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2
votes
0answers
84 views

Calculating the Taylor series, given a functional equation

I have two functions, whose Taylor series about infinity are given by $$ f(z) = \frac{1}{z} + \sum_{n=2}^{\infty} \frac{A_k}{z^k}, \quad g(z)=\frac{1}{z} + \sum_{n=2}^{\infty} \frac{B_k}{z^k} $$ and ...
7
votes
1answer
276 views

A variant of Cauchy-type functional equation conjecture

Let $f:\mathbb{C}\to \mathbb{C}$ be a complex function such that $$|f(x-y)|=|f(x)-f(y)|,\qquad x,y\in\mathbb{C}.$$ Is it true that $$f(x+y)=f(x)+f(y),\qquad x,y\in\mathbb{C}?$$ The answer is ...
0
votes
1answer
197 views

Background on the functional equation $F(x+1)+F(x)=f(x)‎$ [closed]

In the theory of indefinite sums, anti-differences and finite calculus, ‎the following ‎difference ‎functional ‎equation ‎and ‎its ‎solutions ‎are ‎very ‎important: ‎$$‎\bigtriangleup ‎F(x):=F(x+1)-...
5
votes
2answers
372 views

Existence of Solution, System of Equations

Suppose $P(\lambda, i)$ is the probability that a Poisson random variable with average $\lambda$ is equal to $i$, i.e. $\frac{\lambda^i}{e^{\lambda}i!}$ I think the following system of equations ...
2
votes
0answers
94 views

A generalized Cauchy type functional equation

Let $(S,+)$ be an abelian semigroup . Let $f:S \to \mathbb C$ be a function such that for some positive integer $n>1$, $f(x+y)^n=(f(x)+f(y))^n,\forall x,y \in S$. Then is it true that $f(x+y)=f(x)...
10
votes
2answers
136 views

How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?

Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by ...
3
votes
1answer
118 views

Solvability of a certain functional equation in simple $C^*$ algebras

For which simple unital $C^*$ algebras does the following functional equation have a solution: $$ d^2=0,\;{(d+d^*)}^2=1$$ The Calkin algebra and $M_{2n}(\mathbb{C})$ are some examples. It is not ...
0
votes
1answer
87 views

1D functional equation: solve for function with given expected value w.r.t normal density

Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation $$ \begin{split} \mathbb ...
1
vote
1answer
71 views

Equation for a geometrical half surface from folding a flat curved in one direction surface in half [closed]

I cannot not formulate the problem as i do not know how to model this idea. This is an open question not an mathematical exercise so if anybody has a good proposition i'm happy to use it. Sorry in ...
0
votes
2answers
71 views

Find the general solution to the Forsyth/Abel functional equation

Let $R(m,n)$ be defined on all the integers such that $R(m,0)=m, R(0,n)=n, R(m,n)=R(n,m)$ and $R(R(m,n),p)=R(m,R(n,p))$ for all integers $p$. Thus $R$ satisfies the Abel associativity equation. Let $...
1
vote
0answers
51 views

Characterisation of functions for which the Fourier transform commutes with a particular operator

Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...
0
votes
0answers
53 views

Functional equation with Cosine transform

Defining $f(x)$ as a Cosine transform on $\mathbb{R}^+$: $$f(x)=\int_0^{\infty} S(t) e^{-i \lambda \int_1^{t} S(u)^2 \frac{du}{u^2} } \cos(2 \pi xt) dt $$ Can we find a real function $S(t)$ on $\...
3
votes
2answers
262 views

Seeking proof to an asymptotics of a recursion or functional equation

My question on math.stackexchange.com and the continuation by an answer to it gives the two summation expressions for the recursion $$a_n = 1+\frac1{2^n}\sum_{k=0}^n {n\choose k}a_k,\, \forall n\in\...
2
votes
0answers
71 views

Functional equation involving integrals and exponential

Can we find on $\mathbb{R}^+$ a real positive function $f(x)$ (in $C^{\infty}$) such that: $$\int_0^{\infty} f(x) e^{\lambda \int_1^{x} f(t)^2 dt} dx=0$$ where $\lambda$ is a complex number (with $0&...
1
vote
2answers
79 views

A functional equation with a quadratic solution

I have the following problem. I have a function $v(x, \theta)$ that can be expressed in two ways, for all $x, \theta \in \Re$: $v(x, \theta) = u(x - \theta)$, where $u$ is strictly concave and ...
0
votes
0answers
52 views

Existence of a couple of functions solution of a differential equation (with additional constraint)

I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$): $$...
2
votes
2answers
363 views

What is the solution, $f(n)$, of the following functional equation: $mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn)$?

What is the solution, $f(n)$, of the following functional equation: $$mf(m)+nf(n)=(m+n+xmn)f(m+n+xmn) ,$$ where $f$ takes on integer values, $m$ and $n$ are integers, and $x$ is an indeterminate? ...
0
votes
0answers
18 views

DDE-like? equations on spaces of discrete-valued functions [reference request]

I am looking for a reference to an introduction to dynamics of some sort of equations that look like delay differential equations in a quite particular setting. I am not an expert in standard DDEs by ...
3
votes
2answers
359 views

Solution of the functional equation $f(x+1)=g(x)f(x)$

In the paper (http://www.sciencedirect.com/science/article/pii/S0022247X97953439), Webster obtained a unique solution of the functional equation $f(x+1)=g(x)f(x)$ (where $f,g:\mathbb{R}^+\rightarrow \...
1
vote
0answers
100 views

Analytic continuation of $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s}$. Possibly related to Riemann Zeta function $\zeta(s)$?

I'm trying to find the analytic continuation for $\alpha(s)=\sum_{n=a+1}^\infty\frac{1}{(n^2-a^2)^s} ,$ with $a\in \mathbb{N^+}$ and $s<1$. I need most likely only the values for $s=\frac{1}{2}-m$...
3
votes
1answer
137 views

Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? [closed]

Suppose that $f: \mathbb R \rightarrow \mathbb R$ such that $$f(x^3+y^3)=f(x+y)((f(x-y))^2+f(xy)),$$ for all $x,y$ real numbers. Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? I did ...
2
votes
1answer
173 views

Solve functional equation: $\frac{F(x)}{F(1)} = 2F\left(\frac{(1+x)^2}{4a}\right)-F(x/a)$

Solve this functional equation: $$\frac{F(x)}{F(1)} = 2F\left(\frac{(1+x)^2}{4a}\right)-F(x/a)$$ for $F(x)$ where $a > 0$ is a parameter. I know there is a trivial constant solution, $F(x) = 1$. ...
2
votes
0answers
135 views

Functional equation of automorphic L function ( over global function field)

Setting: $X$: a smooth proper curve of genus g over $k=\mathbb{F}_p$ with function field $F$, $X'$: an etale double cover with function field $F'$. $E_\pi$: the coefficient field of $\pi$. Now given ...
3
votes
0answers
142 views

modularity of Theta functions attached to Hecke characters

Let $K/\mathbb{Q}$ be a quadratic imaginary field, and let $\chi$ be a Hecke character on $K$. Using Poisson summation, one can show that the theta function $$ \theta(z):=\sum_{I\subseteq \mathcal{O}...
-1
votes
1answer
98 views

Are there solutions for this functional equation?

The distribution $g(x)$ has the following properties: $$\int_{-\infty}^{\infty}g'(x)f(x)dx=f(\pi g(0)-1/2)-f(\pi g(0)+1/2)$$ for any analytic $f(x)$. How can I find $g(x)$?
3
votes
1answer
214 views

Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a Certain Property

First, define a sequence $F_0,F_1,\dots$ of functions by $$F_0(x,z) = z,$$ $$F_k(x,z)=x\exp\left(F_{k-1}(x,z)\right) \quad \text{for }k\geq1.$$ So, for example, $$F_1(x,z) = x e^z, \quad F_2(x,z)=xe^{...
8
votes
2answers
184 views

Linearizing a power series by conjugation

Let $\mathfrak{I}:=\big\{ \, f:=\sum_{k=0}^\infty f_k z^k \in\mathbb{C}[[z]]\; : \text{s.t. }\; f_0=0 \;\text{ and }\; f_1=1\big\}$. A most basic result about linearization states that, for any $f\...
1
vote
0answers
45 views

Variational Problems with Subsidiary Conditions

I am studying from Gelfand, I.M.; Fomin, S.V., Calculus of variations. Transl. from the Russian and edited by Richard A. Silverman., Mineola, NY: Dover Publications. vii, 232 p. (2000). ZBL0964....
2
votes
0answers
105 views

Which functions $f: \mathbb{R} \to \mathbb{R}$ is injective over some subinterval of $(x,y)$ whenever $x<y$ and $f(x) \ne f(y)$?

Under what conditions on a function $f: \mathbb{R} \to \mathbb{R}$ can we say that given any real numbers $x,y$ with $x<y$ if $f(x) \ne f(y)$ then there is a sub-interval $S_{(x,y)}$ of $(x,y)$ ...
1
vote
0answers
39 views

Is research of the Hyers-Ulam stability of this functional equation already conducted?

The functional equation in question is of the type $f(g(x))=g(f(x))$, where $f$ is the unknown function. Are there existing research already conducted on the Hyers-Ulam stability of this generalized ...
25
votes
4answers
1k views

For a continuous function $f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ does $(f(x)-f(y)) (f(\frac{x+y}{2}) - f(\sqrt{xy}))=0$ imply that $f$ is constant?

Suppose that $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a continuous function such that for all positive real numbers $x,y$ the following is true : $$(f(x)-f(y)) \left ( f \left ( \frac{x+y}{2} \right ) - ...
1
vote
2answers
452 views

Intuition behind the Riemann $\zeta$ functional equation

Let $\Lambda(s)=\pi^{-s/2} \Gamma\left(\frac{s}{2}\right)\zeta(s).$ Then $\Lambda\left(\frac{1}{2} + s\right) = \Lambda\left(\frac{1}{2} - s\right)$ constitutes the functional equation for the Riemann ...
1
vote
0answers
37 views

Necessary additive and multiplicative properties to characterize a mildly growing function

Given $k>1$ what could be the necessary additive and multiplicative property of the minimum smooth growing monotone function $f:\Bbb R\rightarrow\Bbb R$ needed such that $\forall a\geq 2^k+1$ we ...
1
vote
1answer
91 views

Does this equation have an explicit solution? [closed]

For a positive constant $C$: \begin{align} y(x)+C\ln y(x)=f(x). \end{align} At least from specific $f(x)$, such as piece-wise linear function, is there an explicit solution for $y(x)$?
2
votes
0answers
121 views

Reflection Formulas for the $\Gamma$ Function

We have $$\begin{align} &\Gamma\Big(1~+~x\Big)~\cdot~\Gamma\Big(1-x\Big)~=~\frac{\pi x}{\sin\pi x} \\\\ &\Gamma\Big(\tfrac12+x\Big)~\cdot~\Gamma\Big(\tfrac12-x\Big)~=~\frac\pi{\cos\pi x} \\\...
3
votes
0answers
51 views

Solving a nonlinear integral equation for a distribution function

I have an equation of a probability distribution function $F$ on $[0,\infty)$, $$F(x)=e^{-\eta [1-\int_0^x F(x-y)g(y)\mathrm{d}y]},\quad x\in[0,\infty), $$ where $g$ is a probability density function, ...
0
votes
2answers
97 views

A simple equation with a normal distribution [closed]

Let $f$ be the distribution of a normal variable $\mathcal{N}(0,1)$, ie $$ f(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ I have to solve the equation: $$x+y = f(x) - f(y)$$ I was working with mathematica ...
5
votes
1answer
228 views

On a generalization of the classical Cauchy's functional equation

I start with some known preliminaries on the problem: Classical result. The one-dimensional Cauchy functional equation $$ \forall x,y \in \mathbb{R}, \,\,\,f(x+y)=f(x)+f(y) $$ with $f:\mathbb{R}\to \...
8
votes
0answers
131 views

Zappa-Szép products of the group of integers with itself

Since my previous question didn't get much attention and I couldn't make any relevant progress on it, I thought it would be a good idea to "simplify" it by replacing monoids by groups. That is: ...
5
votes
0answers
87 views

Zappa-Szép products of the monoid of integers with itself

Question What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations? $\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\ \bullet ~~~ \...
10
votes
1answer
425 views

A conjecture about certain values of the Fabius function

The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations $$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$ and $$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\...
9
votes
1answer
237 views

Some nice functional equations for $q$-continued fractions

Given $\large q=e^{2\pi i \tau}$. Define, $$\alpha(\tau) = \sqrt2\,q^{1/8}\prod_{n=1}^\infty\frac{ (1-q^{4n-1})(1-q^{4n-3})}{(1-q^{4n-2})(1-q^{4n-2})}$$ $$\beta(\tau) = q^{1/5}\prod_{n=1}^\infty\frac{ ...
4
votes
2answers
437 views

Total progeny of a Galton-Watson branching process - standard textbook question

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly. ...
5
votes
2answers
384 views

On the consistency of the definition of the conductor for automorphic forms

Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_2(F)$, where $F$ is local non-archimedean. The local conudctor associated to $\pi$ can be defined in two usual manners: By its ...
6
votes
0answers
194 views

General solution of the Multiplicative symmetry equation $f(xf(y))=f(f(x)y)$ in nonabelian groups

As we know, the functional equation $f(xf(y))=f(f(x)y)$ was completely solved in abelian groups (by J. G. Dhombres, Solution... $f(x\ast f(y))=f(y\ast f(x))$, Aequationes Math. 15 (1977), 173--193, ...
30
votes
1answer
1k views

$f'=e^{f^{-1}}$, again

This question is a spin-off of this one, in which the OP asks whether there is a solution $f:\mathbb R\to\mathbb R$ of the functional equation (not exactly an ODE) $f'=e^{f^{-1}}$, where $f^{-1}$ is ...
15
votes
3answers
2k views

How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$?

Let $f$ be a function such that :$f:\mathbb{R}\to \mathbb{R}$ and $f^{-1}$ is a compositional inverse of $f$. I would'd like to know how do I solve this class of differential equation : $$\...
5
votes
0answers
321 views

The functional equation $f(x) = qx + qxf(x) - f(x^2)$

A word (i.e., ordered string of letters) is bifix-free provided it has no proper initial string and terminal string that are identical. For example, the word $ingratiating$ has bifix $ing$, but the ...
0
votes
0answers
130 views

calculus of the variations problem

I have the following functional to maximize $$J(P,K) = \int_0^1 \frac{P(x)\mu^{-1}(x)}{A+\int_x^1 P(z)\mu^{-1}(z)(1+K^2(z))\mathrm{d}z}\mathrm{d}x$$ subject to $$\int_0^1 P(x) \mathrm{d} x = 1,$$ $$...
1
vote
2answers
136 views

Cauchy-like functional equation $f(x+g(x)y)=f(x)+f(g(0)y)-f(0)$

I have the functional equation $$ f(x+g(x)y)=f(x)+f(g(0)y)-f(0) $$ where $f$ is positive, monotone increasing and continuous $g$ is continuous and positive The domain of both functions is a closed ...