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Does anyone know how to solve $$f(n+1,m+1) = (m+2)(f(n,m) - f(n,m+1))$$ I'm not too sure how to approach it. I've found some specific results like: $$f(n+1,n) = (n+1)!$$ $$f(n,0) = (-1)^{n+1}$$ $$f(n,...
Bradley2016's user avatar
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14 votes
1 answer
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Let $p$ be a prime such that $2$ is a primitive root of $p$. We want to find a bijective function $f: (\mathbb{Z}/p\mathbb{Z})^× \to (\mathbb{Z}/p\mathbb{Z})^× $ s.t. $$f(2k) = f(k) + f(f(k)) $$ $$f(-...
Adarsh Singh's user avatar
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For prime $p$, let $$ H_p(t)=\sum_{i=0}^{(p-1)/2}\binom{\frac{p-1}{2}}{i}^2t^i$$ be the Deuring polynomial, whose roots correspond to the supersingular elliptic curves of the form $$E_t:\ y^2=x(x-1)(x-...
Dror Speiser's user avatar
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2 answers
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Motivation: I was using actuarial tables to compute the probability, considering two people (of the same sex) aged $x,y$, that one dies before the other (some graphs are here with data from France), ...
Gro-Tsen's user avatar
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Let $F=\mathbb{Q}(\sqrt{D})$ be a real quadratic field, and let $f$ be a Hilbert modular cusp form of weight $(k,k)$ of $\mathrm{SL}_2(\mathcal{O}_F)$. Define the Asai $L$-function as \begin{equation*}...
Misaka 16559's user avatar
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1 answer
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Does $\Phi(s) := 4 \sum_{t=1}^\infty \frac{t}{\sqrt{s}} K_1(2t\sqrt{s})$ satisfy $\Phi(1/s)=s^4\Phi(s)?$ Here $K_1$ is the modified Bessel function. I'm interested in this because I want to further ...
John McManus's user avatar
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$\newcommand\D{\mathcal D}$Let $G(z):=P(Z>z)$ for real $z$, where $Z\sim N(0,1)$. Let $\D$ stand for the set of all continuous strictly decreasing functions from $(0,\infty)$ onto $(0,\infty)$. Is ...
Iosif Pinelis's user avatar
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In this Wikipedia article: we have the following about the beta function approximation: Stirling's approximation gives the asymptotic formula: $$ B(x, y) \sim \sqrt{2\pi} \cdot \frac{x^{x - \frac{1}{2}...
Ilovemath's user avatar
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31 votes
6 answers
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I have a formal power series $\Phi(q,t)$ that satisfies the following functional equation: $$\Phi(q,t)\cdot \Phi(q^{-1},-t) = 1.$$ Is there a nice known family of functions that satisfies identities ...
Nicholas Proudfoot's user avatar
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I'm studying the following differential equation $$ x \frac{\partial^3}{\partial x^3} P[h, x] = \left (x^3 \frac{\partial^3}{\partial x^3} + 3x^2 h \frac{\partial^3}{\partial x^2 \partial h} + ...
Sergii Voloshyn's user avatar
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It's well-known (see e.g. the Wikipedia article) that the Lambert $W$ function can be used to solve equations of the form $$x = a e^{b x}$$ by setting $$x = - \frac{W(-ab)}{b}.$$ This follows from the ...
mt1729's user avatar
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1 answer
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Does there exist a continuous function $ f:\mathbb{R} \to \mathbb{R} $ such that $$ f(x),f(x) + \sqrt{2} , f(x) + x $$ are in $\mathbb{Q}^c$ for all $ x \in \mathbb{Q}^c $? Here, $ \mathbb{Q}^c $ ...
Mohammad Ghiasi's user avatar
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I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
gmvh's user avatar
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I encountered the equation below, encountered a problem that has been bothering me for a long time Does anyone have an idea how to prove it? I would be extremely grateful to you if you come up with an ...
tongjun's user avatar
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I have 2 functions of time $f(t),g(t)$ and a condition for the time-derivative of a third function $h(t)$, say $$\dot{h}(t)=\dot{g}(t)\cos{f(t)},$$ so $h$ is defined provided a value for $h(0)$ (as $h(...
Joan Llobera's user avatar
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Currently, I am working on a random series as follows. Let $\{Y_k\}$ be a sequence of i.i.d. Bernoulli random variables with expectation $p$. Then we define $$ S = \sum_{k=1}^\infty \prod_{\ell=1}^k 2^...
Greenhand's user avatar
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The following problem arose out of a research problem. Let us consider the $n \times n$ matrix valued function $[x_{i,j}(p)]$ (of $p$), satisfying $$ \sum_j x_{i,j}(p) x_{k,j}(p)|x_{k,j}(p)|^{p}= \...
Arun 's user avatar
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For a freely generated countable abelian group $A$ with the trivial action on itself ($a\cdot b = b$) the resulting cohomology groups are well-known and eventually vanish (see e.g. here). Coming from ...
Ollie's user avatar
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I was wondering if it is known that the 3n+1 Collatz conjecture could be reframed as a statement about the set of solutions to a particular equation formulated as the sum of residues. This is ...
thphys's user avatar
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Recently, I was trying to introduce the concept of natural boundaries to a fellow math student, and what greater way to do this than using an example? In particular, I tried to use as an illustration, ...
Prelude's user avatar
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2 votes
1 answer
454 views

I am absolutely not familiar with differential equations. However, I am facing the following differential equation: \begin{equation} a(x)y^{\prime}(x)+b(x)y(x)=c(x)\sqrt{y^{2}(x)+d(x)} \end{equation} ...
Dennis Marx's user avatar
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I am working on some physics problem and got stuck with the following equation: Let $a$ be a very small positive number. Is there a bounded function $F$, $0 \leq F \leq 1$, such that for all $x \in \...
Enumerator's user avatar
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0 answers
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If a real analytic function $f$ is involutive i.e. $f(f(x))=x$ and its Mellin transform can be taken on a section of the real axis, and is analytic for $x>0$, in certain cases can this imply that $\...
J. Zimmerman's user avatar
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I'm not an expert in analysis whatsoever, so I might be posing a well-established question, or even an unanswerable one. Also, any suggestion on changes that might make the problem better are welcome. ...
Juan F. Meleiro's user avatar
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2 votes
1 answer
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I have the below function: $$\pi(x) = \frac{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) \cdot r_1}{s_0\cdot \left(1-\left(\frac{s_1}{s_1+x \cdot \lambda}\right)^{k}\right) ...
Arthur's user avatar
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2 answers
267 views

This question might be trivial, but I didn't find a clean reference and have not attempted to prove it myself yet: Let $f:[0,1]\rightarrow [0,1]$ be a continuous and monotonic function such that $f(0)=...
Pedram's user avatar
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6 votes
1 answer
220 views

In a calculation of some momenta of random matrices (GOE), I encounter a functional equation, in the form of a second-order recursion, $$L(s+1)=L(s)+2s(2s+1)L(s-1).$$ Is it familiar to someone ? Is ...
Denis Serre's user avatar
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Consider a set of integrable functions on the interval $(0,1)$. Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function). In such system the ...
Anixx's user avatar
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I am considering the following integral operator: $$K(\sigma)(\theta)=\int_0^{2\pi} \sigma(\theta') J_0(|e^{i\theta}-e^{i\theta'}|)\,d\theta',$$ where $J_0$ is the Bessel function of order $0.$ I am ...
Didier Felbacq's user avatar
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1 vote
1 answer
177 views

How many functions $f:\mathbb{R}\to R\subseteq\mathbb{R}$ are there such that there is an $S\subseteq\mathbb{R}$ containing an interval containing $1$ such that for any $\lambda\in S$ there exist $\...
gmvh's user avatar
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199 views

Let $f$ be a real analytic (on at least $|x|<2$) and real solution of the functional equation $f(0) = 1,f(x+1) = \exp(f(x))$. For the existence of such $f$, see here. Then $$ f(x) = \sum_n a_n x^n ;...
mick's user avatar
  • 799
3 votes
2 answers
734 views

Basically the title explains most of my question here. Purely out of curiosity (I have no real application), I am wondering if there are any "interesting" functions $f$ where we know of a ...
gdoug's user avatar
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1 vote
2 answers
511 views

I am stuck on a recurrence relation with two variables. I'm familiar with techniques to solve recurrence relations with one variable and looked into ways to solve recurrence relations with multiple ...
user675763's user avatar
  • 43
4 votes
0 answers
93 views

Suppose that for any real number $a$, we have a function $f_a:\mathbb R \to \mathbb R$ or such that $f_a(x)$ is monotonically strictly increasing in $x$ and hence invertible on its image. We also ...
user56834's user avatar
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1 vote
0 answers
179 views

There is a whole theory of finding the invariant polynomials for matrix groups $\Gamma$ acting on the polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$. I would be interested in finding invariant ...
Jan-Willem van Ittersum's user avatar
  • 399
0 votes
2 answers
458 views

I've not yet finished a course in functional analysis so I'm unsure how to go about this, but I've always been fascinated by a simple functional differential equation I concocted for almost no reason. ...
CheeseBlues's user avatar
  • 19
1 vote
2 answers
550 views

I've been interested greatly in the study of functional equations for some time now, I've learnt many different techniques for their solution. Currently I have been studying superfunctions and ...
Anthony Corsi's user avatar
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1 vote
0 answers
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Recall that Hardy's inequality involving distance from the boundary of a convex set $\Omega \subsetneq \mathbb{R}^n ; n \geq 1$, asserts that $$ \int_{\Omega}|\nabla u|^p \, d x \geq\left(\frac{p-1}{p}...
Davidi Cone's user avatar
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1 vote
0 answers
50 views

In the Rätz’s sens of orhtoganality, can we find an exemple of an involution u(different to -Id)such that x orthogonal to y then x orthogonal to u(y)
MOHAMED TALLA's user avatar
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11 votes
2 answers
979 views

Let $f\colon [0,1]\to[0,1]$ be given by $f(x) = 1-\sqrt{1-x^2}$, i.e., the increasing auto-homeomorphism of $[0,1]$ whose graph is a quarter circle centered at $(0,1)$. I am interested in what can be ...
Gro-Tsen's user avatar
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4 votes
0 answers
148 views

Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ : $$f(n) = \frac{n^2 + n + 4}{2}$$ so $$ \begin{split} f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\ f(...
mick's user avatar
  • 799
10 votes
2 answers
694 views

I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...
LuckyJollyMoments's user avatar
  • 474
0 votes
1 answer
136 views

I have a function $f(z) \in [0,\infty]$ that satisfies $-1 < f(z) < 0$. I would like to find the minimal $\gamma$ that satisfies: $$ \int_0^{\gamma} f(z)dz = \log(1+f(0)).$$ Clearly, I cannot ...
nir's user avatar
  • 101
1 vote
0 answers
76 views

Now I have equation $F(x) = x \sum_{k\ge 0} g_k F(x) F(qx) \cdots F(q^{k-1} x)$, I need to get the coefficient of $x^n$ in $F(x)$, can I apply $q$-Lagrange Inversion formula to this? Moreover, I have ...
alpha1022's user avatar
  • 11
8 votes
1 answer
542 views

While talking about tetration with my friend the following idea (re)occured. $$f(f(z)) = z ,\quad f(\exp(z)) = \exp(f(z)) \tag{A}\label{A}$$ or variations of it like the weaker $$f(f(f(f(z)))) = z ,\...
mick's user avatar
  • 799
0 votes
0 answers
334 views

I'm interested in finding numerical approaches to solving functional equations such as f(xy)=f(x)+f(y), where the equations had no derivatives or integrals, and contains arguments involving x and y . ...
Doug Brunson's user avatar
  • 21
4 votes
2 answers
366 views

A symmetric function is a function $f:\mathbb R^n\to \mathbb R$ such that $f(x_1,\ldots,x_n)=f(\sigma(x_1,\ldots,x_n))$ for every permutation $\sigma\in S_n.$ The most commonly encountered symmetric ...
Nick Belane's user avatar
  • 79
2 votes
0 answers
97 views

I wanted to ask if anyone knows of good texts/resources on methods for solving holonomic recurrence relations (if there are any general analytical approaches): $$p_1(n)a(n)+p_2(n)a(n-1)+\dotsb+p_k(n)a(...
Doug Brunson's user avatar
  • 21
4 votes
2 answers
449 views

Let $a_n$ be a sequence of strictly positive real numbers such that $\lim_{n \to \infty}a_n=0$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ that admit primitives (i.e. there exists a function $F:...
Shthephathord23's user avatar
  • 341
9 votes
4 answers
2k views

Find all continuous and bounded functions $g$ with : $$\forall x \in \mathbb R, 4g(x)=g(x+1)+g(x-1)+g(x+\pi)+g(x-\pi).$$ I have posted this question here, but received no answer.
Dattier's user avatar
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