Related papers: A digit reversal property for Stern polynomials

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on $[-1,1]$. The recurrence coefficients…

We prove that the enumerative polynomials of generalized Stirling permutations by the statistics of plateaux, descents and ascents are partial $\gamma$-positive. Specialization of our result to the Jacobi-Stirling permutations confirms a…

We prove that the permutations of $\{1,\dots, n\}$ having an increasing (resp., decreasing) subsequence of length $n-r$ index a subset of the set of all $r$th Kronecker powers of $n \times n$ permutation matrices which is a basis for the…

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal…

We prove a generalization of W.M. Schmidt's theorem related to the Diophantine approximations for a linear form of the type $\alpha_1x_1+\alpha_2x_2 +y$ with {\it positive} integers $x_1,x_2$.

Let $N>1$ and let $\Phi_N(X,Y)\in\mathbb{Z}[X,Y]$ be the modular polynomial which vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. In this note we study the divisibility of the…

We show that all negative powers B_{a,b}^-{s} of the Beta distribution are infinitely divisible. The case b<1 follows by complete monotonicity, the case b > 1, s > 1 by hyperbolically complete monotonicity and the case b > 1, s < 1 by a…

The $q$-binomial coefficients $\qbinom{n}{m}=\prod_{i=1}^m(1-q^{n-m+i})/(1-q^i)$, for integers $0\le m\le n$, are known to be polynomials with non-negative integer coefficients. This readily follows from the $q$-binomial theorem, or the…

Stanley, building on work of Stern, defined an array of numbers by the recurrence $s(n, 2k) = s(n-1, k)$, $s(n, 2k+1) = s(n-1, k) + s(n-1, k+1)$. Stanley showed that, for each positive integer $r$, the sequence $s_n^r:= \sum_k s(n,k)^r$…

We obtain closed form expressions for convolutions of scale transformations within a certain subset of Appell polynomials. This subset contains the Bernoulli, Apostol-Euler, and Cauchy polynomials, as well as various kinds of their…

We prove an important property of the binomial transform: it converts multiplication by the discrete variable into a certain difference operator. We also consider the case of dividing by the discrete variable. The properties presented here…

For the generalized Jacobi, Laguerre and Hermite polynomials $P_n^{(\alpha_n, \beta_n)} (x), L_n^{(\alpha_n)} (x),$\break $H_n^{(\gamma_n)} (x)$ the limit distributions of the zeros are found, when the sequences $\alpha_n$ or $\beta_n$ tend…

For an $n$-bit positive integer $a$ written in binary as $$ a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$ \overleftarrow{a} =…

We prove that the reverse characteristic polynomial $\det(I_n - zA_n)$ of a random $n \times n$ matrix $A_n$ with iid $\mathrm{Bernoulli}(d/n)$ entries converges in distribution towards the random infinite product $\prod_{\ell =…

We determine all pairs of positive integers $(a,b)$ such that $a+b$ and $a \times b$ have the same decimal digits in reverse order: \[ (2,2), (9,9), (3,24), (2,47), (2,497), (2,4997), (2,49997), \ldots \] We use deterministic finite…

Let $p$ be an odd prime. In this paper, we study the permutation behaviour of the reversed Dickson polynomials of the $(k+1)$-th kind $D_{n,k}(1,x)$ when $n=p^{l_1}+3$, $n=p^{l_1}+p^{l_2}+p^{l_3}$, and $n=p^{l_1}+p^{l_2}+p^{l_3}+p^{l_4}$,…

Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2.…

Let \psi(x) be a polynomial with rational coefficients. Suppose that \psi has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x,y\in A and a…

We prove polynomial identities for the N=1 superconformal model SM(2,4\nu) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a…

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…