Positivity and continued fractions from the binomial transformation

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.

Coefficientwise Hankel-total positivity of row-generating polynomials for the $m$-Jacobi-Rogers triangle

“…In fact, log-convexity of many combinatorial sequences can be extended to Stieltjes moment property. See Liu and Wang [58] and Zhu [103] for log-convexity and Liang et al [57], Wang and Zhu [101] and Zhu [107,108] for Stieltjes moment property in combinatorics.…”

“…More and more combinatorial polynomials were proved to have such properties. For example, the Bell polynomials, the classical Eulerian polynomials, the Narayana polynomials of type A and B, Ramanujan polynomials, Dowling polynomials, Jacobi-Stirling polynomials, and so on, are q-log-convex (see Chen et al [20,21], Liu and Wang [58], Zhu [103,104,105,106], Zhu and Sun [113] for instance), 3-q-log-convex (see [107]) and q-Stieltjes moment (see [90,101,107,109]). We refer the reader to [73,74,90,91,110,111,112] for coefficientwise Hankel-total positivity in more indeterminates.…”

“…where J n,k = 0 unless 0 ≤ k ≤ n and J 0,0 = 1. The x-log-convexity, x-log-convexity of higher order and coefficientwise Hankel-total positivity of (J n,0 ) n≥0 have been studied, see [18,69,70,91,101,103,105,107,110] for instance. But the following question is open.…”

Stieltjes moment properties and continued fractions from combinatorial triangles

“…In addition, many log-convex sequences in combinatorics have SM property. We refer readers to Liu and Wang [24] and Zhu [40] for log-convexity and Wang and Zhu [38] and Zhu [44,45] for SM property.…”

On a Stirling-Whitney-Riordan triangle

“…If x is a valuable q, then it has been proved that many famous polynomials have the qlog-convexity, e.g., the Bell polynomials, the classical Eulerian polynomials, the Narayana polynomials of type A and B, Jacobi-Stirling polynomials, and so on, see Liu and Wang [25], Chen et al [11], Zhu [46,47,48,49] for instance. These polynomials also have 3-q-log-convexity, see Zhu [50,52]. In addition, the next is an important criterion for 3-q-log-convexity.…”