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29 March 2024
 
  » arxiv » 1308.2736

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On the $q$-log-convexity conjecture of Sun
Donna Q.J. Dou ; Anne X.Y. Ren ;
Date 13 Aug 2013
AbstractIn his study of Ramanujan-Sato type series for $1/pi$, Sun introduced a sequence of polynomials $S_n(q)$ as given by $$S_n(q)=sumlimits_{k=0}^n{nchoose k}{2kchoose k}{2(n-k)choose n-k}q^k,$$ 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.
Source arXiv, 1308.2736
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