Related papers: The $q$-log-convexity of Domb's polynomials

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…

We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling…

This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of…

Let $P_{n,k}$ be the number of permutations $\pi$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $\pi$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define…

Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the $q$-log-convexity of the Narayana polynomials, and a proof of the second conjecture that the…

In this paper, we mainly study the stability of iterated polynomials and linear transformations preserving the strong $q$-log-convexity of polynomials Let $[T_{n,k}]_{n,k\geq0}$ be an array of nonnegative numbers. We give some criteria for…

We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence $\{\sum_{k=0}^n{n\choose k}^2q^k\}_{n\geq 0}$. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum…

In this paper, we provide criteria for the log-concavity of rows and the strong $q$-log-convexity of the generating functions of rows in more generalized triangles. Additionally, we prove that the bi$^s$nomial transformation not only…

In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of…

In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this…

We prove the reverse ultra log-concavity of the Boros-Moll polynomials. We further establish an inequality which implies the log-concavity of the sequence $\{i!d_i(m)\}$ for any $m\geq 2$, where $d_i(m)$ are the coefficients of the…

In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where $p$ is any prime greater than $3$) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as…

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of…

Polynomial sequence ${P_m}_{m\geq0}$ is $q$-logarithmically concave if $P_{m}^2-P_{m+1}P_{m-1}$ is a polynomial with nonnegative coefficients for any $m\geq{1}$. We introduce an analogue of this notion for formal power series whose…

In this paper we prove the strong $q$-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arraya and a criterion for determining…

We establish discrete and continuous log-concavity results for a biparametric extension of the $q$-numbers and of the $q$-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our…

We give a short proof of the log-concavity of the coefficients of the reduced characteristic polynomial of a matroid. The proof uses an extension of the theory of Lorentzian polynomials to convex cones, and reproves the Hodge-Riemann…

Descent polynomials and peak polynomials, which enumerate permutations with given descent and peak sets respectively, have recently received considerable attention. We give several formulas for $q$-analogs of these polynomials which refine…

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then we have $$…

The Boros-Moll polynomials $P_m(a)$ arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that $P_m(a)$ is 2-log-concave for any…