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538 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 $s_k:=...

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196 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**

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185 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 $(\mathbf{...

**4**

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161 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 ...

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132 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**

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197 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 (...

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95 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 a(n+...

**2**

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56 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 $f(X), g(X) \in \mathbb{F}[X]$....

**2**

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164 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 f(i-1,j-1,...

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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), \...

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171 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 $\mathbb{...

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59 views

### A two variable recurrence relation with conditionals

I have arrived at the combinatorial problem of enumerating certain types of ballot paths. This has led to analyzing the sequence defined by the following recurrence
$$
f(n,m) = \begin{cases} f(n, \...

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61 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**

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54 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 $x$ ...

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153 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**

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162 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 & = & K_{1}\tbinom{...

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101 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 $f(z,m)...

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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

### Multidimensional recurrence relations

There are many methods of solving one-dimensional homogeneous linear recurrence relations, i.e. such of the form
$$ a_n = \sum_{k=1}^{m}\alpha_ka_{n-k}.$$
Most widespread use linear algebra or ...

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132 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 \frac{r-(k-1)}{r+b-(k-1)...

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75 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”.
...

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52 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. ...

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78 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 i}{(...