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8
votes
0answers
525 views

Is this 2x2 determinant sequence positive and increasing?

Let $X_1,X_2,X_3$ be a three discrete (integer and non-negative valued) random variables with local probabilities $a_k:=\mathbb{P}(X_1=k)$, $b_k:=\mathbb{P}(X_2=k)$, $c_k:=\mathbb{P}(X_3=k)$ and ...
6
votes
0answers
179 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 ...
6
votes
0answers
159 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 ...
4
votes
0answers
110 views

Puzzling behaviour of a recursively defined sequence of functions

This question arose in connection with another problem that I described earlier in Constructing a generating function using a series with all negative and positive powers of a variable. I had certain ...
3
votes
0answers
123 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
0answers
195 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
0answers
93 views

Factorization of linear recurrences

For each (commutative unitary) ring $R$, let $\mathfrak{R}(R)$ be the set of all linear recurrences over $R$, that is, the set of all sequences $(a(n))_{n \geq 0}$ in $R$ such that $$a(n+k) = r_1 ...
2
votes
0answers
55 views

The characteristic polynomial of the product of two linear recurrences

Let $\mathbb{F}$ be a field and let $(a_n)_{n \geq 0}$, $(b_n)_{n \geq 0}$ be two linear recurrences with terms in $\mathbb{F}$ and respective characteristic polynomials $f(X), g(X) \in ...
2
votes
0answers
163 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
0answers
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
votes
0answers
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 ...
1
vote
0answers
42 views

How to solve the following bivariate recurrence?

$$F(n,r) = (1-w(r))F(n-1, r) + w(r-1)F(n-1, r-1)$$ where $w(r)$ is monotonically non-increasing in $r$ and $0 \leq w(r) \leq 1$ with $0 \leq r$ Initial condition: \begin{eqnarray} F(0, r) & = ...
1
vote
0answers
140 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
0answers
149 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
0answers
99 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
0answers
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 ...
0
votes
0answers
45 views

Asymptotic growth of $f(x) = 2f(x/2 + x/\ln(x))$

In this paper on the Akra–Bazzi theorem, Tom Leighton mentions in his remark at the bottom of page 8, that a function on the positive reals satisfying $f(x) = 2f(x/2 + x/\ln(x))$ for large enough $x$ ...
0
votes
0answers
48 views

Multivariate recurrence relation

Consider the recurrence relation \begin{equation} \mathbb{I}(m<M)[- k_{\rm on} c_{m,n} + (m+1) k_{\rm off} c_{m+1,n}] + \mathbb{I}(m>0)[-k_{\rm off} m \, c_{m,n} + k_{\rm on} c_{m-1,n} ] + ...
0
votes
0answers
115 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
0answers
74 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
0answers
45 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
0answers
75 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 ...