Questions tagged [difference-equations]
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.
86 questions
2
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3
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Reference for relation between $ \left( \frac{1}{x} D \right)^{n+1} $ and the Bessel coefficients
Background
In the OEIS sequence A001498, the coefficients of the Bessel polynomials are described. They adhere to the formula $$ a(n, k) = \frac{(n+k)!}{\left(2^k \ (n-k)! \ k! \right)} \tag{1} \label{...
- 5,065
1
vote
0
answers
276
views
Increasing monotonicity of the ratio beween two adjacent terms of a sequence defined by a recursive relation
For $n\in\mathbb{N}$, let $b_1=\frac{1}{3}$, $b_2=\frac{13}{90}$, $b_3=\frac{251}{2835}$, and
\begin{equation}\label{Wang-Fei-Recur}
b_{n+1}=\frac{1}{n+1}\Biggl[\frac{n+1}{2n+3}-\sum_{k=0}^{n-1}\frac{(...
- 1,209
2
votes
2
answers
247
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Proof of convergence of replicator type II dynamics
One of the evolution equations used in evolutionary game theory is the Replicator type II dynamics
$$
x_i(t+1)=x_i(t)\frac{(Ax(t))_i}{x^T(t)Ax(t)}
$$
where $A$ is an $M\times M$ payoff matrix with ...
- 85
2
votes
0
answers
80
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Strongly difference-closed field
Given a field $F$ and an automorphism $\phi:F\to F$, according to the definition in Kedlaya's book, a strongly difference closed field means that any dualizable (i.e., admits a dual) difference module ...
8
votes
1
answer
263
views
Sheaf theoretic approach to difference equations
Differential equations can be reformulated using the language of $\mathcal{D}$-modules. Is there a sheaf-theoretic approach to $q$-difference equations?
- 1,593
2
votes
2
answers
323
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Great literature on discrete dynamical systems and/or qualitative theory of difference equations
I am asking for the great literature on topics of discrete dynamical systems and/or qualitative theory of difference equations especially aimed on pure mathematicians. Could you please give me some ...
3
votes
1
answer
408
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Asymptotic behavior of a recursion
Let $x_n(0)=1$,
$$
x_n(N+1) = \frac{1}{N+1}\sum_{k=0}^N \sum_{j=1}^n x_j(k)x_{n+1-j}(N-k) + \frac{10}{N+1} x_{n+1}(N) , \quad\quad N\ge 0 .
$$
So the recursion is on $N$, and at each level, we compute ...
- 25.2k
0
votes
0
answers
122
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General solution of partial difference equation that generates Eulerian numbers
I have a question on the partial difference equation
$$f(n+1, k) = (k+1) f(n,k) + (n+1-k)f(n,k-1)$$
where $(k, n) \in \mathbb{Z}^2$.
It is well known, that under some boundary conditions this equation ...
6
votes
1
answer
228
views
Solving a specific difference equation
Dynamics
I have the following recursive discrete dynamics for $t=1,2,\dots,d$ where we can assume $d>100$ or even $d\to\infty$ if needed.
\begin{align}
x_{1} =1,\quad \quad
x_{2} = \frac{1+\...
- 673
14
votes
3
answers
2k
views
Solve recurrence relation
I am trying to figure out how to solve recurrence relations of type
$$f(n) = af(n-1) + (bn + c)f(n-2)$$
I will be grateful for any tips or books.
- 169
1
vote
0
answers
126
views
Pontryagin's maximum principle for discrete systems: reference request for general case [closed]
I am reading the articles:
Optimal control for systems described by difference systems, Hubert Halkin, Advances in Control Systems, Vol 1, Academic Press, New York-London, 1964, Pages 173-196, ...
- 111
4
votes
1
answer
352
views
Double q-analog of Pochhammer
Has the function
$$(z;q_1,q_2)_\infty := \prod_{n_1,n_2=0}^\infty (1-z \, q_1^{n_1} q_2^{n_2}), \quad |q_1|,|q_2|<1$$
been studied in the math literature? For example, does it obey any difference ...
- 525
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votes
2
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573
views
Change of variable formulas in discrete calculus?
Crossposted from MSE.
In discrete calculus one defines the $h$-difference operator $$\Delta_h[f(x)] = f(x+h) - f(x)$$
and we often define $\Delta = \Delta_1.$ We can similarly define the indefinite ...
- 123
2
votes
0
answers
95
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Coefficient growth upper bound of a recurrence relation
Consider the recurrence relation one can obtain from the radial Schrödinger equation for the hydrogen atom, where $\psi(r)=\sum_{n=0}^{\infty} a_nr^n$:
$$(n+3)(n+2)a_{n+2}+2a_{n+1}=2|E|a_n, n\geq0$$
...
- 121
5
votes
1
answer
313
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Polynomial solutions to a difference equation
This question may look unmotivated, but is connected with continued fractions for $\pi^2$.
Let $n$ be a nonnegative integer, and consider the difference equation
$$(x+2n+4)(x+n+1)P(x+1)-(x-1)(x+n-1)P(...
- 14k
0
votes
1
answer
219
views
Solutions of complex linear difference equations
I'm wondering what the solutions of complex linear difference equations like
\begin{equation}
f(z+\eta_1)+f(z+\eta_2)+\ldots+f(z+\eta_n)=0,\ \ \ \eta_1 \cdots\eta_n \in \mathbf{C}
\end{equation}
look ...
- 3
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votes
2
answers
611
views
What is the inverse of a triangular matrix whose nonzero elements are binomial coefficients? What is the closed-form solution to a recursive relation?
Let
\begin{equation*}
\begin{split}
M_m
&=\begin{pmatrix}
-\binom{1}{0} & \binom{2}{0} &-\binom{3}{0} &\dotsm & (-1)^{m-1}\binom{m-1}{0} & (-1)^m\binom{m}{0}\\
0 & \binom{2}...
- 1,209
2
votes
0
answers
133
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What are the limits of what the theory of time-scale calculus can capture?
Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article 1 and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
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1
vote
0
answers
202
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Time-scale calculus (an similar approaches - measure chains) on more general "time" sets
Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article [1] and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
- 429
2
votes
1
answer
380
views
Difference equation satisfied by discrete harmonic functions on square lattice
A function $f:\mathbb Z^2 \rightarrow \mathbb R$ is said to be discrete harmonic if it satisfies the discrete Laplacian equation
$$
\Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~.
$$
...
- 521
1
vote
0
answers
63
views
Help with a surface of delay differential equations
This question is difficult for me to phrase, as it's very much outside of my mathematical purview. This is a question which intersects directly with my research, but as I work predominantly in ...
- 398
2
votes
1
answer
80
views
High order difference operator applied to 1/u
I need a formula for $\Delta^k \frac 1 u$, where $u(x)$ is a strictly positive function, $\Delta^k$ is the difference operator defined recursively as $\Delta^k=\Delta^1 \Delta^{k-1}$ and $\Delta^1 u(x)...
- 9,142
0
votes
1
answer
175
views
Unique zero solution to a difference equation via Laplace transform
We want to prove that
the unique solution to the following difference equation is the null one:
$$
au(x)+b\mathbf{1}_{(0,\frac{1}{2})}(x)u(x+\frac{1}{2})+c\mathbf{1}_{(\frac{1%
}{2},1)}(x)u(x-\frac{1}{...
- 617
1
vote
0
answers
431
views
Eigenvalues of an (almost) pentadiagonal symmetric Toeplitz matrix
I am looking for analytic expressions for the eigenvalues of matrices of the form
$$A = \begin{bmatrix}
6 & -4 & 1 & 0 & 0 & 0 & 0 \\
-4 & 6 & -4 & 1 & 0 &...
- 41
1
vote
0
answers
199
views
Uniform lower bound for the distance between terms of a linear recurrence sequence
Let $(u_k)_{k=0}^\infty$ be a non-degenerate linear recurrence sequence of algebraic numbers. Denote by $\alpha_1,\dots,\alpha_m$ its characteristic roots, and suppose that $\alpha := \underset{i}{\...
- 11
1
vote
0
answers
50
views
Prove monotonicity of a system of difference equations
I have the following system of difference equations. Fix any $(\alpha,K,p)\in\mathbb R_+\times\mathbb R_+\times (0,1)$. Let $u_0=1, \nu_0=0$. The sequence $(a_t,w_t,\nu_t,u_t)_{t\geq 1}$ is defined as ...
- 389
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votes
0
answers
112
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Rational zeta series and differential-difference equations
In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$
A variation of the above identity arises by ...
- 5,065
-1
votes
1
answer
183
views
Is this recurrent sequence decreasing?
Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $...
0
votes
1
answer
647
views
How to solve this conditional recurrence relation?(two variable and conditions)
I am trying to solve the following recurrence relation
$4 \leq n \;\; \; \; \; \;$ and $\; \; \; \; 2\leq i \leq \lfloor{\frac{n}{2}}\rfloor$
$F(2i,n)=$
$\begin{cases}
\frac{1}{2(2i)-5}F(2i-2,...
- 3
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votes
0
answers
132
views
discrete Fourier transform for composition of differential operators on a grid
This question pertains to stability analysis of finite difference methods using the discrete Fourier transform.
Suppose I have a convection diffusion equation of the form:
(1) $\hspace{.5in}u_t + \...
- 121
2
votes
3
answers
586
views
Does this deceptively simple nonlinear recurrence relation have a closed form solution?
Given the base case $a_0 = 1$, does $a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor}$ have a closed form solution? The sequence itself is divergent and simply goes {$1, 2, 2+\frac{1}{2}, ...
- 397
1
vote
0
answers
119
views
General solution for first-order difference equation
I have the following first-order difference equation
$$y_{t} = \frac{x_{t}}{1-\rho L} + \epsilon_{t}$$
where $L$ denotes the backshift operator, i.e., $L(x_{t}) = x_{t-1}$. I can obtain a solution ...
- 111
1
vote
1
answer
279
views
On difference identities and $[K:F]$
Let $(K,\sigma)$ be a difference field of characteristic $0$, i.e. equiped with field morphism $\sigma:K\rightarrow K$. Assume that $K$ satisfy a non-trivial univariate difference polynomial identity $...
- 1,585
1
vote
0
answers
58
views
How to change difference equation time steps when rearranging?
I am using difference equations to solve SDOF systems.
I have the system $$m\ddot{y_i}+c\dot{y_i} + ky_i = x_i$$
Using the difference equation results for the derivatives,
I am meant to end up with
...
- 11
5
votes
1
answer
1k
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A strange two-variable recursion
In some work I was doing with a colleague the following function of two natural number variables, defined by a recursion, came up and we have no clue how to solve it. Any suggestions or improvements ...
- 40.1k
6
votes
0
answers
460
views
Hrushovski's proof of the Manin-Mumford Conjecture
For my master's thesis, I am studying Hrushovski's model-theoretic proof of the Manin-Mumford Conjecture. Among the references I have used are the following:
Lecture notes 'Model Theory of Difference ...
- 61
2
votes
3
answers
317
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Difference equation and formal series
For a given formal series $g(x)=\sum_{k=0}^\infty g_k x^k$ I would like to find a formal series $f(x)=\sum_{k=0}^\infty f_k x^k$ such that they satisfy the difference equation
$$
f(x+1)-f(x)=g(x).
$$
...
- 1,343
8
votes
3
answers
789
views
Combinatorial identity with connection coefficients and falling factorial $\langle i x\rangle_n$
Let $j, k ,n$ be nonnegative integers such that $0 \leq j, k \leq n \leq k +j $. Pick integer $m$ such that $0 \leq m \leq k + j - n$.
Let $\langle x \rangle_m$ denote the falling factorial $x(x-1)\...
- 1,247
2
votes
1
answer
190
views
A problem from linear algebra and difference equations
Let $A$ be a linear second-order difference operator acting on the space of complex sequences as
$$(Af)_{n}=f_{n-1}+a_{n}f_{n}+f_{n+1}, \quad n\in\mathbb{Z},$$
where $a_{n}\in\mathbb{C}$. Further, let ...
- 2,208
1
vote
0
answers
289
views
Bounds for Discrete Poisson Kernel of a Square
I am having difficulty in proving the lower bound of the discrete Poisson kernel of a square denoted as $H$ described below. It is stated in Gregory F. Lawler's Randomm Walk and the Heat Equation as ...
- 2,363
1
vote
1
answer
226
views
boundedness of a nonlinear recursive sequence
Consider a real sequence $(x_k)$ for $k=0,1,2,\dots,N$ as $x_0=1$ and for $k>0$
$$ x_k=x_{k-1}+\frac{\gamma}{N}x_{k-1}^2,\qquad (\gamma>0).$$
I wonder to show that the sequence is bounded as $N\...
- 105
2
votes
0
answers
405
views
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
$$
f(n,m) = \begin{cases} f(n, \...
- 305
9
votes
3
answers
717
views
Finite realization of irrational transfer functions
In the field of digital signal processing, linear time-invariant systems play a distinguished role. These are the systems for which there exists an impulse response, a function $h:\mathbb{Z}\to\...
- 8,631
1
vote
1
answer
506
views
Integer Polynomial solutions to functional equation
Recently I came across a functional equation which always has a polynomial with integer coefficients solution.
Let
$$
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).
$$
Problem:
...
- 2,193
2
votes
1
answer
147
views
how to solve this equation? [closed]
$Q=-{\frac {q \left( {{\rm e}^{{\it nb}\,{\it nv}\,\theta\,{\it Td}}}-1
\right) }{1-{{\rm e}^{\theta\,{\it Td}}}} \left( 1-{{\rm e}^{{\frac {
Q\theta\,{\it Td}}{p}}}} \right) \left( 1-{{\rm e}^{{\...
- 21
3
votes
0
answers
197
views
Deriving Milne's predictor of order four from extrapolation polynomial [closed]
I am trying to derive the following Milne's predictor formula of order four for the differential equation $\frac{dy}{dx}=f(x,y)$ from an extrapolation polynomial of degree four. $$y_{n+1} = y_{n-3}+\...
- 131
2
votes
1
answer
136
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, ...
- 45
5
votes
2
answers
564
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
votes
0
answers
133
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 .
- 21
5
votes
1
answer
556
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 ...
- 6,132