Questions regarding functions defined recursively, such as the Fibonacci sequence.

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256 views

Linear multivariate recurrences with constant coefficients

In the theory of univariate linear recurrences with constant coefficients, there is a general method of solving initial value problems based on characteristic polynomials. I would like to ask, if any ...
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226 views

Solving two-parameter linear recurrence with different initial values

I'm looking for a method to solve the well-known recurrence ralation with different initial values. The one-parameter linear recurrence is easier. Let's take a peek: $a_0 = \alpha$; $a_1 = \beta$; ...
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171 views

How to solve the recursive relation in Kalman filter?

I was wondering how to solve the Kalman filter's recursive equation (also see the appendix at the end of this post) for the estimated state $\hat{\textbf{x}}_{n|n}$ at time $n$, over discrete times ...
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72 views

Construction of polynomials with non-commutative elements.

I have a simple set of polynomials which I know how to construct for each integer $n$, but I havn't been able to write them down in terms of concrete sums and products. For $n\in\mathbb N_+$, we have ...
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445 views

recurrence relation with non constant coefficients

I'm trying to solve a second order differential equation and I got a recurrence. Can someone help to solve $$n(n-1+q)a_{n}-a_{n-3}+e\cdot a_{n-2}=0$$ where $q$, $e$, and $a_{0}$ are some real numbers ...
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68 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 = ...
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260 views

Bell-like recurrence

Let $$A(n)=\sum_{k=0}^{n-1}\binom{n}{k}A(k)+n!,\quad A(0)=1$$ $$B(n)=\sum_{k=0}^{n-1}\binom{n}{k}B(k)-n!-n!\sum_{k=1}^{n}\frac{1}{k!},\quad B(0)=-1.$$ I'm interested in computing $S(n)=A(n)+B(n)$ ...
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52 views

How do I derive this recursive function?

I have a function $S_t = a*Y_t+(1-a)*S_{t-1}$ Where $a\in (0,1)\cap \mathbb{Q}$, $t$ represents a unit of time and $Y_t$ is the value at a time period $t$ I am trying to find $dS_t \over dt$ I ...
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21 views

Using a Recursion Tree to solve the recurrence $T(n) = \sqrt n T(\frac{n}{2}) + 10n$?

I am attempting to solve the above recurence by giving tight $\Theta$ bounds. Assume that the logs here are all base 2! To solve a recursion tree as far as I understand, I need two things. The ...
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12 views

Is there a way to express a closed form for a partial derivative of this recurrence relation?

Here's the relation: if $n\ge j:$ then $$ \sigma(n,j,d) = d \cdot\left( \log j-\sigma\left(\frac{n}{j}, 1+d, d \right)\right)+\sigma(n, j+d, d )$$ And here's the terminating condition if $n < j$ ...
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30 views

Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...
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22 views

Explicit formula for inverse function for n-th partial sum of harmonic series

Originally, I was trying to work out an explicit formula for a function with the property that $f(x + \frac{1}{f(x)}) = f(x) + k$ for some $k$. This is inspired by a thing I did with a metronome ...
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23 views

Asymptotic behaviour of two dependent recursive sequences

I have two sequences whose relation is described in the following recurrence relations: $ p_{k + 1} = p_k + \frac{1}{2s_k}$ $ s_{k + 1} = s_k + \frac{s_k}{p_{k+1}}$ (when $p_0=2, s_0 = ...
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43 views

Recurrence relation for Binary String Question

I have a question which has been a little stumped. I'm pretty sure I know the answer, but don't know how to prove it to be true. Here it is: "Given an infinite length random binary string, what is ...
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32 views

Question about finite sums and integer recursions.

Let $n$ be a positive integer and let $g(n)$ be a given strictly increasing integer function such that $0<g(n)<n$ for all $n$. Also the sequence $ |g(n) - n|$ is unbounded as $n$ grows. Let ...
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33 views

Complex numbers product and ratio, prove this relation.

Define a table $T$ as follows: $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = (2+3 i) \sum _{i=1}^{n-1} T(n-i,k-1)+(5+7 i) \sum _{i=1}^{n-1} T(n-i,k) \; \text{else} \; T(n,k) = 0$$ Then take rows ...
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31 views

Possible to determine if a more 'compact' solution to a linear recurrence exists?

Given a linear a recurrence relation. It is possible to express a solution in terms of summations, products, and the coefficients which appear in the recurrence. For example, in the case of a single ...
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42 views

General Solution to Linear Difference Equation: Is this correct

Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + ...
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Need help with these recurrence relations

I had received some challenging recurrence last week, I did most of them except this and also one of its kind. It states Given $a_0=0$ and $a_1=1$, solve these recurrence relations: ...
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25 views

What [precisely] is special, algebraically, about fundamental Pell solutions?

I'm asking for a list of algebraic identities which "uniquely identify" (or nearly so) the fundamental solution (or those immediately around it) of a Pell equation. As a concrete [numerical] example, ...
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42 views

Finite geometric sequence with a ratio greater than 1

I am trying to solve the recurrence relation: $ t(n)= 3T(n/2)+Cn$ (if $n>2$) $t(2)=C$ (otherwise) I know that there is the Master Theorem but I am trying to use the tree method. This ...
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53 views

Proof that $\forall x \in \mathbb{R}, \forall n \in \mathbb{N}: (f(x))^n = f(nx)$ only has solutions of the form $f(x) = c^x$ for some constant $c$

I am looking for a proof (or a counterexample I suppose) that the recurrence equation (is that the right term here?) $(f(x))^n = f(nx)$ only has a solution of the form $ f(x)=c^x$ for some ...
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32 views

Solving a recurence system

Find a function $f(x,y) : \mathbb{N}^2 \to \mathbb{N}^2$ such that: $1 + f(x+1,y) - f(x,y) = a$ $1 + f(x, y+1) - f(x,y) = b$ $k + f(x-k, y) - f(x,y) = c \space \forall k \leq x$ $k + f(x,y-k) - ...
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37 views

Analytic Function Derived From Recursive Reverse Taylor Series?

Given the following recursive relation: $a_0 = 1,$ $a_n = a_{n-1}(p-2q)+2(-p)^n$ is there a simple function that has this as its Taylor series, i.e. $f(x) = \sum_{n=0}^{\infty} \frac{a_n}{n!}x^n$ ...
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36 views

Infinitely many primes in second-order recurrence

I just wondered about the following question: Suppose that we are given a homogeneous second-order recurrence relation, $x_{n+2}+ax_{n+1}+bx_n=0$ for all $n\in\mathbb{N}$. Can we choose integers ...
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28 views

Deriving recurrence relations, very stuck!

Going through past papers for my exams and cannot figure this one out, does anyone know how to do these? The Bessel functions of integer order, Jn(x), are described by a generating function of the ...
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43 views

Monotonicity of a recursively defined sequence $s_{n+1} = \sqrt{4s_n -1}$

I am trying to answer this question: Let $s_1 = k$ and $s_{n+1} = \sqrt{4s_n -1}$. For what values of $k$ will the sequence $s_n$ be monotone increasing? I know the definition of monotone increasing ...
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63 views

Solving a Recurrence Relation With Summation and Tau Function

How can I solve the following: $$ T(n) = \sum_{i = 1}^{d(n) - 2}T(v_i) + \sum_{i = d(n) - 1}^{n - 1}c + c' $$ Where $d(n)$ is the Tau function, and v is the set of values dividing n. e.g. $d(18) = ...
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48 views

Solution to a recursion relation.

Let $\beta >0 $. The question is to solve a following recursion: \begin{equation} P^{(j+2)}(\beta) = \frac{\imath}{2} \left[ \left((-1+\beta) j - 1\right) P^{(j+1)}(\beta) + ...
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33 views

Solve recurrence $a[n,k]=(2m-2k)\;a[n-1,k+1]+k\;a[n-1,k]$

I'm trying to count how many vectors of size $n$ there are, given that the elements of the vector are integers from the range $\{-m,m\}-\{0\}$ (zero is excluded), and there are no pair of elements ...
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23 views

Analyzing a recurrence model: equilibriums, stability and periodic behavior.

In orer to increase my knowledge in math I decided to analyze the following recurrence relation (logistic growth in ecology) $$N(t+1) = N(t) (1 + r(1-\frac{N(t)}{K}))$$ I found the equilibriums by ...
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61 views

May someone tell me where I went wrong when finding a closed-form solution to a recursion?

Let $r(n)=\left\lceil\frac{n}{4}\right\rceil$, where $k \in \mathbb{N}$. Note that this can be rewritten as a recursion $R(n)=R(n-4)+1$, where, $R(0)=0, R(1)=1, R(2)=1, R(3)=1$. Since this recursion ...
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39 views

Recurrence of a function

Consider the following recurrence: $$ T(n) = 2T(n/2)+n\lg n$$ Let’s use a base-case $T (2) = 2$ and let’s assume $n$ is a power of $2$. (a) “guess and prove by induction” method, considering the ...
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38 views

Algorithms: Recurrence

Here's a problem that I am struggling with... If two algorithms A and B both solve the same problem. On an input of size $n$ Algorithm $A$ breaks it into $5$ pieces of size $n/2$, recursively solves ...
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432 views

Recurrence relation question

A country uses as currency coins with values of 1 peso, 2pesos, 5 pesos, and 10 pesos and bills with values of 5 pesos, 10 pesos, 20 pesos, 50 pesos, and 100 pesos. a) Find a recurrence relation for ...
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82 views

What is common between difference operators and recurent relations?

They say that solutions of recurrence relations are combinations of exponential functions, the series like [1 a a^2 a^3 and etc]. I know that the difference operators have a matrix like ...
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47 views

How to derive recursive equation for expected discounted utility function?

I am trying to mathematically represent expected discounted utilities when the length of a lifetime is uncertain. $V_0$ is the expected discounted utility at $t=0$ and can be represented as such: ...
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43 views

Help Solving a Recurrence Relation with an Inverse Term

I am having a hard time generating a characteristic polynomial for a recurrence relation I thought of the other day, $a_n = a_{n-1} + \frac1{a_{n-1}}$. I am pretty familiar solving basic recurrence ...
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154 views

Help with find recurrence relation running time.

Write a recurrence relation describing the worst case running time of each of the following algorithms, and determine the asymptotic complexity of the function defined by the recurrence relation. ...
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80 views

substitution method for recurrence tree

If one has recurrence function - $$T(k) = 2T(k-1)+ \frac 1k$$ how can one determine the upper and lower bounds? For upper bound, I try use 'substitution method' and drew out recurrence tree, ...
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61 views

How to solve the recurrence relation $f(n) = 1 - (1 - f(n - 1) \times (1 - p)) ^ 2$ to find a closed-form solution?

A friend of mine gave me a math problem whose answer turned out to be $$f(n) = 1 - (1 - f(n - 1) \times (1 - p)) ^ 2$$ for some fixed $p$. I'm trying to find a closed-form solution to the ...
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45 views

Closed form of an inhomogeneous non-constant recurrence relation

I try to find a closed form for the $f_k$'s (at least for $f_1$) satisfying the following recurrence relation for any fixed $n>0$: $f_0 = 0$, $f_n = 0$, and $f_k = \frac{n}{2k}(f_{k-1} + f_{k+1} + ...
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65 views

looking for explanation behind solution for a 1st order recurrence relation.

In lecture, we covered 1st order recurrence relations and came up with a solution by inspection. I sort of see that we're finding the next term in the sequence by multiplying the initial condition by ...
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196 views

Recurrent relation for number of ways to get a balanced n-binary tree

In answering a question related to binary trees, I came up with the following recurrent relation: Base cases: $$ f \left (1 \right ) = 1 $$ $$ f \left (2 \right ) = 2 $$ Recurrent relations: $$ f(n) ...
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33 views

Whether recursive relationship is a different version of principle of mathematical induction?

In connection with the question I can't get satisfied with such 'so on' type logic. Is there a better way to solve it? and the responses recieved I would like to know whether ...
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59 views

Solving a recurrence involving binomials.

Does anybody know how to solve the following recurrence? Maybe with generating functions? Any hint? $t(n) = 1 + \frac{1}{2^{n-1}} \sum_{i=0}^{n-1} {n \choose i} t(i)$
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36 views

Is there any way to solve the following recurrence relation in 2-dim with different boundary conditions?

I was trying to solve the following recursion problem. It seems like because of the different nature of the boundary conditions it is getting strange although I know the solution exists. The problem ...
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66 views

Deriving this recursive expression for Riemann Prime Counting Function?

Why does this work? $f(n,k,1)=0$ $f(n,k,j)= \frac{1}{k} - f(\lfloor\frac{n}{j}\rfloor, k+1, \lfloor\frac{n}{j}\rfloor) + f(n,k,j-1)$ Here, f(n,1,n) computes the Riemann Prime Counting Function. ...
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93 views

Scaling for characteristic polynomial of sequence of growing matrices

This is a follow-up question to Limit of sequence of growing matrices. There I was considering a sequence of matrices defined by $$ K_L = \left[H\otimes I_{2^{L-2}}\right]\left[I_2 \otimes ...
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60 views

Bounding a sequence defined recursively

Let $\{x_n\}$ be a sequence of positive numbers and $\alpha \in (0,1)$. Let $y_0 := x_n$ and $$ y_k := (\alpha \,y_{k-1} + 1-\alpha) x_{n-k} $$ for $k=1,2,\dots,n-1$. Is it possible to give a sharp ...