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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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        1answer
        103 views

        Nonlinear recurrence

        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_{...
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        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 ...
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        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-...
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        163 views

        An iterative argument involving $f(n + 1) - f(n) $

        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 >...
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        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\...
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        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 ...
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        70 views

        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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        200 views

        Solving Linear Matrix Recurrences

        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\ \...
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        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 ...
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        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}, ...
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        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 ...
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        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 ...
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        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 ...
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        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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