相关论文: A short proof of Mathar's 2021 recurrence conjectu…

For the OEIS sequence A001711, contributed by N. J. A. Sloane long before the on-line era and identified there as the diagonal $T(n+4, 4)$ of a generalized-Stirling triangle, R. J. Mathar contributed in February 2020 the conjectured order-2…

For the OEIS sequence A025166, defined by $a(n) = -n!\,2^{n}\,L_{n}(1/2)$ where $L_{n}$ is the Laguerre polynomial of degree $n$, R.~J.~Mathar contributed in February 2013 the conjectured order-2 P-recursive recurrence \[ a(n) + (-4n+3)\,…

For the OEIS sequence A176677, defined by the quadratic convolution recurrence $a(0) = a(1) = 1$ and $a(n+1) = \sum_{p=0}^n a(p) a(n-p) - 1$ for $n \ge 1$, R.~J.~Mathar contributed in March 2016 the conjectured order-4 P-recursive…

For the OEIS sequence A214615, defined by $a(n) = M_{n}(1)$ where $M_{n}$ is the $n$-th Meixner polynomial satisfying $M_{n+1}(x) = x\,M_{n}(x) - n^{2}\,M_{n-1}(x)$, R.~J.~Mathar contributed on 6~March 2013 the conjectured order-2…

For the OEIS sequence A002627, defined by the inhomogeneous first-order recurrence $a(n) = n\,a(n-1) + 1$ with $a(0) = 0$, R.~J.~Mathar recorded in February 2014 the conjectured second-order homogeneous recurrence \[ a(n) - (n+1)\,a(n-1) +…

For the OEIS sequence A032123, the number of length-$2n$ black-and-white strings with $n$ black beads, considered up to reversal, R. J. Mathar contributed in November 2013 the conjectured order-5 P-recursive recurrence \[ \begin{aligned}…

For the OEIS sequence A348410, P. Bala recorded in February 2022 two equivalent closed forms, $a(n) = [x^{n}] ((1-x)(1-x^2))^{-n}$ and a single-index binomial sum. R. J. Mathar (October 2021) and V. Kotesovec (November 2021) each…

We study log-concavity properties of real sequences $(a_n)_{n \ge 0}$ satisfying a $d$-th order linear recurrence whose coefficients are linear functions of $n$; the so-called P-recursive (or holonomic) sequences. Writing the recurrence in…

We study the recurrence coefficients of the monic polynomials $P_n(z)$ orthogonal with respect to the deformed (also called semi-classical) Freud weight \begin{equation*} w_{\alpha}(x;s,N)=|x|^{\alpha}{\rm…

We prove an asymptotic formula for the recurrence coefficients of orthogonal polynomials with orthogonality measure $\log \bigl(\frac{2}{1-x}\bigr) {\rm d}x$ on $(-1,1)$. The asymptotic formula confirms a special case of a conjecture by…

In 1951 paper \cite{Ki} Kippenhahn conjectured that if the characteristic polynomial \ $P_A(x_1,x_2,x_3)=\mbox{det}(x_1A_1+x_2A_2-x_3I)$, \ where $A_1$ and $A_2$ are $n\times n$ Hermitian matrices, has a repeated factor in the polynomial…

This paper studies the monic semi-classical Laguerre polynomials based on previous work by Boelen and Van Assche \cite{Boelen}, Filipuk et al. \cite{Filipuk} and Clarkson and Jordaan \cite{Clarkson}. Filipuk, Van Assche and Zhang proved…

Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n$-th Fourier coefficient of $f$. The second author…

In this paper, we present a new method for the analysis of piecewise dynamical systems that are similar to the Collatz conjecture in regard to certain properties of the commutator of their sub-functions. We use the fact that the commutator…

We prove that the symmetric-cube coefficients $A_n=(-27)^n[z^n]\,_2F_1(1/3,1/3;1;z)^3$ satisfy the supercongruence $A(mp)\equiv A(m) \bmod p^4$ for every prime $p\geq 5$ and every $m\geq 1$. The proof rests on three ingredients: (i) the…

We study the dependence of recurrence coefficients in the three-term recurrence relation for orthogonal polynomials with a certain deformation of the $q$-Laguerre weight on the degree parameter $n$. We show that this dependence is described…

In 1971, by induction on $n$ and using a two-term linear recurrence relation, Graham and Pollak got a beautiful formula $$\det(D_n)=-(n-1)(-2)^{n-2}$$ on the determinant of distance matrix $D_n$ of a tree $T_n$ on $n$ vertices. The…

We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev\'e equation. The first example deals with Freud weights $|x|^\rho \exp(-|x|^m)$ on the real line,…

Let $R=\mathcal{O}_{\Q(\sqrt{d})}$ for $d<0$, squarefree, $d\neq -1,-3$. We prove Lehmer's conjecture for associated reciprocal polynomials of $R$-matrices; that is, any noncyclotomic $R$-matrix has Mahler measure at least…

In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the…