scholarly journals On the Relation between Pincherle's Polynomials and the Hypergeometric Function

1920 ◽  
Vol 39 ◽  
pp. 58-62 ◽  
Author(s):  
Bevan B. Baker
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Hypergeometric Function ◽  
The Difference ◽  
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1. The Pincherle polynomials are defined as the coefficients in the expansion of {1 − 3 tx + t3}−½ in ascending powers of t. If the coefficient of tn be denoted by Pn(x), then the polynomials satisfy the difference equationand Pn(x) satisfies the differential equation

scholarly journals The evaluation of certain continued fractions

1937 ◽  
Vol 30 ◽  
pp. vi-x
Author(s):  
C. G. Darwin
Keyword(s):  
Difference Equation ◽  
Hypergeometric Function ◽  
Difference Equations ◽  
Continued Fractions ◽  
Continued Fraction ◽  
The Difference ◽  
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1. If the approximate numerical value of e is expressed as a continued fraction the result isand it was in finding the proof that the sequence extends correctly to infinity that the following work was done. First the continued fraction may be simplified by setting down the difference equations for numerator and denominator as usual, and eliminating two out of every successive three equations. A difference equation is thus formed between the first, fourth, seventh, tenth … convergents , and this equation will generate another continued fraction. After a little rearrangement of the first two members it appears that (1) implies2. We therefore consider the continued fractionwhich includes (2), and also certain continued fractions which were discussed by Prof. Turnbull. He evaluated them without solving the difference equations, and it is the purpose here to show how the difference equations may be solved completely both in his cases and in the different problem of (2). It will appear that the work is connected with certain types of hypergeometric function, but I shall not go into this deeply.

On discrete harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 194-206 ◽  
Author(s):  
H. A. Heilbronn
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Difference Equation ◽  
Harmonic Functions ◽  
The Difference ◽  
Discrete Harmonic Functions ◽  
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A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the difference equationThis equation can be considered as the direct analogue either of the differential equationor of the integral equationin the notation normally employed to harmonic functions.

A Singular Perturbation Problem and A Neutral Differential-Difference Equation

1974 ◽  
Vol 17 (1) ◽  
pp. 77-83
Author(s):  
Edward Moore
Keyword(s):  
Difference Equation ◽  
Taylor Series Expansion ◽  
Linear Difference Equation ◽  
Constant Matrix ◽  
Perturbation Problem ◽  
Constant Coefficients ◽  
Explicit Formulae ◽  
Differential Difference Equation ◽  
The Difference ◽  
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Vasil’eva, [2], demonstrates a close connection between the explicit formulae for solutions to the linear difference equation with constant coefficients(1.1)where z is an n-vector, A an n×n constant matrix, τ>0, and a corresponding differential equation with constant coefficients(1.2)(1.2) is obtained from (1.1) by replacing the difference z(t—τ) by the first two terms of its Taylor Series expansion, combined with a suitable rearrangement of the terms.

scholarly journals On a Difference Equation due to Stirling

1917 ◽  
Vol 36 ◽  
pp. 40-60
Author(s):  
Eleanor Pairman
Keyword(s):  
Difference Equation ◽  
Factorial Series ◽  
The Difference ◽  
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In 1730 there was published Stirling's Methodus Differentialis, and in it (Prop. VIII., p. 44) he considers the Difference Equationand shows that it is satisfied by an inverse factorial series

scholarly journals The summability of formal solutions of functional equations

1967 ◽  
Vol 7 (2) ◽  
pp. 141-144
Author(s):  
J. D. Gray
Keyword(s):  
Difference Equation ◽  
Functional Equations ◽  
Formal Solution ◽  
Principal Solution ◽  
The Difference ◽  
Image Position ◽  
Formal Solutions

In 1923 Nörlund [1] considered the difference equationand showed that the formal solution of (1)obtained by iteration, although in general divergent, is in fact Abel summable to a solution of (1). He writes the arbitary constant c asand thus the “principal solution” of the difference equation is

scholarly journals solutions of period three for a non-linear difference equation

1984 ◽  
Vol 25 (4) ◽  
pp. 451-462 ◽  
Author(s):  
A. Brown
Keyword(s):  
Mathematical Model ◽  
Difference Equation ◽  
Exact Solutions ◽  
Linear Difference Equation ◽  
Critical Values ◽  
Simple Mathematical Model ◽  
Frequency Dependent Selection ◽  
Stable Solutions ◽  
The Difference ◽  
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AbstractThe paper uses the factorisation method to discuss solutions of period three for the difference equationwhich has been proposed as a simple mathematical model for the effect of frequency dependent selection in genetics. Numerical values are obtained for the critical values of a at which solutions of period three first appear. In addition, the interval in which stable solutions are possible has been determined. Exact solutions are given for the case a = 4 and these have been used to check the results.

scholarly journals Non-existence of invariant circles

1984 ◽  
Vol 4 (2) ◽  
pp. 301-309 ◽  
Author(s):  
John N. Mather
Keyword(s):  
Dynamical System ◽  
Difference Equation ◽  
Rigorous Proof ◽  
Numerical Evidence ◽  
Invariant Circles ◽  
The Difference ◽  
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AbstractThe dynamical system associated to the difference equationhas been studied numerically by several authors. On the basis of numerical evidence, they conclude that there exists a numberk0≈ 0.97 such that there are homotopically non-trivial invariant circles for |k|≤k0and there are none for |k|>k0. In this note, we give a simple rigorous proof that there are none for |k|>.

scholarly journals A nonlinear difference equation with two parameters

1985 ◽  
Vol 26 (4) ◽  
pp. 430-451 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
General Problem ◽  
Iteration Process ◽  
Nonlinear Difference Equation ◽  
Cubic Equation ◽  
The Difference ◽  
The Stability ◽  
Image Position ◽  
Two Parameters ◽  
Parameter Problem

AbstractThe paper is mainly concerned with the difference equationwhere k and m are parameters, with k > 0. This equation arises from a method proposed for solving a cubic equation by iteration and represents a standardised form of the general problem. In using the above equation it is essential to know when the iteration process converges and this is discussed by means of the usual stability criterion. Critical values are obtained for the occurrence of solutions with period two and period three and the stability of these solutions is also examined. This was done by considering the changes as k increases, for a give value of m, which makes it effectively a one-parameter problem, and it turns out that the change with k can differ strongly from the usual behaviour for a one-parameter difference equation. For m = 2, for example it appears that the usual picture of stable 2-cycle solutions giving way to stable 4-cycle solutions is valid for smaller values of k but the situation is recersed for larger values of k where stable 4-cycle solutions precede stable 2-cycle solutions. Similar anomalies arise for the 3-cycle solutions.

scholarly journals A non-linear difference equation with two parameters. II

1985 ◽  
Vol 27 (2) ◽  
pp. 145-166 ◽  
Author(s):  
A. Brown
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Anomalous Behaviour ◽  
Three Solutions ◽  
Unstable Solutions ◽  
The Difference ◽  
Set Up ◽  
The Stability ◽  
Image Position ◽  
Two Parameters

AbstractThe paper discusses solutions of period 4 for the difference equationwhere k and m are real parameters, with k > 0. For given values of k and m there are at most three solutions with period 4 and equations are set up to determine the elements of these solutions and the stability of each solution. Only real solutions are considered. The procedure that is used to find these solutions allows unstable solutions to be identified as well as stable solutions.In a previous paper, solutions of period 2 and period 3 were examined for this equation and there was evidence of anomalous behaviour in the way the stability intervals occurred. Some preliminary information about solutions of period 4 was mentioned in the discussion. The present paper provides more complete results, which confirm the anomalous behaviour and give a better idea of how the stability criterion changes for different families of solutions. These results are used to indicate the variety of behaviour that can be found for one-parameter systems by imposing suitable conditions on m and k.

scholarly journals The Properties of a new Orthogonal Function Associated with the Confluent Hypergeometric Function

1924 ◽  
Vol 43 ◽  
pp. 117-130
Author(s):  
George E. Chappell
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Hypergeometric Function ◽  
Special Form ◽  
Confluent Hypergeometric Function ◽  
Orthogonal Function ◽  
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The function whose properties are discussed in this note, is a special form of Whittaker's Confluent Hypergeometric Function, Wkm(z). It is the general solution of the Differential Equationand can be obtained in the form of a series, terminating only when k is half a positive odd integer, viz.,

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