scholarly journals A new method for solving the linear recurrence relation with nonhomogeneous constant coefficients in some cases

2021 ◽  
Vol 2113 (1) ◽  
pp. 012070
Author(s):  
Ben-Chao Yang ◽  
Xue-Feng Han
Keyword(s):  
Recurrence Relation ◽  
Computer Programming ◽  
Recursive Relation ◽  
New Method ◽  
Linear Recurrence ◽  
General Proof ◽  
Constant Coefficients ◽  
Linear Recurrence Relation ◽  
Almost All ◽  
General Method

Abstract Recursive relation mainly describes the unique law satisfied by a sequence, so it plays an important role in almost all branches of mathematics. It is also one of the main algorithms commonly used in computer programming. This paper first introduces the concept of recursive relation and two common basic forms, then starts with the solution of linear recursive relation with non-homogeneous constant coefficients, gives a new solution idea, and gives a general proof. Finally, through an example, the general method and the new method given in this paper are compared and verified.

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.

Summation of a slowly convergent fourier series occurring in a fluid motion problem

1956 ◽  
Vol 237 (1208) ◽  
pp. 17-27
Keyword(s):  
Asymptotic Expansion ◽  
Hypergeometric Function ◽  
Recurrence Relation ◽  
Fluid Motion ◽  
Linear Recurrence ◽  
In Series ◽  
Motion Problem ◽  
Extensive Class ◽  
Linear Recurrence Relation ◽  
General Method

Methods are described for the numerical evaluation of a slowly convergent sine series S ( 0 ) (0⩽0⩽1/2n), in which the coefficients satisfy a linear recurrence relation; a connexion is established between S ( 0 ) and the hypergeometric function. A general method is presented for the evaluation of an extensive class of power series in a complex variable, based on the asymptotic expansion of the coefficients in descending inverse factorials. The proof is given of a lemma on the asymptotic expansion of products of gamma functions, which is fundamental in the application of the method to hypergeometric series. The method is applied to S ( 0 ) in two ways: in the first, use is made of the connexion with the hypergeometric function; in the second, the general coefficient in the series is expanded directly by solving the recurrence relation in series. Another method of calculating S ( 0 ), founded on Euler’s transformation of series, can be used over a range given approximately by 0°⩽0⩽70°.

scholarly journals A Reciprocity Theorem for Monomer-Dimer Coverings

2003 ◽  
Vol DMTCS Proceedings vol. AB,... (Proceedings) ◽  
Author(s):  
Nick Anzalone ◽  
John Baldwin ◽  
Ilya Bronshtein ◽  
Kyle Petersen
Keyword(s):  
Recurrence Relation ◽  
Infinite Sequence ◽  
A Priori ◽  
Perfect Matchings ◽  
Two Dimensions ◽  
Linear Recurrence ◽  
Rectangular Grid ◽  
Combinatorial Model ◽  
Constant Coefficients ◽  
Linear Recurrence Relation

International audience The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics.It has only been exactly solved for the special case of dimer coverings in two dimensions ([Ka61], [TF61]). In earlier work, Stanley [St85] proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp [P01], Stanley's result concerns the unique way of extending $N(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $N(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) = \varepsilon_{m,n} N(m,n)$ where $\varepsilon_{m,n}=1$ unless $m \equiv 2(\mod 4)$ and $n$ is odd, in which case $\varepsilon_{m,n}=-1$. Furthermore, Propp's method was applicable to higher-dimensional cases.This paper discusses similar investigations of the numbers $M(m,n)$, of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an $m$ by $n$ rectangular grid. We show that for each fixed $m$ there is a unique way of extending $M(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $M(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients.We show that $M(m,n)$, a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.

scholarly journals Matrices determined by a linear recurrence relation among entries

2003 ◽  
Vol 373 ◽  
pp. 89-99 ◽  
Author(s):  
Gi-Sang Cheon ◽  
Suk-Geun Hwang ◽  
Seog-Hoon Rim ◽  
Seok-Zun Song
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

scholarly journals Linear recurrences in the degree sequences of monomial mappings

2008 ◽  
Vol 28 (5) ◽  
pp. 1369-1375 ◽  
Author(s):  
ERIC BEDFORD ◽  
KYOUNGHEE KIM
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Degree Sequences ◽  
Integer Matrix ◽  
Linear Recurrences ◽  
Monomial Map ◽  
Linear Recurrence Relation

AbstractLet A be an integer matrix, and let fA be the associated monomial map. We give a connection between the eigenvalues of A and the existence of a linear recurrence relation in the sequence of degrees.

General anharmonic oscillators

1978 ◽  
Vol 364 (1717) ◽  
pp. 265-275 ◽  
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Behaviour ◽  
Recurrence Relation ◽  
Quantum Number ◽  
Anharmonic Oscillator ◽  
Linear Recurrence ◽  
Anharmonic Oscillators ◽  
Anharmonicity Constant ◽  
Linear Recurrence Relation

The eigenvalue problem of the general anharmonic oscillator (Hamiltonian H 2 μ ( k, λ ) = -d 2 / d x 2 + kx 2 + λx 2 μ , ( k, λ ) is investi­gated in this work. Very accurate eigenvalues are obtained in all régimes of the quantum number n and the anharmonicity constant λ . The eigenvalues, as functions of λ , exhibit crossings. The qualitative features of the actual crossing pattern are substantially reproduced in the W. K. B. approximation. Successive moments of any transition between two general anharmonic oscillator eigenstates satisfy exactly a linear recurrence relation. The asymptotic behaviour of this recursion and its consequences are examined.

Convolution Identities for Stirling Numbers of the First Kind

Integers ◽  
2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Takashi Agoh ◽  
Karl Dilcher
Keyword(s):  
Recurrence Relation ◽  
Stirling Numbers ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

AbstractWe derive several new convolution identities for the Stirling numbers of the first kind. As a consequence we obtain a new linear recurrence relation which generalizes known relations.

scholarly journals A Reciprocity Theorem for Domino Tilings

10.37236/1562 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
James Propp
Keyword(s):  
Recurrence Relation ◽  
Reciprocity Theorem ◽  
Algebraic Methods ◽  
Linear Recurrence ◽  
Combinatorial Interpretation ◽  
Domino Tilings ◽  
Linear Recurrence Relation

Let $T(m,n)$ denote the number of ways to tile an $m$-by-$n$ rectangle with dominos. For any fixed $m$, the numbers $T(m,n)$ satisfy a linear recurrence relation, and so may be extrapolated to negative values of $n$; these extrapolated values satisfy the relation $$T(m,-2-n)=\epsilon_{m,n}T(m,n),$$ where $\epsilon_{m,n}=-1$ if $m \equiv 2$ (mod 4) and $n$ is odd and where $\epsilon_{m,n}=+1$ otherwise. This is equivalent to a fact demonstrated by Stanley using algebraic methods. Here I give a proof that provides, among other things, a uniform combinatorial interpretation of $T(m,n)$ that applies regardless of the sign of $n$.

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