Related papers: Solving Skolem problem for negative indexed $k-$ge…

A generalization of the well-known Fibonacci sequence is the $k$-Fibonacci sequence with some fixed integer $k\ge 2$. The first $k$ terms of this sequence are $0,0, \ldots, 1$, and each term afterwards is the sum of the preceding $k$ terms.…

Given a prime $p$ and a positive integer $k$, let $\mathrm{M}_{n}(\mathbb{Z}/p^{k}\mathbb{Z})$ be the ring of $n \times n$ matrices over $\mathbb{Z}/p^{k}\mathbb{Z}$. We consider the number of solutions $X \in…

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum_{i,j=0}^n\binom ni^m\binom nj^m\binom{i+j}ix^{i+j}\ \ (m,n=0,1,2,\ldots).$$ For example, we show that $$\sum_{k=0}^{p-1}S_k^{(0)}(x)\equiv\frac…

In this note we associate a sequence of non-negative integers to any convergent series of positive real numbers and study this sequence for the series $\sum_{n \geq 1} n^{-k}$ where $k$ is an integer $\geq 2$.

Motivated by the study of the distribution of zeros of generalized Bessel-type functions, the principal goal of this paper is to identify new research directions in the theory of multiplier sequences. The investigations focus on multiplier…

We first interpret Pell's equation satisfied by Chebyshev polynomials for each degree t, as a certain Positivstellensatz, which then yields for each integer t, what we call a generalized Pell's equation, satisfied by reciprocals of…

In this paper, generalized Pell graphs $\Pi _{n,k}$, $k\ge 2$, are introduced. The special case of $k=2$ are the Pell graphs $\Pi _{n}$ defined earlier by Munarini. Several metric, enumerative, and structural properties of these graphs are…

We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…

We begin with the observation that the signed generalized Stirling polynomials $P_k(m,x)$, which occur in a generalization of Malmsten's integral, reduce to the falling factorials when $k=m$. The structure of these generalized Stirling…

For n=1,2,3,... let p_n be the n-th prime. We mainly show that p_n>n+sum_{k=1}^n p_k/k for all n>124, and sum_{k=1}^n kp_k<n^2p_n/3 for all n>30.

Let $k$ be an algebraically closed field of characteristic $p > 3$. Let $A$ be an abelian surface over $k$. Fix an integer $n \geq 1$ such that $p \nmid n$ and let $K^{[n]}$ be the $n$-th Generalized Kummer Variety associated to $A$. In…

If $ p_k(a,m,n) $ denotes the number of partitions of $n$ into $k$th powers with a number of parts that is congruent to $ a $ modulo $m,$ then $p_2(0,2,n)\sim p_2(1,2,n)$ and the sign of the difference $p_2(0,2,n)- p_k(1,2,n)$ alternates…

The aim of this paper is to prove a general version of Pl\"unnecke's inequality. Namely, assume that for finite sets $A$, $B_1, ... B_k$ we have information on the size of the sumsets $A+B_{i_1}+... +B_{i_l}$ for all choices of indices…

Let ${\cal P}_n^c$ denote the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let $$D^+ := \{z \in \mathbb{C}: |z| \leq 1, \, \, \Im(z) \geq 0\}$$ be the closed upper half-disk of the complex plane. For…

It has been shown that zeros of Kac polynomials $K_n(z)$ of degree $n$ cluster asymptotically near the unit circle as $n\to\infty$ under some assumptions. This property remains unchanged for the $l$-th derivative of the Kac polynomials…

For every positive integer $n$, we find a complete classification for planar graphs according to the collection of numbers of common neighbours for every $n$-tuple of distinct vertices. Our results expand the literature on planar graphical…

For any positive integers $m$ and $\alpha$, we prove that $$\sum_{k=0}^{n-1}\epsilon^k(2k+1)A_k^{(\alpha)}(x)^m\equiv0\pmod{n}, $$ where $\epsilon\in\{1,-1\}$ and $$…

For a regular compact set $K$ in $C^m$ and a measure $\mu$ on $K$ satisfying the Bernstein-Markov inequality, we consider the ensemble $P_N$ of polynomials of degree $N$, endowed with the Gaussian probability measure induced by $L^2(\mu)$.…

Poly-Bernoulli numbers $B_n^{(k)}\in\mathbb{Q}$\,($n \geq 0$,\,$k \in \mathbb{Z}$) are defined by Kaneko in 1997. Multi-Poly-Bernoulli numbers\,$B_n^{(k_1,k_2,\ldots, k_r)}$, defined by using multiple polylogarithms, are generations of…

We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \sum_{k=0}^n A_k B_k(z)$, where $\{A_k\}_{k=0}^{\infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\{B_k\}_{k=0}^{\infty}$ are…