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 ...

- 91k

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 ...

- 91k

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 ...

- 14.8k

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
...

- 31.5k

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 ...

- 155k

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 ...

- 1,247

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 ...

- 387

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 ...

- 393

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 ...

- 5,040

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}...

- 39.1k

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+...

- 85.1k

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) =
\...

- 27.5k

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 $u(x)=x$ ...

- 155k

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 ...

- 85.1k

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{...

- 27.5k

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 ...

- 155k

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 ...

- 17.5k

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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