12 votes
Accepted

Integer Polynomial solutions to functional equation

Note that $L(f):=(2x+1)^2f(x+1)-4x(x+n+1)f(x)$ is a linear operator which maps the $\mathbb{Q}$-linear space $\pi_n$ of polynomials $h(x)\in \mathbb{Q}[x]$ of degree at most $n$ to itself. Assume that ...
10 votes
Accepted

Combinatorial Identity with Connection Coefficients and Falling Factorial $\langle i x\rangle_n$

It is very probable that what is written below is the simplification of Darij's argument. I use the notation $x^{\underline{n}}=x(x-1)\dots(x-n+1)$ [as in Knuth's books] for the falling factorial, and ...
9 votes
Accepted

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 ran a small computation: The following plot is created as follows: For each point $r(\cos \theta, \sin \theta)$, I use $x_{-1} = 0$, $x_0 = r$ as initial values, and $\theta$ as the parameter. The ...
8 votes

Combinatorial Identity with Connection Coefficients and Falling Factorial $\langle i x\rangle_n$

This is correct. Let me prove a more general fact: Theorem 1. Let $i$, $j$ and $n$ be three nonnegative integers such that $i\leq n$ and $j\leq n$. Let $P\in\mathbb{Q}\left[ X\right] $ be a ...
8 votes
Accepted

Does this deceptively simple nonlinear recurrence relation have a closed form solution?

The sequence $a_n$ for $n\geq 1$ has the following formula: $$a_n=\left\lfloor \sqrt{2n}+\tfrac{1}{2}\right\rfloor +\frac{\left\lfloor \frac{1}{2} \left(\sqrt{8 n-7}+1\right)\right\rfloor-\left\lfloor ...
6 votes
Accepted

A strange two-variable recursion

First, we note that $f(m,n) - f(m,n-1)$ does not depend on $n$, so for a fixed $m$ we can write $f(m,n) = \alpha_m n + \beta_m$ for some $\alpha_m,\beta_m$ which depend only on $m$. Imposing the ...
3 votes

Does this deceptively simple nonlinear recurrence relation have a closed form solution?

I have decided to call this sequence $\Theta_n$ for the triangular-harmonic number sequence because it clearly has properties related to both triangular and harmonic numbers. I had been studying it ...
3 votes

Does this deceptively simple nonlinear recurrence relation have a closed form solution?

A similar, possibly simpler closed form is the following: set $$b_n=\left\lfloor\frac{1+\sqrt{8n-7}}{2}\right\rfloor,$$ then $$a_n=\frac{b_n+1}{2}+\frac{n-1}{b_n}.$$ It is not hard to derive this from ...
3 votes
Accepted

Finite realization of irrational transfer functions

There is at least one (rather trivial) way to formalize "implementability" that has the property that the only implementable functions are rational. Suppose that the implementation is defined as a ...
  • 54.3k
3 votes
Accepted

A problem from linear algebra and difference equations

I'm going to use the notation defined above here. When a letter is used alone, it denotes the entire sequence. Claim: If $C^{[j]}(f, g) = 0$ for all $j \leq k$, then $g^{[k]}$ is a linear ...
  • 4,706
3 votes

Difference equation and formal series

In general, this question makes sense at a formal level near $\infty$ only, i.e. for power series involving negative powers of $x$. Setting $z:=\frac{1}{x}$, the equation becomes a so-called ...
3 votes

Difference equation and formal series

The equation $f(x+1)-f(x)=g(x)$ can be viewed as a "discrete derivative", hence a solution $f(x)$ can not be expected to be unique. \begin{align} f(x+1)-f(x)&=\sum_{j\geq0}f_j\sum_{k=0}^j\binom{j}...
3 votes

Boundedness of solutions of a difference equation

Here is just an idea, which may or may not work. Suppose that $\beta=\lambda>0$. Let $t_n:=z_n/\beta$ and $c:=\alpha/\beta^2$. Then the dynamics can be rewritten as $$(!)\qquad t_{n+1}=\frac{c+t_n+...
2 votes

closed form solution of the following iterative equation?

There is no solution to the problem in the first version of the OP. Proof: We have to consider two cases: Case a) at least one of the $P_j$ is zero The let $k$ be the smallest index for which $...
2 votes

closed form solution of the following iterative equation?

If you set $\alpha=1$ and impose the condition that $P_{2i}=P_{2i+1}$, then you get the sequence $P_0,P_2,P_4,... = 1,1/2,0,1/4,-1/4,0,1/4,1/8,-3/8,-1/8,1/8,0,...$. Define $Q_{2i}=P_{2i}-P_{2i-2}$. ...
  • 27.6k
2 votes
Accepted

How to solve this conditional recurrence relation?(two variable and conditions)

Under the assumption that $F(2i,n) = 0$ when $i<4$ or $2i>n$, the first and second cases in the recurrence become partial cases of the third one. Hence, the recurrence reduces to $$F(2i,n) = \...
2 votes

High order difference operator applied to 1/u

You can apply the formula $$\Delta^p f(x)=\sum_{k=0}^p{p\choose k}(-1)^{p-k}f(x+k)$$ to $f(x)=1/u(x)$. Further simplification will need knowledge how $u(x)$ depends on $x$. For example, if $u(x)=x$ ...
1 vote
Accepted

Unique zero solution to a difference equation via Laplace transform

Your condition can be rewritten as the system of three equations: $$au(x)+bu(x+1/2)=0\ \forall x\in(0,1/2), \tag{1}$$ $$au(x)+cu(x-1/2)=0\ \forall x\in(1/2,1), \tag{2} $$ $$au(1/2)=0. \tag{3} $$ In ...
1 vote
Accepted

Is this recurrent sequence decreasing?

Since $x_0=0$, it will be convenient to do summation starting from $t=0$. Denoting $r:=1-p-q$, we have \begin{split} S_n &= \frac1n\sum_{t=0}^n \left(pr^{2t} + (q-p)r^{t} - q\right) \\ &=\frac{...
1 vote
Accepted

On difference identities and $[K:F]$

The answer is yes and appears in R. Cohn's book 'Difference algebra' Lemma II p. 201.
  • 1,525
1 vote
Accepted

Sum of difference equation involving hypergeometric functions 1F0

The solution you give for $a_{n+1} = \frac{nb-x}{(n+1)b} a_n$ has a missing factor, I think it should read $$a_n=-\frac{b}{x}\frac{(-x/b)_n}{n!}.$$ Then the sum over $n$ equals $$\sum_{n=1}^\infty ...
1 vote
Accepted

boundedness of a nonlinear recursive sequence

You can view this difference equation as the Euler method for the IVP $y'=\gamma y^2$, $y(0)=1$, on the interval $0\le t\le 1$, using a grid of width $1/N$ and setting $x_k=y(k/N)$. By solving the ...
1 vote
Accepted

Derivative in terms of finite differences

This is a special case of well-known general formula for the n-th derivative via differences and Stirling numbers. Cf. good books on finite differences, e.g. of Gelfond or Jordan.
  • 1,386
1 vote

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

The equation $$h(\vec x)\cdot A(\vec x)=\sum_{i=1}^m A(\vec x - \vec e_i)$$ represents a general translation on the function $A(\vec x)$ and can be stated introducing the operator $$h(\vec x)=\exp\...
  • 1,622
1 vote

Discrete version of Ito's lemma

Proposition 1.13.1 of this book. The result is based on the Clarke-Ocone formula and is a discrete-time (as opposed to a simple process in continuous time approach, as posted by most of the others). ...
  • 4,973
1 vote

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

After this question popped up again, it seemed to me to scream out for a use of the Fourier Transform (FT). I have decided to post this as an answer, since this approach is transparent and provides a ...
  • 306

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