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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 $P_{n+1}(q)P_{n-1}(q)-P_n(q)^2$ in $q$ is nonnegative.

Observe that $$\sum_{k=0}^n\binom{n+k}k=\binom{2n+1}{n+1}$$ by the Chu-Vandermonde identity. Motivated by this, for $m,n\in\mathbb N=\{0,1,2,\ldots\}$ we introduce the polynomial $$S^m_n(q):=\sum_{k=0}^n\binom{n+k}k^mq^k.$$

Based on my computation via Mathematica, I have formulated the following conjecture.

Conjecture. For any integer $m\ge2$, the polynomial sequence $(S^m_n(q))_{n\ge0}$ is $q$-log-convex.

We know that there are lots of studies on log-convexity of integer sequences and $q$-log-convexity of polynomial sequences.

QUESTION. Does the above conjecture hold? Can one prove it via the known methods?

Your comments are welcome!

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  • $\begingroup$ Is "conconvex" a typo? $\endgroup$
    – Daniel Weber
    Commented Sep 21 at 5:56
  • $\begingroup$ Yes, that's a typo. I have corrected it. $\endgroup$
    – Zhi-Wei Sun
    Commented Sep 21 at 6:11
  • $\begingroup$ Recall that those numbers $f_n=\sum_{k=0}^n\binom nk^3\ (n\in\mathbb N)$ are called Franel numbers. Define $f_n(q)=\sum_{k=0}^n\binom nk^3 q^k$ for $n\in\mathbb N$. I also conjecture that the polynomial sequence $(f_n(q))_{n\ge8}$ is $q$-log-convex. $\endgroup$
    – Zhi-Wei Sun
    Commented Sep 21 at 6:18

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