A New Recurrence Relation and Related Determinantal form for Binomial Type Polynomial Sequences

2016 ◽  
Vol 13 (6) ◽  
pp. 4001-4017 ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Elisabetta Longo
Keyword(s):  
Recurrence Relation ◽  
Polynomial Sequences ◽  
Binomial Type ◽  
Determinantal Form

Polynomial Sequences of Binomial Type Path Integrals

2002 ◽  
Vol 6 (1) ◽  
pp. 45-56 ◽  
Author(s):  
Vladimir V. Kisil
Keyword(s):  
Path Integrals ◽  
Polynomial Sequences ◽  
Binomial Type

scholarly journals On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Lucas Numbers ◽  
Polynomial Sequences ◽  
Fibonacci And Lucas Numbers ◽  
Linear Recurrence Relation ◽  
Humbert Polynomials ◽  
General Method

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

scholarly journals Determinant Representations of Polynomial Sequences of Riordan Type

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Sheng-liang Yang ◽  
Sai-nan Zheng
Keyword(s):  
Recurrence Relation ◽  
Characteristic Polynomial ◽  
Chebyshev Polynomials ◽  
General Term ◽  
Polynomial Sequence ◽  
Polynomial Sequences ◽  
Principal Submatrix ◽  
Riordan Array

In this paper, using the production matrix of a Riordan array, we obtain a recurrence relation for polynomial sequence associated with the Riordan array, and we also show that the general term for the sequence can be expressed as the characteristic polynomial of the principal submatrix of the production matrix. As applications, a unified determinant expression for the four kinds of Chebyshev polynomials is given.

scholarly journals Sequences of Non-Gegenbauer-Humbert Polynomials Meet the Generalized Gegenbauer-Humbert Polynomials

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-18 ◽  
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
Recurrence Relation ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Polynomial Sequences ◽  
Linear Recurrence Relation ◽  
The Relationship ◽  
Humbert Polynomials

Here, we present a connection between a sequence of polynomials generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known transfer formulas between non-Gegenbauer-Humbert polynomials and generalized Gegenbauer-Humbert polynomials are given. The applications of the relationship to the construction of identities of polynomial sequences defined by linear recurrence relations are also discussed.

scholarly journals An algebraic exposition of umbral calculus with application to general linear interpolation problem: A survey

2014 ◽  
Vol 96 (110) ◽  
pp. 67-83 ◽  
Author(s):  
Francesco Costabile ◽  
Elisabetta Longo
Keyword(s):  
Interpolation Problem ◽  
Linear Interpolation ◽  
Umbral Calculus ◽  
General Linear ◽  
Numerical Examples ◽  
Polynomial Sequences ◽  
Sheffer Sequences ◽  
Determinantal Form ◽  
Systematic Exposition ◽  
Sheffer Polynomial

A systematic exposition of Sheffer polynomial sequences via determinantal form is given. A general linear interpolation problem related to Sheffer sequences is considered. It generalizes many known cases of linear interpolation. Numerical examples and conclusions close the paper.

scholarly journals Polynomial Sequences of Binomial-Type Arising in Graph Theory

10.37236/3702 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Jonathan Schneider
Keyword(s):  
Graph Theory ◽  
Large Class ◽  
Higher Dimensions ◽  
Polynomial Sequence ◽  
Chromatic Polynomials ◽  
Grid Graphs ◽  
Polynomial Sequences ◽  
Fixed Set ◽  
Binomial Type ◽  
Open Question

In this paper, we show that the solution to a large class of "tiling'' problems is given by a polynomial sequence of binomial type. More specifically, we show that the number of ways to place a fixed set of polyominos on an $n\times n$ toroidal chessboard such that no two polyominos overlap is eventually a polynomial in $n$, and that certain sets of these polynomials satisfy binomial-type recurrences. We exhibit generalizations of this theorem to higher dimensions and other lattices. Finally, we apply the techniques developed in this paper to resolve an open question about the structure of coefficients of chromatic polynomials of certain grid graphs (namely that they also satisfy a binomial-type recurrence).

Sebuah Generalisasi Baru Barisan Fibonacci-Lucas

2019 ◽  
Vol 2 (2) ◽  
Author(s):  
Musraini M Musraini M ◽  
Rustam Efendi ◽  
Rolan Pane ◽  
Endang Lily
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Lucas Sequence ◽  
Binet’S Formula

Barisan Fibonacci dan Lucas telah digeneralisasi dalam banyak cara, beberapa dengan mempertahankan kondisi awal, dan lainnya dengan mempertahankan relasi rekurensi. Makalah ini menyajikan sebuah generalisasi baru barisan Fibonacci-Lucas yang didefinisikan oleh relasi rekurensi B_n=B_(n-1)+B_(n-2),n≥2 , B_0=2b,B_1=s dengan b dan s bilangan bulat  tak negatif. Selanjutnya, beberapa identitas dihasilkan dan diturunkan menggunakan formula Binet dan metode sederhana lainnya. Juga dibahas beberapa identitas dalam bentuk determinan.   The Fibonacci and Lucas sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this paper, a new generalization of Fibonacci-Lucas sequence is introduced and defined by the recurrence relation B_n=B_(n-1)+B_(n-2),n≥2, with ,  B_0=2b,B_1=s                          where b and s are non negative integers. Further, some identities are generated and derived by Binet’s formula and other simple methods. Also some determinant identities are discussed.

scholarly journals Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 804
Author(s):  
Ioannis K. Argyros ◽  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Approximate Solution ◽  
Convergence Analysis ◽  
Banach Spaces ◽  
Recurrence Relation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Existence And Uniqueness ◽  
Numerical Illustration ◽  
Type Method ◽  
Local Convergence Analysis

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

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