The recurrences tag has no wiki summary.

**5**

votes

**0**answers

121 views

### Degenerate linear recurrence sequences

Let $(u_n)_{n \geq 0}$ be a linear recurrence given by
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n-k} \quad \forall n \geq k ,$$
where $u_0, \ldots, u_{k-1}, a_1, \ldots, a_k \in \mathbb{Z}$. We recall ...

**5**

votes

**0**answers

144 views

### Non-crystallographic cluster algebras

Background
Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed ...

**3**

votes

**0**answers

100 views

### ``Occasional'' Laurent phenomenon

This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?).
He asked ...

**3**

votes

**0**answers

178 views

### Arithmetical properties of certain recurrence relations

Consider the following recurrence relation:
$$(4i+6j+1)(2i+3j)a_{i,j}=3j a_{i+1,j-1}-2i a_{i-1,j}+32i(i-1) a_{i-2,j+1}, i\ge0, j\ge0,$$
$$a_{0,0}=1.$$
This equation appeared in the article ...

**2**

votes

**0**answers

152 views

### Master theorem for probabilistically inspired recurrences

Is there a general solution for multi-variable recurrences of the following form which come from Markov chain analysis?
$f(i,j,k) = p_1 f(i-1, j,k) + p_2 f(i, j-1,k) + p_3 f(i,j,k-1) + p_4 ...

**2**

votes

**0**answers

93 views

### Perturbation analysis for three term recurrences

Jacobi polynomials, denoted by $J^{(\alpha,\beta)}_n$, on $[-1,1]$ satisfy a three term recurrence
$$ J_{n+1}^{(\alpha,\beta)}(x) = (A_n+B_nx)J^{(\alpha,\beta)}_n + C_nJ_{n-1}^{(\alpha,\beta)}(x), ...

**2**

votes

**0**answers

159 views

### Partitions of central sets via dynamical systems

In the book "Recurrence in Ergodic Theory and Combinatorial Number Theory", 1981, Furstenberg introduced the notion of central sets.
He proved in Theorem 8.8 that in each finite partition of ...

**1**

vote

**0**answers

59 views

### Number of digits in $n$-th term of generalized Fibonacci/Narayana sequence

Let $d$ be a nonnegative integer, and let the sequence ${F_d(n)}$ be defined as follows:
$F_d(n) = 1$ for $n = 0, 1, \ldots, d$
$F_d(n) = F_d(n-1) + F_d(n-1-d)$ for $n>d$
For $d=0$ the sequence ...

**1**

vote

**0**answers

83 views

### Definition of primitive divisor of a Lucas sequence

If $\{a_n\}_{n=1}^\infty$ is a sequence of integers, then the definition of primitive divisor of one of its terms is quite natural:
Def. 1. A prime number $p$ is a primitive divisor of $a_n$ if $p ...

**1**

vote

**0**answers

125 views

### Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$
\begin{array}{cccccccccc}
1 & = & ...

**1**

vote

**0**answers

74 views

### Convergence and summation of power towers or Ackermann functions

I deleted this at StackExchange as it seems advanced or not very relevant for there and I want to ask here.
Let z be a complex number, $Im(z)\neq 0$ or $Re(z)<4$, m be a natural number, let ...

**1**

vote

**0**answers

99 views

### Trying to get an idea of the maths I could use for this optimization problem

Firstly, apologies if some of the notation or terminology is odd, or if I am defining functions that have standard notation associated with them already - I am not familiar with the concepts in this ...

**0**

votes

**0**answers

43 views

### Finding a stochastic differential equation as limit of a discrete stochastic equation

I'm dealing with the following problem:
Choose $Z_0 \in [0,1]$ and define a process governed by the following discrete stochastic equation:
$Z_{k+1}-Z_k=P_k(1-2Z_k)$
where $P_k=0$ with probability ...

**0**

votes

**0**answers

90 views

### 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 ...

**0**

votes

**0**answers

62 views

### For which recurrence relations is it decidable whether a formal power series has a maximal zero coefficient?

In this MSE question, I asked whether one can prove that a generating function has infinitely many coefficients equal to zero. The answer given (and accepted) to that rather broad question was “No”.
...

**0**

votes

**0**answers

43 views

### A question on bivariate recurrence

Is there a way to get a closed form of this recurrence (albeit approximate):
$$A(n ,k ) = {1 \over {d^k}}A(n-1, k-1) + (1 - {1 \over {d^k}})A(n-1,k)$$
Where, $n,k \ge 0$ and $d \ge 2$ are integers. ...

**0**

votes

**0**answers

72 views

### Bessel function stable recurrence relation

I want to compute hypergeometric 1f1 function using
I can't use direct computation of Bessel functions due to complexity. I want to use BesselJ recurrence relation:
But forward recursion is ...

**0**

votes

**0**answers

100 views

### Bound a sum of a serie defined by a recursive integer function

I'm using a recursive function $f: \mathbb{N} \rightarrow \mathbb{N}$, that is defined as
\begin{equation}
f(n)=\lceil \log(f(n-1)) \rceil +f(n-1)
\end{equation}
where $f(1)=F\in \mathbb{N}$, and ...

**0**

votes

**0**answers

73 views

### Approximate closed-form solution for a recurrence

Find an (approximate) closed-form solution for $S(m, b)$.
$$S(m,b)=\sum_{i=0}^{\lfloor (e-1)/2\rfloor}{e \choose i}S(m-1, b-i) \quad +
\sum_{i=\lfloor (e-1)/2\rfloor+1}^{\min(b,e)}{e\choose ...

**0**

votes

**0**answers

107 views

### Recursive relation using successor function

What is the recursive relation for
H(m)=2^(m^2)
using successor function
recursive relation for multiplication:
mult(x,0)=0;
mult(x,S(y))=add(x,mult(x,y))
recursive relation for addition:
add(x,0)=x;
...