The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
1answer
60 views

recursive equation to solve( similar to combinatorics)

I have this recursive equation: $$\begin{align*} F(m,n)=F&=F(m,n-1)+F(m-1,n)-F(m-1,n-1-m) \\\\ F(m,0)&=F\left(m,m(m+1)/2\right)=1\\ F(m,i)=0&=0\text{ if i<0, i> }i<0\text{ or ...
3
votes
0answers
104 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 ...
6
votes
0answers
129 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 ...
0
votes
1answer
58 views

Stationary distribution of random walk alias solving uncountably many linear equations [closed]

Let us have interval $I = (i_1,i_2)$, function $f_1 : I \mapsto I$, function $f_2 : I \mapsto I$. Let $x_0$, $x_1$, $x_2$, ... be series of random variables from interval $I$ denoting random walk. ...
1
vote
0answers
69 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
2answers
102 views

Transforming a recurrence to the product of two other recurrences

This question deals with sequence: $$a_0 = 0,\ a_1 = 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6$$ The question is about establishing that $a_n$ is a composite number (except some finite cases). In one ...
0
votes
0answers
45 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
0answers
92 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 ...
6
votes
1answer
260 views

Polynomial recurrence relation covering the integers (and then Gaussian integers)

Say that a polynomial recurrence relation (my terminology) for $f_i$ is: $k$ initial conditions setting $f_1,\ldots,f_k$ to integers ($\in \mathbb{Z})$. A recurrence equation of the form $f_i =$ a ...
1
vote
0answers
86 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 ...
0
votes
0answers
63 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”. ...
10
votes
3answers
274 views

Limiting probabilities for two-player game drawing random uniform numbers

Consider this simple 2-person game I just made up: Player A goes gets to draw a uniform U[0,1] number up to X times. At any time, he may either keep his number, or draw a brand new uniform number. ...
4
votes
2answers
241 views

Strong divisibility of Lucas sequences

Let $a$ and $b$ be relatively prime integers and let $u_n$ be their associate Lucas sequence, i.e., the second order linear recurrence sequence satisfying $u_0 = 0$, $u_1 = 1$ and $u_{n+2} = au_{n+1} ...
0
votes
0answers
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. ...
9
votes
2answers
281 views

Faster formula to compute sum over partitions

Let $f$ be a function from the positive integers to the real numbers (or some ring...). Let $$(\star) \quad F(n) = \sum_{n_1 \leq \cdots \leq n_j\atop n_1 + \cdots + n_j = n} f(n_1) \cdots f(n_j), $$ ...
1
vote
0answers
128 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 & = & ...
3
votes
0answers
181 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 ...
1
vote
0answers
86 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 ...
0
votes
0answers
77 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 ...
2
votes
1answer
126 views

General four-term recurrence relations

I was perusing through this paper and I was wondering if there is any literature on general four term recurrence relations. Specifically, say a recurrence of the form ...
8
votes
1answer
492 views

Hamming weight of Fibonacci numbers

The Hamming weight $w(n)$ is the number of 1s in $n$ when written in binary. Is there some effective bound on Fibonacci numbers $F_n$ with $w(F_n)\le x$ for a given $x$? Clearly only $F_0=0$ has ...
1
vote
1answer
203 views

Infinitely many sufficiently large powers in linear recurrences

Edit Aaron solved the original question with the fourth order $$ a(n)=n2^n+\frac{(-1)^n-1^n}{2} $$ trying to make the question harder. Let $a(n)$ be a linear recurrence with constant coefficients, of ...
10
votes
3answers
542 views

Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form: $$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...
1
vote
1answer
235 views

Does there exist a recurrence relation which cannot be written in a closed form? [closed]

I'm looking for an example of a recurrence relation where it provably does not have a closed form. Does such a thing exist? For example, I believe $S(n, k)$, or the Stirling numbers of the second ...
0
votes
1answer
140 views

Solving a two dimensional non-homogenous linear recurrence

How one can solve the following recurrence: \begin{align} X[i,0] &=0 \quad \forall i =1,\ldots, m\\ X[m,n] &= a_n X[m,n-1]+b_n \sum_{i=k_m}^{m-1}X[i,i] +c_n \end{align} Where $a_i\ge 1 ,~ 0 ...
4
votes
2answers
280 views

Asymptotic growth of recurrence relation $x_n=\min\limits_{n_1+n_2=n}(a(x_{n_1}+x_{n_2})+2n_1n_2)$

Suppose $a>0$ . Define $$x_n=\min_{n_1+n_2=n}(a(x_{n_1}+x_{n_2})+2n_1n_2) \text{ with } x_1=0.$$ Can we find $r>0$ such that there exists two positive constant $c_1,c_2$ such that ...
22
votes
4answers
1k views

Multiplying by irrational numbers in combinatorial problems

This is getting no attention on stackexchange. Everybody knows that the number of derangements of a set of size $n$ is the nearest integer to $n!/e$. It had escaped my attention until last week, ...
7
votes
1answer
1k views

Beyond Collatz: A $5n+1$ conjecture? [closed]

Let $$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
5
votes
0answers
146 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 ...
0
votes
0answers
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 ...
1
vote
1answer
223 views

Closed form solution to an iterative equation.

$f(n+1) = f(n) + f(n)^{a}$ where $a \in (0,1)$ and $n \ge 1$ with $f(1) = m$. If $a=0$, we see $f(n) = m + n - 1$ and if $a=1$, we see $f(n) = 2^{n-1}m$. So the recursion seems to interpolate ...
0
votes
1answer
232 views

Giving a general term of a recursive function, and upper bound for it

Let a constant $B \ge 1$, and let $l_1 = 0$, $b_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$. Let $l_{t+1} = l_t + 1$ if $b_i < B$, and $l_{t+1} = l_t$ otherwise Let ...
6
votes
3answers
730 views

References on techniques for solving equations with discontinuous functions such as floor and ceiling?

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good ...
2
votes
0answers
155 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 ...
4
votes
3answers
889 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
2answers
709 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
1answer
286 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
2answers
106 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 ...
1
vote
0answers
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 ...
2
votes
0answers
94 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), ...
0
votes
0answers
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; ...
2
votes
0answers
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 ...
6
votes
5answers
475 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
3answers
598 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
2answers
882 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
1answer
612 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
1answer
194 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 = ...
2
votes
4answers
2k views

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 ...