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

In his study of Ramanujan-Sato type series for $1/π$, 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.

No reviews yet.

No comments yet.