On the $q$-log-convexity conjecture of Sun

TL;DR

This paper proves Sun's conjecture that the sequence of polynomials $S_n(q)$ is $q$-log-convex by developing a new criterion for self-reciprocal polynomials and applying it to confirm the conjecture.

Contribution

The paper introduces a sufficient condition for $q$-log-convexity of self-reciprocal polynomials and uses it to prove Sun's conjecture.

Findings

Confirmed Sun's $q$-log-convexity conjecture for $S_n(q)$

Developed a new criterion for $q$-log-convexity of self-reciprocal polynomials

Extended Liu and Wang's method to generate $q$-log-convex sequences

Abstract

In his study of Ramanujan-Sato type series for $1/\pi$, Sun introduced a sequence of polynomials $S_n(q)$ as given by $$S_n(q)=\sum\limits_{k=0}^n{n\choose k}{2k\choose k}{2(n-k)\choose n-k}q^k,$$ and he conjectured that the polynomials $S_n(q)$ are $q$-log-convex. By imitating a result of Liu and Wang on generating new $q$-log-convex sequences of polynomials from old ones, we obtain a sufficient condition for determining the $q$-log-convexity of self-reciprocal polynomials. Based on this criterion, we then give an affirmative answer to Sun's conjecture.