Difference equations, including linear and nonlinear equations, discrete version of topics in analysis, partial difference equations, oscillation theory, periodic solutions, almost periodic solutions, bifurcation theory, stability theory.

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A two variable recurrence relation with conditionals

I have arrived at the combinatorial problem of enumerating certain types of ballot paths. This has led to analyzing the sequence defined by the following recurrence $$ f(n,m) = \begin{cases} f(n, \...
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0answers
43 views

Finite realization of irrational transfer functions

In the field of digital signal processing, linear time-invariant systems play a distinguished role. These are the systems for which there exists an impulse response, a function $h:\mathbb{Z}\to\...
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1answer
340 views

Integer Polynomial solutions to functional equation

Recently I came across a functional equation which always has a polynomial with integer coefficients solution. Let $$ L_n(x)=(2 x+1)^2f(x+1)-4x(x+n+1)f(x)-((2 n+1)!!)^2\prod_{i=1}^n(x+i). $$ Problem: ...
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1answer
82 views

how to solve this equation? [closed]

$Q=-{\frac {q \left( {{\rm e}^{{\it nb}\,{\it nv}\,\theta\,{\it Td}}}-1 \right) }{1-{{\rm e}^{\theta\,{\it Td}}}} \left( 1-{{\rm e}^{{\frac { Q\theta\,{\it Td}}{p}}}} \right) \left( 1-{{\rm e}^{{\...
3
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0answers
69 views

Deriving Milne's predictor of order four from extrapolation polynomial [closed]

I am trying to derive the following Milne's predictor formula of order four for the differential equation $\frac{dy}{dx}=f(x,y)$ from an extrapolation polynomial of degree four. $$y_{n+1} = y_{n-3}+\...
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0answers
24 views

numerical differentiation of sum of one-dimensional sinusoids with angular frequency close to Nyquist one

Suppose that $f(t) = \sum_i C_i e^{i\omega_i t}$, and $f$ is sampled at certain sampling angular frequency $\omega_s$. All $\omega_i$s are very close to $\omega_s/2$, and thus standard finite ...
2
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1answer
42 views

Sum of difference equation involving hypergeometric functions 1F0

I'm trying to prove the sum of a sequence given by $a_{n+1} = \frac{nb-x}{(n+1)b} a_n$ with $a_1 = 1$. This gives the solution $a_n = \frac{(-x/b)_n}{n!}$. When trying to work out what this sums to, ...
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2answers
179 views

closed form solution of the following iterative equation?

is it possible to obtain a closed-form solution w.r.t. ${P_j:\forall j}$ (or in terms of special functions) for the following equations: $\alpha P_0=P_1$, $\alpha<1$ $\alpha P_j=P_{j+1}+P_{j+2}+\...
2
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0answers
62 views

Geometric unfolding of a difference equation

Does anyone know a link to the (superb) slides of a talk given by Zeeman in 1996, I think (with same title as this question)? It used to be at www.math.utsa.edu/ecz/gu.html .
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1answer
256 views

Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...
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1answer
79 views

Solving the difference equation $h(\vec x)\cdot A(\vec x)=\sum_{i=1}^m A(\vec x - \vec e_i)$

I am trying to solve the following difference equation: $$h(x_1,x_2,x_3,x_4) \cdot A(x_1,x_2,x_3,x_4)=A(x_1-1,x_2,x_3,x_4)+A(x_1,x_2-1,x_3,x_4)+A(x_1,x_2,x_3-1,x_4)+A(x_1,x_2,x_3,x_4-1)$$ with ...
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0answers
22 views

Bifurcations in flows on 2-dimensional torus

I am doing research on bifurcations which appear in flows on the 2-dimensional torus, in particular on such which do not appear in flows on $\mathbb{R}^2$. Can anyone provide some references on ...
4
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1answer
315 views

for which values of $\theta$ does this equation $x_{n+1}=\cos(\theta)x^2_{n}-\sin(\theta)x^2_{n-1}$ have bounded solutions?

I would like to investigate the global behavior of the following equation : $$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$$ where $A(\theta)= \cos(\theta)$ and $B(\theta) =\sin(\theta)$ are nonnegative parameters ...
2
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0answers
63 views

Discrete “difference” equations that involve changes in both shift and scale

A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance: $y[n] = x[n] + y[n-1]$ $Y(z) = X(z) + Y(z) \...
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2answers
896 views

Boundedness of solutions of a difference equation

Is there someone who can show me how I can prove this conjecture? Or at least show me how to do the first implication ? Conjecture: Assume $\alpha,\beta, \lambda \in [0,\infty)$. Then every ...
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0answers
134 views

Help solving a recurrence relation

For solving a related probability problem, I need to solve the following recurrence relation:- $q(r,b,L)=\frac{b}{r+b}q(r,b-1,L)+\sum_{k=1}^{L}\frac{r}{r+b}\times\cdots \times \frac{r-(k-1)}{r+b-(k-1)...
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0answers
79 views

Rational dynamical system with nonnegative paramaters

let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
2
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1answer
150 views

a second order difference equation related to a real polynomials which seems to have only real roots

I am seeking solutions to the following difference equation: $$2c_k-c_{k-1}-c_{k+1}=\ln(k+A)-\ln(k+B)$$ where $A>B>0$. This equation is related to a real polynomial (see here) which I want to ...
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1answer
172 views

Generic way to solve f(x+1) - f(x) = g(x) when g(x) is given [closed]

All I have been looking around for a general way to solve the problem of $f(x+1) - f(x) = g(x)$, where $g(x)$ is given. Has this problem been studied before? If there does not exist such a general ...
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1answer
255 views

Non-linear 1st order difference equation

I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$ I have tried various substitutions, simplifications ...
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2answers
222 views

Resource on Infinite Systems of Difference Equations

I have asked this question previously at Math.stackexchange, but it seems to receive little attention there. In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer ...
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0answers
286 views

Differential Equations vs Difference Equations

My question is: Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain? I mean if I know a solution to the difference ...
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1answer
184 views

Vortex equations on cylinder

Solutions to the vortex equations for a closed Riemann surface are well known (moduli space is a symmetric power). What do we know about solutions on surfaces with boundary or non compact surfaces? In ...
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1answer
231 views

nonlinear delay differential equation

Consider the delay differential equation: $ y_x(x) = \sqrt{y(x-\bar{x})} $ where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter. This equation does not seem to have a ...
3
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2answers
348 views

delay differential equation

I'm looking for exact solutions, if such exist, for the following non-linear delay differential equation (DDE): $ y_x(x) = A y(x-1)^a $ where $ 0 < a < 1 $ and $ A > 0 $ are given constants....
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1answer
493 views

Is exponent of discrete-analytic function also discrete-analytic?

Lets define a discrete analytic function such a function that is equal to its Newton series: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ Is function $g(x)=e^{f(x)}$ also ...
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4answers
838 views

What are other applications of difference equations in other branches of mathematics ?

What are some of interesting results that arise from using difference equations in number theory , Combinatorics or any other field ?
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Are there existing resources on modular-esque recurrence relations?

Does anyone know where I would be able to get information on analyzing a class of polynomial recurrence relations of a form like this? $\begin{align*} f_{n,k}(x) & =a(x)f_{n-1,k}(x)+b(x)f_{n-2,k}(...
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1answer
725 views

Why is there no formula for partial sums of some simple series?

I'm pretty sure that the sequences like $F_n=\sum_{k=1}^n \frac 1k$ are not traces of elementary functions on positive integers (take any reasonable definition of "elementary" you want, just make sure ...
3
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1answer
292 views

Difference equation $A(n,x)=p(x)A(n-1,x-1)+q(x)A(n-1,x)$

I asked this question on MSE, but didn't get enough information. If it is a violation of some norms, let me know, I'll delete it. I'm having problem solving this difference equation. Initially I ...
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1answer
257 views

Extension of polynomial functions

Let $P$ be an interval in $\mathbf{R}$, $n \in \mathbf{N}$. Assume that a function $f: P \rightarrow \mathbf{R}$ satisfies $\Delta^{n+1}_h f(x)=0$ for every $x \in P$ and every $h>0$ such that $x+(...
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Are these two functions equal?

The question here is sparked by the discussion inside this question about indefinite sum(antidifference) of tan(x). A proposed solution was a function $$f_1(x)=ix-\psi _{e^{2 i}}^{(0)}\left(x+\frac{\...
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2answers
620 views

Discrete-analytic functions

I do not know if such concept already exists but lets consider functions which are equal to its Newton series. We know that functions which are equal to their Taylor series are called analytic, so ...
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2answers
184 views

Is there a solution/approximation for the non-linear difference equation $c_n = c_{n-1}+c_{\lceil \alpha n \rceil}$, where $0 < \alpha < 1$?

Is there a solution/approximation for the non-linear difference equation $c_n = c_{n-1}+c_{\lceil \alpha n \rceil}$, where $0 < \alpha < 1$?
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4answers
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Shannon's communication paper and finite differences

In Shannon's 1948 paper "A Mathematical Theory of Communication", early on he derives the equation $$N(t)=N(t-t_1)+N(t-t_2)+\ldots+N(t-t_n).$$ He then says "according to a well-known result in finite ...
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1answer
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Smooth and analytic structures on low dimensional euclidian spaces

So it is relatively easy to show that there exists only one smooth structure on the real line $\mathbb{R}$. So here are 2 natural questions: Q1: Up to equivalence, is there only one real analytic ...
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When does a triangle of numbers have a zero row sum?

Suppose we have a triangle of numbers defined by the recurrence relation $$\left| n \atop k \right| = f(n,k) \left| n-1 \atop k \right| +g(n,k) \left| n-1 \atop k-1 \right| + [n=k=0],$$ for some ...
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0answers
246 views

A second order non-linear difference equations

I am trying to explicitly solve [ if possible and if the solution exists ] a second order non-linear difference equation of the form : $ a _ {n+2} ^2 + a_{n} ^2 + K = \lambda a_{n+1} $, where K ...
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4answers
1k views

Laplace's summation formula

I recently came across the following formula, which is apparently known as Laplace's summation formula: $$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{...
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Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?

If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
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3answers
2k views

Solving a general two-term combinatorial recurrence relation

What is known about explicit (not necessarily closed-form) solutions to the recurrence $$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$ with initial condition $R_0^0 = 1$ and ...
2
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1answer
189 views

Elementary proof of bounds on discrete derivative applied to $(1+n)^s$

I would like to show that for $s \in \mathbb{R}$ and a nonnegative integer $k$ $$ \triangle^k ((1+n)^s) \lesssim (1+|n|)^{s-k} $$ where $\triangle$ is the discrete derivative, i.e. $\triangle^1 ((1+n)...
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1answer
2k views

Solving partial difference equation

I am trying to solve the following partial difference equation: $$A_k^{n+1}=(k+1)A_{k+1}^n+(n+2-k)A_{k-1}^n $$ with initial condition: $$\begin{cases} A_0^0&=1\\ A_1^0&=1 \end{cases}$$ I ...
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1answer
316 views

[Numerical Mathemtics] How to solve hexagonal central differences

I want to simulate a 2d linear wave equation on a circle ($\displaystyle\frac{\partial^2 z(x,y,t)}{\partial t^2}=v^2\cdot\left(\displaystyle\frac{\partial^2 z(x,y,t)}{\partial x^2}+\displaystyle\frac{\...
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5answers
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Discrete version of Ito's lemma

Could anyone give me some references where I could find (a) discrete version(s) of Ito's lemma (b) a proof how it converges to the continuous form in the limit (c) its usage within stochastic ...
12
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3answers
865 views

Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$?

Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$? (Asked by bcross at math.iuiui.edu on the Q&A board at JMM.)