The recurrences tag has no usage guidance.

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### 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 ...

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**3**answers

993 views

### Linear Recurrence Relations in 2 Variables with Variable Coefficients

Consider the following recurrence relation:
$$-2a_{n,m} +a_{n-1,m}+a_{n,m-1}=0,$$
where $a_{n,m} \in \mathbb{C}.$I would like a purely combinatorial way to understand the subspace of solutions to this ...

**2**

votes

**2**answers

1k views

### Methods for solving two variable recurrence

I have a recurrence
$$f(i,j) = 1+ \frac{N-i}{N} f(i,j-1) + \frac{i}{N} f(i-1, j)$$
$$f(i,0) = 0$$
$$f(0,j) = j$$
I would like to compute $f(N,M)$ in terms of N and M. The system is defined for ...

**3**

votes

**1**answer

345 views

### Books request on nonlinear recurrence relations.

Hi, do you have some sort of a bibliography on advanced techniques in recurrence equations, such as nonlinear ones and others?
As I see it recurrence equations are quite similar to differential ...

**1**

vote

**2**answers

113 views

### A recursive Double sequence related to uniform Cardinal B-spline

Given a sequence $A_n(k)$ defined as follows:
$A_0(0)=1$, $A_0(k)=0$ for all nonzero integers $k$ and
$$A_{n}(k)=(n+1-k)^2A_{n-1}(k-1)+2(n(n+1)-k^2)A_{n-1}(k)+(n+1+k)^2A_{n-1}(k+1)$$
for all positive ...

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101 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 ...

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99 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**

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**0**answers

168 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 ...

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**5**answers

494 views

### Uniqueness of values in recurrence relations

Given an integer $k > 1$, define the sequences $X(k,n), Y(k,n)$ as follows:
$a=4k-2,$ $y_0 = 1,$ $y_1 = a + 1,y_n = ay_{n-1} - y_{n-2}$
$b = 4k + 2,$ $ x_0 = 1,$ $x_1 = b - 1,$ $x_n = ...

**16**

votes

**3**answers

671 views

### A question on the Laurent phenomenon

This question is motivated by my answer to 109955. It gives a
recurrence relation satisfied by a function $P(n)$ whose terms a
priori are rational functions (of three variables) with complicated
...

**2**

votes

**2**answers

1k views

### How to solve a specific multivariate recurrence relation (or general ones)

How do you solve this recurrence (or multivariate recurrences in general)? Note that $p\in[0,1]$ and $n\in\mathbb{N}$ are given constants, where $np\leq 1$.
...

**2**

votes

**1**answer

701 views

### Three term recurrence relation.

For given $n,\ell\in\mathbb N_0$, I am interested in studying the following recursion relation for some $\mu\in\mathbb R$:
$$\sqrt{-1} \tfrac{j(\ell-j+1)(n-j+1)(n+j+1)}{2(2j-1)(2j+1)} a_{j-1} - ...

**1**

vote

**1**answer

197 views

### On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that
$$
A_0 = A_1 = 0;\quad A_n = ...

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**4**answers

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### Recurrence T(N)=T(N/LOGN)+1 [closed]

I'm stuck on the following reccurrence :
T(N)=T(N/LOGN)+1,IF N>2
T(N)=0,if 0<=N<=2
I need a function T(N) for all N>0
Is there some method for solving ...