# Questions tagged [recurrences]

The recurrences tag has no usage guidance.

151
questions

**0**

votes

**0**answers

389 views

### Two conjectural identities involving $\zeta(3)$ and the golden ratio $\phi$

Let $\phi$ be the famous golden ratio $\frac{\sqrt5+1}2$, and let
$\zeta(3)=\sum_{n=1}^\infty\frac1{n^3}.$
In 2014, in the paper
Zhi-Wei Sun, New series for some special values of $L$-functions, ...

**4**

votes

**1**answer

103 views

### Nonlinear recurrence

I encounter the following recurrence
\begin{equation} \tag{1}\label{eq:1}
h_{j+1} = h_{j} ( 1 - 1/j - c h_j ), \quad j \geq j_0,
\end{equation}
with $h_{j_0}>0$, $c>0$ and $0< 1-1/j_0 - c h_{...

**6**

votes

**0**answers

80 views

### A question about the span of a sequence of polynomials satisfying a linear recurrence

Let F be a finite field and A(n) in F[t], n in N, be defined by a linear recurrence with coefficients in F[t], together with initial conditions. Is there a decision procedure for determining whether ...

**0**

votes

**0**answers

99 views

### How to define $``\ll"$ in higher dimension?

Fix $C>0,$ we say $n \sim m $ if $|n-m| < C$ $ (n, m \in \mathbb Z)$ and $n\ll m$ if $n-m \leq C$ and $n\gg m$ if $n-m \geq C.$
Let $n_1, n_2, n_3, n_4 \in \mathbb Z$.
Assume that $|n_1-...

**1**

vote

**1**answer

163 views

### An iterative argument involving $f(n + 1) - f(n) $

I am working with an argument involving an inequality of the form:
$$ f(n + 1) \leq f(n) + C (f(n))^{1 - \frac{1}{\gamma}} \qquad(\ast)$$
where $f$ is a positive function, $\gamma > 0$ and $C >...

**0**

votes

**0**answers

19 views

### Recurrence relation for the asymptotic expansion of an ODE

I want to solve for the asymptotic solution of the following differential equation
$$ \left(y^2+1\right) R''(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$
as $y\...

**3**

votes

**1**answer

190 views

### A 2d recurrence equation representing a step-free continuous-time Markov process on N^2 with frequency-dependent rates

$\textbf{Background:}$ Consider a particle in $\mathbb{N}^2$ starting at a point $(x,y)$ that can only move one step at a time along the lattice. The particle moves each up or down in continuous time ...

**0**

votes

**1**answer

70 views

### Solving an recursive sequence [closed]

I have an recursive sequence and want to convert it to an explicit formula.
The recursive sequence is:
$f(0) = 4$
$f(1) = 14$
$f(2) = 194$
$f(x+1) = f(x)^2 - 2$

**4**

votes

**1**answer

200 views

### Solving Linear Matrix Recurrences

Question:
Are there standard techniques available for solving the following kind of linear matrix recurrence relations:
$$M_1,\cdots,M_k\ \in\ \mathbb{R}^{m\times n}$$
$$ A_1,\cdots,A_k\ \...

**0**

votes

**0**answers

106 views

### Sum of products of binomials satisfies recurrence relation?

i need to know if the sequence $(a_n)_{n \geq 0}$ defined by $$a_n =\sum_{s=3n}^{16n} C_{15n}^{s-3n} \;
C_{14n}^{s-2n} \; C_{12n+s}^{12n} $$ satisfies a recurrence relation ( type sequence Apery) or ...

**2**

votes

**3**answers

310 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}, ...

**3**

votes

**4**answers

178 views

### Solution of a 2D Recurrence sequence

Can we solve the following recurrence relation:
$$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$
with $a_{0,n}=a_{m,0}=0$? If not, can we get an estimate of the growth of $a_{m,n}?$
I encountered this ...

**4**

votes

**1**answer

141 views

### A 2nd order recursion with polynomial coefficients

I'm hoping to find "exact" an solution to the following simple recursion:
$q_m(j+1) = m \cdot q_m(j) + j(j+m)\cdot q_m(j-1)$
with initial data $q_m(0) = 1$, $q_m(1)=m$, where $m \geq 0$ is an ...

**-1**

votes

**1**answer

113 views

### If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is

Let $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer.
In a previous post I asked if $p_n(a,b)$ was a ...

**5**

votes

**3**answers

307 views

### Enumeration of lattice paths of a specific type

One of the approaches to "Special" meanders led (in particular) to the following question:
What is the number $a_{m,n}(\ell)$ of $\ell$-step paths from $(1,1)$ to $(m,n)$ using the ...