Related papers: Linear transformations and strong $q$-log-concavit…

We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some…

With the notion of mode-$j$ Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff-Hopf and the Perron-Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of…

Infinitesimal conformal transformations of $R^n$ are always polynomial and finitely generated when $n>2$. Here we prove that the Lie algebra of infinitesimal conformal polynomial transformations over $R^n$, $n>1$, is maximal in the Lie…

We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan…

We prove that $\sum_{k=0}^{q-1}\binom{2k}{k}\equiv q^2\pmod{3q^2}$ if q>1 is a power of 3, as recently conjectured by Z.W. Sun and R. Tauraso. Our more precise result actually implies that the value of $(1/q^2)\sum_{k=0}^{q-1}\binom{2k}{k}$…

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…

Let $\mathbb{F}$ be a field, and let $q\in\mathbb{F}$. The $q$-deformed Heisenberg algebra is the unital associative $\mathbb{F}$-algebra $\mathcal{H}(q)$ with generators $A,B$ and relation $AB-qBA=I$, where $I$ is the multiplicative…

For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod…

We consider a $(q,y)$-analogue of Laguerre polynomials $L^{(\alpha)}_n(x;y;q)$ for integral $\alpha\geq -1$, which turns out to be a rescaled version of Al-Salam--Chihara polynomials. A combinatorial interpretation for the $(q,y)$-Laguerre…

Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy the following two conditions: There exists a basis for…

Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positive integers and let $k\in\mathbb{N}_+$ be fixed. The function $p_\mathcal{A}(n,k)$ counts the number of partitions of $n$ with parts in the multiset…

One possible data encryption scheme is related to stream ciphers, which use a sufficiently long pseudo-random sequence. To increase the cryptographic strength of the cipher, linear shift algorithms (generated by linear recurrent sequences…

Let $m_{\lambda }$ be the monomial symmetric functions, $ \lambda $ being an integer partition of $n\in \mathbb{N}^{\ast }$. For the specialization corresponding to the $q$-deformation of the exponential, we prove that each $m_{\lambda }$…

In this paper we study the joint convexity/concavity of the trace functions \[ \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] where $A$ and $B$ are positive definite matrices and $K$ is any…

Let $q$ be a prime power and $n$ and $r$ be positive integers. It is well known that the linearized binomial $L_r(x)=x^{q^r}+ax\in\mathbb{F}_{q^n}[x]$ is a permutation polynomial if and only if $(-1)^{n/d}a^{{(q^n-1)}/{(q^{d}-1)}}\neq 1$…

The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums…

We introduce bivariate versions of the continuous q-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence…

We prove a lemma, which we call the Order Ideal Lemma, that can be used to demonstrate a wide array of log-concavity and log-convexity results in a combinatorial manner using order ideals in distributive lattices. We use the Order Ideal…

Linear operators preserving the direct sum of polynomial rings P(m)\oplus P(n) are constructed. In the case |m-n|=1 they correspond to atypical representations of the superalgebra osp(2,2). For |m-n|=2 the generic, finite dimensional…

The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of ``A General Geometric Fourier Transform`` in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which…