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0
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0answers
51 views

Where can I find out more about such functional equations?

Such as this one $$f(x) = x + f(\frac{x}{1-x})$$ or that one $$f(x) = x + f(\frac{1}{1-x})$$ I did some digging on my own: here, here and here but I was curious if there's any existing info on the ...
1
vote
1answer
105 views

Linear functions

Let $(f_1, f_2, \ldots, f_n)$ be an $n$-tuple of functions mapping non-negative integers to non-negative integers. Let $m$ be a positive integer.Suppose there exists a function $f$ apping non-negative ...
-1
votes
1answer
95 views

Finding functional equations that a given function satisfies

Suppose we're given a function, for example a function $f:\mathbb{C}\rightarrow \mathbb{C}$ such that $f(x)=ax+b $ with $a,b \in \mathbb{C} $. I would like to know which functional equations are ...
3
votes
0answers
172 views

Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$. Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$ Therefore $$\sum\limits_{k=1}^\infty ...
3
votes
1answer
79 views

Functional equations associated with addition theorems for elliptic functions

I'm trying to read the article "Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups" by Bukhshtaber,V. M. Russian Mathematical Surveys(1990),...
0
votes
1answer
84 views

When are/ if recursive identities used? [closed]

Due to the fact that $$\Gamma(x)\Gamma(1-x)\sin \pi x=\pi$$ is this necessarily true: $$\Gamma(x)\Gamma(1-x)\sin(\Gamma(1+x)\Gamma(1-x)\sin(\cdots))=\pi$$ Is this at all used as a tactic to create ...
1
vote
1answer
59 views

Inverting two paraboloid relations

Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations: $$ k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right| $$ $$ ...
5
votes
0answers
164 views

Solving a Laurent polynomial functional equation

I'm considering a set of functional equations: For a given $\phi(x)\in\mathbb {Z}[x,\frac {1}{x}] $ with $\phi(x)=\phi(\frac{1}{x}) $, $f(x)f(\frac {1}{x})+\phi (x)g(x)g(\frac {1}{x})=1, $ where $f(x)...
3
votes
0answers
82 views

continuous linear recurrent relations

For a function $f:\mathbb{R}\rightarrow \mathbb{R}$ denote $f_0(x)=x$, $f_n(x)=f(f_{n-1}(x))$. Assume that $f$ satisfies a functional equation $$f_n(x)+a_{n-1} f_{n-1}(x)+\dots+a_0 f_0(x)\equiv 0$$ ...
1
vote
3answers
188 views

Existence of a square root of a functional equation

We can define the iterates $f^{n+1}=f\circ f^n$ for a given smooth map $f:X\to X$, where $X$ could be a finite interval, the real line $\mathbb{R}$, or the circle $S^1$, or any general smooth manifold....
7
votes
2answers
153 views

Characterizing when matrices are 'dissipative'

An $n$ by $n$ matrix A is said to be dissipative with respect to a norm $\|\cdot \|$ if for all $x$ and $t\geq 0$, we have $\|e^{At}x\|\leq\|x\|$. Two matrices $A$ and $B$ are said to be jointly ...
4
votes
1answer
176 views

Cauchy-like functional equation $f(h(y)x+y)=g(y)f(x)+f(y)$

I am looking for the solution to the following two variable functional equation: (*) $f(h(y)\cdot x+y)= g(y)f(x)+f(y)$ where: $h$ is some given continuous function, $f, g,$ unknown functions on ...
1
vote
0answers
105 views

Homomorphic Commutator? Equation

So I was considering the following functional equation: Given $H :\Bbb{C}^2 \rightarrow \Bbb{C} $ find $\theta: \Bbb{C}^2 \rightarrow \Bbb{C}$ such that $$ \theta(H(a,b), H(c,d)) = H(\theta(a,c), ...
5
votes
0answers
94 views

Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form $$ g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{...
0
votes
0answers
87 views

Existence of a solution to $xf(x) = \int_0^1 k(x,y) yf(y) dy$

Crosspost - I asked very similar question on math.sx. Let $X = (0,1)\times (0,1)$ and $k\colon X \to \mathbb{R}$ be a Lebesgue measurable non-negative function such that $$ \int_0^1 k(x,y) dy = 1$$ ...
0
votes
0answers
76 views

Energy Oscillations in a One Dimensional Crystal

Good day! Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially ...
2
votes
1answer
195 views

Survey on functional equations and inequalities

Where can I find a comprehensive survey monograph on functional equations and inequalities from sketch to current research trends with some focus on applications (both inside and outside mathematics)?
2
votes
0answers
128 views

Distributive law between Kleisli triples

A distributive law of a monad $S$ over a monad $T$ is a natural transformation $l : T S \to S T$ such that: $l \circ T \eta^S = \eta^S T$ $l \circ \eta^T S = S \eta^T$ $\mu^S T \circ S l \circ l S = ...
0
votes
1answer
72 views

Functions with special separability

Suppose we have differentiable functions $F$, $f_1, \dots, f_n$, and $g_1, \dots, g_n$ satisfy the following relation $$ F(x+y) = \sum_{i=1}^n f_i(x) g_i(y).$$ What are the possible forms of $F$?
0
votes
1answer
102 views

Uniqueness of solutions of functional equations [closed]

A solution to $f(2x)=\alpha f(x)$ with a boundary condition $f(\beta) = \beta$ is $$ f(x) = \left( \frac{\beta}{\alpha^{\log_2 \beta}} \right) \alpha^{\log_2 x}. $$ Do we know whether or not the ...
3
votes
1answer
159 views

Existence of solution for this set of polynomial equations

We are given a number $n$ and a vector $p=(p_1,p_2,\ldots,p_r)$, where $$p_1\geq p_2 \geq \ldots \geq p_r > 0 ; \ \ \ \ \sum_{i\in [r]} p_i \leq 1$$ I'm interested in proving that a solution for ...
5
votes
2answers
328 views

Does this equation has a closed-form solution for $t$? ($(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i)$)

We are given $n\in \mathbb N^+$ and $p\in[\frac{1}{2},\frac{n+1}{n+2}]$. Our goal is to find $t\in[0,1]$ such that $$(1-p)\sum_{i=0}^{n}t^i = p\sum_{i=0}^{n}(1-t)^i$$ Is there a closed-form ...
3
votes
2answers
334 views

Consistent price index

This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem. Let $Y=({\...
2
votes
1answer
161 views

Counterexample for the Generalized Associativity Equation

The Generalized Associativity Equation is given by $$ F(G(x,y),z)=K(x,H(y,z)),$$ where the functions $F,G,H$ and $K$ are all from $\mathbb{R}^2$ to $\mathbb{R}$. In his book "Lectures on Functional ...
4
votes
0answers
71 views

Archimedean $\varepsilon$-factors

Let $K$ be either $\bf R$ or $\bf C$. Let $p$ and $q$ be integers with $p \leq -1$, $q \geq 0$, and $p+q=-1$. Consider the Hodge structure $M = M(p,q)$ over $K$ with coefficients in $\bf R$, defined ...
1
vote
0answers
54 views

Family of (Cumulative Distribution) Functions

I'm looking for a 2 (or more)-parameter family of functions $F$ with the following properties: For each $f \in F$, $f(0)=0$, $f(1)=1$, and $f$ is (weakly) increasing. $F$ is closed under products. $...
2
votes
0answers
173 views

Solve this functional equation with respect to $f$

Let $v\not= 1$ be a real number. Let $f(s)$ be real analytic on an open interval containing $v$ and $1$, with a zero of order $m\ge 1$ at $s=1$. My question is: Can we solve this functional equation ...
2
votes
1answer
341 views

A functional inequality

$g:[0,1]\to[0,1]$ continuously differentiable and increasing such that for all integers $t>0$ and for all $r\in(0,1)$, $g(r^{t+1})>g(r)\cdot g(r^t)$. Does this imply that for all $r\in(0,1)...
0
votes
2answers
161 views

Solving a functional equation

I would like to consider the following simple problem. I want to find two functions $f,g : \mathbb R \to \mathbb R$ such that, being given a collection $(h_v)_{v\in V}$ of real functions indexed by ...
4
votes
1answer
439 views

solution of functional equation $f^{\circ k}(x) = x$

The equation $f^{\circ k}(x) = \mathrm{Id}$ for $x\in E$ is called the Babbage equation and the general solution is given in the following way [M. Kuczma, Functional equations in a single variable]: ...
5
votes
2answers
226 views

General additive function of probability

Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that: $H$ is continuous, $H$ is symmetric w.r.t. the order of its arguments, ...
2
votes
0answers
127 views

A $GL_1$ Voronoi formula

I want a functional equation for the function defined by the Dirichlet series, $$ D(s,a/q)= \sum_{n=1}^\infty \frac{e^{2\pi i n a/q}}{n^s}. $$ which sends $s$ to $1-s$ and preferably sends $a$ to $\...
0
votes
1answer
270 views

Is there any mathematical study about |a-b|=f(|g(a)-g(b)|)? or does there exist f() and g() satisfy this equation? [closed]

The problem is just as the title. It is clear that the linear function $f(x)=kx$ and $g(x)=(1/k)x$ can meet it. Is there any other function pairs f(x) and g(x) can meet this equation? or the equation $...
11
votes
4answers
691 views

Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form: $$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in \mathbb{R},\;\text{and}\;f^0=\text{...
1
vote
2answers
188 views

Unusual Differential Equation for CDF

Consider the following differential equation $$F(cx) = F(x) + x F'(x)$$ for $c>1$. Does this differential equation belong to a some well known class? Is there a way to find all the solutions $...
1
vote
1answer
193 views

How to solve for the nonlinear functional equation? [closed]

I got a nonlinear functional equation like: $f(x) = g(x) + h(f(Ax))$, where $A$ is a constant, $x$ is a scalar, $g()$ and $h()$ are given. The objective is to solve for the expression of $f(x)$. ...
2
votes
1answer
123 views

A Recursive Maximization Problem

Let $A\ge B>0$ be real constants. I say that a function $f:[0,1]\rightarrow[0,1]$ satisfies the $(A,B)$-condition if for all $p\in [0,1]$, the expression $$q(A-Bp-Bf(q))$$ is maximized (not ...
0
votes
1answer
258 views

An Integral Functional Equation

Let $f$ be a non-negative function supported and integrable on the positive real axis, such that $$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$ where $c[p]$ a number (functional) dependent on function $...
4
votes
1answer
512 views

A Differential Equation with Nested Functions

This was posted to Math Stackexchange, but got no useful answers, and the more I think about it, the harder it seems. I would like to know whether there exists a differentiable function from the (...
2
votes
0answers
115 views

A Convolution Integral Equation

Is there any close-form solution for a function $f(t)$ satisfied the below equation: $f(t)=g(t)+\frac{1}{t^2}(h(t)*f(t))$. Operator $*$ is convolution integral, and $g(t)$ and $h(t)$ are known ...
2
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0answers
365 views

The functional equation of Hofstadter's Q sequence

Hofstadter's Q sequence is defined by $Q(1) = Q(2) = 1$ and $Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n \geq 3$. So far hardly anything on this sequence has been proved -- not even that $Q(n)$ is well-...
2
votes
2answers
378 views

Functional equations

What are the general solutions of the functional equations? $$ f(x,y)+f(y,z)=\frac{1}{f(x,z)} $$ $$ f(x,y)f(y,z)f(x,z)=1 $$
0
votes
1answer
277 views

Some functional equations in two variables

I have two questions. i) Does there exist a function $\varphi:\mathbb{R}\to\mathbb{R}$ for which the functional equation $$ |f(x)-f(y)|=\frac{1}{|\varphi(x)-\varphi(y)|} $$ has a solution $f:\mathbb{...
0
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0answers
106 views

Solving Multivariate and multi power Equations

There are 84 equations, $r-A_DD^5-5A_DD^4D_i-(a_{d_i}+2b_{d_i}D_i)+(1-\alpha)\lambda_i=0,$ $A_L/L-r-(A_L/L^2)L_i-(a_{l_i}+2b_{l_i}L_i)-\lambda_i=0,$ $(1-\alpha)D_i-L_i=0$ where $i=1,\cdots,28$, $...
11
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2answers
514 views

Series defined by a fixed-point functional equation

Description I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here ...
2
votes
0answers
157 views

Critical case linear autonomous functional differential equation

I am looking for asymptotic ($t\to\infty$) behavior of the general solution $g(t)$ to a following linear functional differential equation $$ \text{(1)} \quad\quad\quad g'(t)-g'(t-T)=-g(t) $$ with ...
27
votes
3answers
992 views

Rational functions with a common iterate

Let $f$ and $g$ be two rational functions. To avoid trivialities, we suppose that their degrees are at least $2$. We say that they have a common iterate if $f^m=g^n$ for some positive integers $m,n$, ...
5
votes
1answer
369 views

Hahn-Banach theorem with real extended valued function

Hello to everyone, My problem is the following: I have this version of the Hahn-Banach theorem: Let V be a vector space and let $p:V\rightarrow \mathbb{R}$ be any convex function. Let $W$ be a vector ...
0
votes
3answers
186 views

References for functional equations in more general settings than the reals

Hi there - I'm Manny, a soon to be MSC thesist. I'm looking for a subject to write my thesis about - and recently I was caught by functional differential equations. Is there any neat reference for ...
3
votes
2answers
1k views

Techniques to solve equations involving a definite integral [closed]

Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, $\alpha\in\...