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For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since $$6=1+1+4=1+2+3=2+2+2.$$ QUESTION. ...

A sequence of polynomials $$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$ with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...