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Positivity and continued fractions from the binomial transformation

Part of: Combinatorics Sequences and sets Elementary number theory

Published online by Cambridge University Press:  18 March 2019

Bao-Xuan Zhu
Bao-Xuan Zhu*
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, PR China (bxzhu@jsnu.edu.cn)
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Abstract

Given a sequence of polynomials $\{x_k(q)\}_{k \ges 0}$, define the transformation

$$y_n(q) = a^n\sum\limits_{i = 0}^n {\left( \matrix{n \cr i} \right)} b^{n-i}x_i(q)$$
for $n\ges 0$. In this paper, we obtain the relation between the Jacobi continued fraction of the ordinary generating function of yn(q) and that of xn(q). We also prove that the transformation preserves q-TPr+1 (q-TP) property of the Hankel matrix $[x_{i+j}(q)]_{i,j \ges 0}$, in particular for r = 2,3, implying the r-q-log-convexity of the sequence $\{y_n(q)\}_{n\ges 0}$. As applications, we can give the continued fraction expressions of Eulerian polynomials of types A and B, derangement polynomials types A and B, general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. In addition, we also prove the strong q-log-convexity of derangement polynomials type B, Dowling polynomials and Tanny-geometric polynomials and 3-q-log-convexity of general Eulerian polynomials, Dowling polynomials and Tanny-geometric polynomials. We also present a new proof of the result of Pólya and Szegö about the binomial convolution preserving the Stieltjes moment property and a new proof of the result of Zhu and Sun on the binomial transformation preserving strong q-log-convexity.

Keywords

log-convexitystrong q-log-convexityk-q-log-convexitytotal positivityhankel matrixcontinued fraction

MSC classification

Primary: 05A10: Factorials, binomial coefficients, combinatorial functions
Secondary: 05A20: Combinatorial inequalities 05A30: $q$-calculus and related topics 11A55: Continued fractions 11B83: Special sequences and polynomials
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 831 - 847
Copyright
Copyright © Royal Society of Edinburgh 2019 

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