The difference-equations tag has no wiki summary.

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### 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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### 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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### 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 ...

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### 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}} \\ ...

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### 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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**1**answer

163 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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**1**answer

243 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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### 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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222 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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**1**answer

158 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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189 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 ...

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**2**answers

307 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 ...

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**1**answer

479 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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### 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) & ...

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695 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 ...

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### 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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**1**answer

254 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 ...

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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 ...

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### 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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### 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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### 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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### 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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### What is known about fractions in difference equations?

I'm using the following book as reference:
** Walter G. Kelly and Allan C. Peterson Difference Equations Second Edition, Harcourt Academic Press, 2001. **
I gather from the book and from research ...

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### 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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### 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) - ...

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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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### 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 ...

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### 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 ...

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### 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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### [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 ...

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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 ...

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### 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.)