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# Questions tagged [recurrences]

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### Two conjectural identities involving $\zeta(3)$ and the golden ratio $\phi$

Let $\phi$ be the famous golden ratio $\frac{\sqrt5+1}2$, and let $\zeta(3)=\sum_{n=1}^\infty\frac1{n^3}.$ In 2014, in the paper Zhi-Wei Sun, New series for some special values of $L$-functions, ...
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I encounter the following recurrence \begin{equation} \tag{1}\label{eq:1} h_{j+1} = h_{j} ( 1 - 1/j - c h_j ), \quad j \geq j_0, \end{equation} with $h_{j_0}>0$, $c>0$ and $0< 1-1/j_0 - c h_{... 0answers 80 views ### A question about the span of a sequence of polynomials satisfying a linear recurrence Let F be a finite field and A(n) in F[t], n in N, be defined by a linear recurrence with coefficients in F[t], together with initial conditions. Is there a decision procedure for determining whether ... 0answers 99 views ### How to define$\ll"$in higher dimension? Fix$C>0,$we say$n \sim m $if$|n-m| < C (n, m \in \mathbb Z)$and$n\ll m$if$n-m \leq C$and$n\gg m$if$n-m \geq C.$Let$n_1, n_2, n_3, n_4 \in \mathbb Z$. Assume that$|n_1-...
1answer
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I am working with an argument involving an inequality of the form: $$f(n + 1) \leq f(n) + C (f(n))^{1 - \frac{1}{\gamma}} \qquad(\ast)$$ where $f$ is a positive function, $\gamma > 0$ and $C >... 0answers 19 views ### Recurrence relation for the asymptotic expansion of an ODE I want to solve for the asymptotic solution of the following differential equation $$\left(y^2+1\right) R''(y)+y\left(2-p \left(b_{0} \sqrt{y^2+1}\right)^{-p}\right) R'(y)-l (l+1) R(y)=0$$ as$y\...
1answer
190 views

### A 2d recurrence equation representing a step-free continuous-time Markov process on N^2 with frequency-dependent rates

$\textbf{Background:}$ Consider a particle in $\mathbb{N}^2$ starting at a point $(x,y)$ that can only move one step at a time along the lattice. The particle moves each up or down in continuous time ...
1answer
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### Solving an recursive sequence [closed]

I have an recursive sequence and want to convert it to an explicit formula. The recursive sequence is: $f(0) = 4$ $f(1) = 14$ $f(2) = 194$ $f(x+1) = f(x)^2 - 2$
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Question: Are there standard techniques available for solving the following kind of linear matrix recurrence relations: $$M_1,\cdots,M_k\ \in\ \mathbb{R}^{m\times n}$$ $$A_1,\cdots,A_k\ \... 0answers 106 views ### Sum of products of binomials satisfies recurrence relation? i need to know if the sequence (a_n)_{n \geq 0} defined by$$a_n =\sum_{s=3n}^{16n} C_{15n}^{s-3n} \; C_{14n}^{s-2n} \; C_{12n+s}^{12n} $$satisfies a recurrence relation ( type sequence Apery) or ... 3answers 310 views ### Does this deceptively simple nonlinear recurrence relation have a closed form solution? Given the base case a_0 = 1, does a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor} have a closed form solution? The sequence itself is divergent and simply goes {1, 2, 2+\frac{1}{2}, ... 4answers 178 views ### Solution of a 2D Recurrence sequence Can we solve the following recurrence relation:$$a_{m,n} = 1 + \frac{a_{m,n-1}+a_{m-1,n}}{2}$$with a_{0,n}=a_{m,0}=0? If not, can we get an estimate of the growth of a_{m,n}? I encountered this ... 1answer 141 views ### A 2nd order recursion with polynomial coefficients I'm hoping to find "exact" an solution to the following simple recursion: q_m(j+1) = m \cdot q_m(j) + j(j+m)\cdot q_m(j-1) with initial data q_m(0) = 1, q_m(1)=m, where m \geq 0 is an ... 1answer 113 views ### If p_n(a,b) is a rational number (or integer) for 3 consecutive values of n then every p_n(a,b) is Let a and b be two real numbers and p_n(x,y) the polynomial:$$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i}, where $n$ is a positive integer. In a previous post I asked if $p_n(a,b)$ was a ...
3answers
307 views

### Enumeration of lattice paths of a specific type

One of the approaches to "Special" meanders led (in particular) to the following question: What is the number $a_{m,n}(\ell)$ of $\ell$-step paths from $(1,1)$ to $(m,n)$ using the ...

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