scholarly journals Parity Theorems for Statistics on Domino Arrangements

10.37236/1977 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark A. Shattuck ◽  
Carl G. Wagner
Keyword(s):  
Generating Functions ◽  
Lucas Polynomials ◽  
Special Values

We study special values of Carlitz's $q$-Fibonacci and $q$-Lucas polynomials $F_n(q,t)$ and $L_n(q,t)$. Brief algebraic and detailed combinatorial treatments are presented, the latter based on the fact that these polynomials are bivariate generating functions for a pair of statistics defined, respectively, on linear and circular domino arrangements.

scholarly journals Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
GwangYeon Lee ◽  
Mustafa Asci
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Pascal Matrix ◽  
Riordan Arrays ◽  
Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

Fibonacci and Lucas polynomials

1981 ◽  
Vol 90 (3) ◽  
pp. 385-387 ◽  
Author(s):  
B. G. S. Doman ◽  
J. K. Williams
Keyword(s):  
Difference Equation ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Linear Difference Equation ◽  
Second Order ◽  
Lucas Numbers ◽  
Order Linear ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

The Fibonacci and Lucas polynomials Fn(z) and Ln(z) are denned. These reduce to the familiar Fibonacci and Lucas numbers when z = 1. The polynomials are shown to satisfy a second order linear difference equation. Generating functions are derived, and also various simple identities, and relations with hypergeometric functions, Gegenbauer and Chebyshev polynomials.

scholarly journals Symmetric and generating functions of generalized (p,q)-numbers

2021 ◽  
Vol 48 (4) ◽  
Author(s):  
Nabiha Saba ◽  
◽  
Ali Boussayoud ◽  
Abdelhamid Abderrezzak ◽  
◽  
...  
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Lucas Polynomials ◽  
Binet’S Formula

In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.

scholarly journals Generating functions of planar polygons from homological mirror symmetry of elliptic curves

2020 ◽  
Vol 6 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Jonas Kaszian ◽  
Jie Zhou
Keyword(s):  
Elliptic Curves ◽  
Theta Function ◽  
Generating Functions ◽  
Mirror Symmetry ◽  
Rational Functions ◽  
Mock Theta Function ◽  
Homological Mirror Symmetry ◽  
Jacobi Theta Function ◽  
Special Values

Abstract We study generating functions of certain shapes of planar polygons arising from homological mirror symmetry of elliptic curves. We express these generating functions in terms of rational functions of the Jacobi theta function and Zwegers’ mock theta function and determine their (mock) Jacobi properties. We also analyze their special values and singularities, which are of geometric interest as well.

scholarly journals On the (p,q)–Chebyshev Polynomials and Related Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 136 ◽  
Author(s):  
Can Kızılateş ◽  
Naim Tuğlu ◽  
Bayram Çekim
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Lucas Polynomials

In this paper, we introduce ( p , q ) –Chebyshev polynomials of the first and second kind that reduces the ( p , q ) –Fibonacci and the ( p , q ) –Lucas polynomials. These polynomials have explicit forms and generating functions are given. Then, derivative properties between these first and second kind polynomials, determinant representations, multilateral and multilinear generating functions are derived.

Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials and their properties

2020 ◽  
pp. 2150076
Author(s):  
Gamaliel Cerda-Morales
Keyword(s):  
Recurrence Relation ◽  
Generating Functions ◽  
Third Order ◽  
Lucas Polynomials ◽  
Binet Formula

In this paper, the Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are defined. Then Binet formula and generating functions of these numbers are given. Also, some summation identities for Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials are obtained by using the recurrence relation satisfied by them. Then, some linear and quadratic relations are given between Gaussian third-order Jacobsthal and Gaussian third-order Jacobsthal–Lucas polynomials.

Generating Functions for Ultraspherical Functions

1968 ◽  
Vol 20 ◽  
pp. 120-134 ◽  
Author(s):  
B. Viswanathan
Keyword(s):  
Differential Equation ◽  
Generating Functions ◽  
Special Values ◽  
Image Position

The ultraspherical function1.1for |1 — x| < 2 is a solution of the differential equation1.2This equation has two independent solutions; of the two, only Pn(λ)(x) is analytic at x = 1, aside for some special values of λ, which we shall not consider.

scholarly journals On Generalized Jacobsthal and Jacobsthal-Lucas polynomials

2016 ◽  
Vol 24 (3) ◽  
pp. 61-78 ◽  
Author(s):  
Paula Catarino ◽  
Maria Luisa Morgado
Keyword(s):  
Generating Functions ◽  
Special Kind ◽  
Generalized Polynomials ◽  
Lucas Polynomials ◽  
Tridiagonal Matrices

AbstractIn this paper we introduce a generalized Jacobsthal and Jacobsthal-Lucas polynomials, Jh,nand jh,n, respectively, that consist on an extension of Jacobsthal's polynomials Jn(𝑥) and Jacobsthal-Lucas polynomials jn(𝑥). We provide their properties and a generalization of the usual identities. We also present, for each one of these generalized polynomials, their ordinary generating functions and matrices. In the last part of the paper, we present some special kind of tridiagonal matrices whose entries are elements of these generalized polynomials.

scholarly journals Generating Functions of Binary Products of Tribonacci and Tribonacci Lucas Polynomials and Special Numbers

2022 ◽  
pp. 237-262
Author(s):  
Hind Merzouk ◽  
Ali Boussayoud ◽  
Kasi Viswanadh V. Kanuri
Keyword(s):  
Generating Functions ◽  
Lucas Polynomials

In this paper, we introduce a new operator defined in this paper, we give some new generating functions of binary products of Tribonacci and Tribonacci Lucas polynomials and special numbers.

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