convex. The sequences $\{M_n\}_{n=0}^\infty$, $\{F_n\}_{n=0}^\infty$, $\{F_n^{(3)}\}_{n=0}^\infty$ and $\{F_n^{(4)}\}_{n=0}^\infty$ are $k$-log-convex for any $k\geq1$, except for a constant number, depending on $k$, of terms at the beginning. The conjecture extends the paper's proved strict 2-log-convexity results to all iterates of the log-convexity operator; the precise status of the asserted infinite and arbitrary-order properties is not established in the supplied text.