Related papers: Ap\'ery-like sequences defined by four-term recurr…

We investigate numerical semigroups generated by any quadratic sequence with initial term zero and an infinite number of terms. We find an efficient algorithm for calculating the Ap\'ery set, as well as bounds on the elements of the Ap\'ery…

Sequences with good randomness properties are quite important for stream ciphers. In this paper, a new class of quaternary sequences is constructed by using generalized cyclotomic classes of $\mathbb{Z}_{2p^m}$ $(m\geq1)$. The exact values…

In this work we study the relationship between several combinatorial formulas for type $A$ spherical Whittaker functions. These are spherical functions on $p$-adic groups, which arise in the theory of automorphic forms. They depend on a…

Turyn-type sequences, TT(n), are quadruples of {+,-1}-sequences (A;B;C;D), with lengths n,n,n,n-1 respectively, where the sum of the nonperiodic autocorrelation functions of A,B and twice that of C,D is a delta-function (i.e., vanishes…

The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.

Roughly speaking, a recurrence relation is nested if it contains a subexpression of the form ... A(...A(...)...). Many nested recurrence relations occur in the literature, and determining their behavior seems to be quite difficult and…

Using a simple Vi\`ete-like formula for $\pi$ based on the nested radicals $a_k = \sqrt{2 + a_{k-1}}$ and $a_1 = \sqrt{2}$, we derive a set of the recurrence relations for the constant $1$. Computational test shows that application of this…

Let $p$ be a prime greater than $3$ and let $a$ be a rational p-adic integer. In this paper we try to determine $\sum_{k=1}^{[p/3]}\binom{3k}ka^k\pmod p$, and real the connection between cubic congruences and the sum…

We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by…

The Ap\'ery polynomials are given by $$A_n(x)=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2x^k\ \ (n=0,1,2,\ldots).$$ (Those $A_n=A_n(1)$ are Ap\'ery numbers.) Let $p$ be an odd prime. We show that…

We first show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form $p_{n+1}(z) = Q_k(z)p_{n}(z)+ \gamma p_{n-1}(z)$ as $n$ $\rightarrow$ $\infty$, where the coefficient…

As a continuation of the authors and Wakatsuki's previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that for any integral models of the space of binary cubic…

Based on prototypical example of Al.Zamolodchikov's recursion relations for the four point conformal block and using recently proposed Alday-Gaiotto-Tachikawa (AGT) conjecture, recursion relations are derived for the generalized…

We prove that the leading coefficient of the "error" terms of NRS(2) applied to a cubic polynomial $f(z)$ with starting point $(-\frac{a_1}{a_2}, -\frac{a_1}{a_2})$ are positive-coefficient rational functions in the zeros of $f(z)$. We…

Only one three-term recurrence relation, namely, $W_{r}=2W_{r-1}-W_{r-4}$, is known for the generalized Tribonacci numbers, $W_r$, $r\in\mathbb{Z}$, defined by $W_{r}=W_{r-1}+W_{r-2}+W_{r-3}$ and \mbox{$W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}$},…

The main objective of this paper is to evaluate six new Ap\'ery-like series of weight $5$ in closed form. These series involve harmonic numbers and exhibit the characteristic reciprocal central binomial coefficient structure. Generating…

In a recent paper, Bilu et al. studied a conjecture of Marques and Lengyel on the $p$-adic valuation of the Tribonacci sequence. In this article, we study the $p$-adic valuation of third order linear recurrence sequences by considering a…

For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$,…

In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality…

We consider the real eigenvalues of the elliptic Ginibre matrix indexed by the non-Hermiticity parameter $\tau \in [0,1]$, and present a Harer-Zagier type recursion formula for the even moments in the form of an $11$-term recurrence…