Modular Resolutions by Polyseries

Chaos, Solitons & Fractals, 2009

Let hðxÞ be a polynomial with real coefficients. We introduce hðxÞ-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these hðxÞ-Fibonacci polynomials. We also introduce hðxÞ-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ðxÞ that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.

We give an answer to a recent question of Prodinger, which consists of finding q-analogues of identities related to Fibonacci and Lucas polynomials.

arXiv: Combinatorics, 2019

We show that the Catalan-Schroeder convolution recurrences and their higher order generalizations can be solved using Riordan arrays and the Catalan numbers. We investigate the Hankel transforms of many of the recurrence solutions, and indicate that Somos $4$ sequences often arise. We exhibit relations between recurrences, Riordan arrays, elliptic curves and Somos $4$ sequences. We furthermore indicate how one can associate a family of orthogonal polynomials to a point on an elliptic curve, whose moments are related to recurrence solutions.

1995

A remarkable sequence of polynomials is considered. These polynomials in q describe in particular the number of solution to the equation X 2 = 0 in triangular n × n matrices over a field ‫ކ‬q with q elements. They have at least three other important interpretations and a conjectural explicit expression in terms of the entries of the Catalan triangle. Résumé Nous considérons une suite remarquable de polynômes. Ces polynômes en q décrivent en particulier le nombre de solutions de l'équation X 2 = 0 dans les matrices n × n sur un corps ‫ކ‬q ayant q elements. Ils ont au moins trois autres interprétations importantes et une forme explicite conjecturale en termes des entrées du triangle de Catalan.

Discrete Applied Mathematics, 2018

Recently, by using methods of hypercomplex function theory, the authors have shown that a certain sequence S of rational numbers (Vietoris' sequence) combines seemingly disperse subjects in real, complex and hypercomplex analysis. This sequence appeared for the first time in a theorem by Vietoris (1958) with important applications in harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic functions (Ruscheweyh/Salinas, 2004). A non-standard application of Clifford algebra tools for defining Clifford-holomorphic sequences of Appell polynomials was the hypercomplex context in which a one-parametric generalization S(n), n ≥ 1, of S (corresponding to n = 2) surprisingly showed up. Without relying on hypercomplex methods this paper demonstrates how purely real methods also lead to S(n). For arbitrary n ≥ 1 the generating function is determined and for n = 2 a particular case of a recurrence relation similar to that known for Catalan numbers is proved.

In this paper we shall find a new connection between nth degree polynomial mod p congruence with n roots and higher order Fibonacci and Lucas sequences. We shall first discuss about the recent work been done in sequences and their connection to polynomial congruence and then find out new relations between particular recurrence relation and the congruence of the sequences

Colloquium Mathematicum

Let a and b be integers with b(a 2 + 4b) = 0. Let u0 = 0, u1 = 1, and un = aun−1 + bun−2 for n ≥ 2. Let v0 = 2, v1 = a, and vn = avn−1 + bvn−2 for n ≥ 2. Using algebraic identities we will prove some results, including the following ones. For integers n ≥ 0 and k ≥ 1, u 2 n+3k = (v 2k + (−b) k)u 2 n+2k − (−b) k (v 2k + (−b) k)u 2 n+k + (−b) 3k u 2 n v 2 n+3k = (v 2k + (−b) k)v 2 n+2k − (−b) k (v 2k + (−b) k)v 2 n+k + (−b) 3k v 2 n u 3 n+4k = (v 3k + (−b) k v k)u 3 n+3k − (−b) k (v 4k + (−b) k v 2k + 2(−b) 2k)u 3 n+2k + (−b) 3k (v 3k + (−b) k v k)u 3 n+k − (−b) 6k u 3 n v 3 n+4k = (v 3k + (−b) k v k)v 3 n+3k − (−b) k (v 4k + (−b) k v 2k + 2(−b) 2k)v 3 n+2k + (−b) 3k (v 3k + (−b) k v k)v 3 n+k − (−b) 6k v 3 n. These results generalize some results of Gould (1963), Zeitlin and Parker (1963), Bicknell (1972), and Prodinger (1997). 1. Motivation Definition 1.1. Let a and b be integers with b(a 2 + 4b) = 0. Let u 0 = 0, u 1 = 1, and for n ≥ 2, u n = au n−1 + bu n−2. Let v 0 = 2, v 1 = a, and for n ≥ 2, v n = av n−1 + bv n−2. These sequences were originally studied by Lucas [L]. The Binet formulas state that for all n ≥ 0,

Southeast Asian Bulletin of Mathematics

In this paper, we obtain some new modular equations of degree 2 for the ratios of Ramanujan’s theta-function and also establish the general formulas for their explicit evaluations. As an application, we establish some new modular relations for Ramanujan–Gollnitz-Gordon continued fraction H(q) with H(q^n/2), Ramanujan–Selberg continued fraction V (q) with V (q^n/2) and Eisenstein continued fraction E(q) with E(q^n/2)for n = 3; 5 and 7.

arXiv (Cornell University), 2015

In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (A176898) S n = (6n 3n)(3n 2n) 2(2n n)(2n+1) , and the binomial coefficients 3n n and 4n 2n. As an application, we address several conjectures of Z. W. Sun on congruences of sums involving S n and we prove a cubic residuacity criterion in terms of sums of the binomial coefficients 3n n conjectured by Z. H. Sun.

We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate strong connection between Euler and Bernoulli numbers and entries of inverses of certain lower triangular matrices built of binomial coe¢ cients. In other words we interpret Euler and Bernoulli numbers in terms of modi…ed Pascal matrices.