Questions tagged [special-functions]
Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.
99
questions
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votes
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Pochhammer symbol of a differential, and hypergeometric polynomials
I have a minor result which I'm sure has come up somewhere before but I can't seem to find it.
Consider a confluent hypergeometric function of the form
$$\newcommand{\ff}{{}_1F_1}
\ff(b+k;b;z)\...
9
votes
2
answers
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views
Сlosed formula for $(g\partial)^n$
The objective is to obtain a closed formula for:
$$
\boxed{A(n)=\big(g(z)\,\partial_z\big)^n,\qquad n=1,2,\dots}
$$
where $g(z)$ is smooth in $z$ and $\partial_z$ is a derivative with respect to $z$.
...
34
votes
2
answers
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Representations of $\zeta(3)$ as continued fractions involving cubic polynomials
$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$
where $k\in\mathbb Q$ and $p$ is a ...
6
votes
1
answer
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views
On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence
In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
1
vote
2
answers
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views
Recurrence for the sum
Let $m\geq 2$ be a fixed integer.
Let
$$f(n):=\begin{cases}
mf\left(\frac{n}{m}\right),&\text{if $n\mod m = 0$;}\\
1,&\text{otherwise}
\end{cases}$$
then if we have
$$a(n):=\begin{cases}
1,&...
75
votes
3
answers
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Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?
Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs.
I encountered other analogies as well.
$${\begin{aligned}\gamma &=\int _{0}^{1}\int _{...
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votes
5
answers
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Groups, quantum groups and (fill in the blank)
In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...
4
votes
1
answer
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views
$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$
The Fejer-Jackson inequality as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
I conjecture that the inequality as follows holds:
$$\sum_{...
40
votes
7
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How should an analytic number theorist look at Bessel functions?
(And a related question: Where should an analytic number theorist learn about Bessel functions?)
Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
28
votes
1
answer
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Can we use the Rogers-Ramanujan cfrac to parameterize the Fermat quintic $x^5+y^5=1$?
Define $\color{blue}{q=e^{2\pi i \tau}}$ and Dedekind eta function $\eta(\tau)$. Note: I found these relations empirically, but their consistent forms suggest they can be rigorously proven.
I. $p=2$...
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votes
2
answers
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How to prove Lambert's W function is not elementary?
Liouville's theorem gives such a proof for antiderivatives of functions like $e^x/x$ or $e^{x^2}$, and differential Galois theory extends that to Bessel functions, say. But what tools exist for ...
15
votes
2
answers
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A tantalizing Gamma quotient to challenge the Rohrlich-Lang Conjecture
The Rohrlich-Lang Conjecture for polynomial relations in Gamma values predicts that all polynomial relations between Gamma values over $\mathbb Q$ come from the functional equations satisfied by the ...
11
votes
1
answer
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Has anyone seen this series?
I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...
7
votes
1
answer
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Eigenvectors of a matrix with entries involving combinatorics
In the question Eigenvalues of a matrix with entries involving combinatorics No_way asked about eigenvectors of $n\times n$ matrix $M$ with entries \begin{eqnarray*}
M_{ij}=(-1)^{i+j}F(n, l, i, j),
\...
7
votes
2
answers
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Duality of eta product identities: a new idea?
Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs:
let's call two eta product identities $\sum\limits_{i=1}^r a_iP_i=...
4
votes
2
answers
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views
Finding a sextic analogue to the solvable octic $\frac{(x + 1)^6(x^2 + x + 7)}x = -k^3$ where $e^{(\pi/3)\sqrt{d}}\approx k^3+41.999999999999\dots$
I. Degree 8
Assume the $j_i$ to be free parameters. The following octics in $x$ belong to $8T43,$ have group $\text{PGL}(2,7)$, and order $2\times168 = 336.$
\begin{align}
{j_1}\; &=\frac{(x^2 + ...
4
votes
4
answers
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What or where is the series expansion of the function $\ln\bigl(\frac{\tan x}{x}-1\bigr)$ or $\ln(\tan x-x)$ around $x=0$?
It is known that
\begin{equation*}
\tan x=\sum_{k=1}^{\infty}\frac{2^{2k}\bigl(2^{2k}-1\bigr)}{(2k)!}|B_{2k}|x^{2k-1}, \quad |x|<\frac{\pi}{2}
\end{equation*}
and
\begin{equation*}
\ln\tan x=\ln x+\...
4
votes
1
answer
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Extension of the Jacobi triple product identity
The Jacobi triple product and the mathematical identity of it is:
$$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$
I would ...
3
votes
1
answer
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About eigen-functions of the Gaussian kernel
If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
3
votes
2
answers
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Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$? [closed]
$$e^x=\sum \limits_{k=0}^\infty \frac{x^k}{k!}$$
We can rewrite the equation as
$$e^x=\sum \limits_{k=0}^\infty \frac{x^k}{ \Gamma(k+1)} \tag{1}$$
because $x!=\Gamma(x+1)$ where $x$ is non-negative ...
2
votes
0
answers
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views
Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?
I. Two recurrence relations
The first one was also discussed in this MO post. We have the similar,
\begin{align}
(n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...
1
vote
0
answers
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Multivariate solution to Lambert W / product-log function
Consider solving the following system for $x$
\begin{align*}
a - b e^{\theta x} - cx = 0
\end{align*}
According to your favorite computer algebra program, one possible (and the simplest) is
\begin{...
1
vote
0
answers
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Ask for a proof of an inequality involving the Bernoulli numbers
Let $B_k$ be the Bernoulli numbers and let
\begin{equation}
T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1.
\end{equation}
Prove the inequality
\begin{equation*}
\frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
43
votes
6
answers
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Why are hypergeometric series important and do they have a geometric or heuristic motivation?
Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia ...
40
votes
1
answer
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Curious $q$-analogues
Consider the Fibonacci polynomials
$$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j} $$
and the Lucas polynomials
$$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \...
28
votes
4
answers
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The function $\sum_{0}^{\infty} x^n/n^n$
The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...
28
votes
2
answers
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A 14th and 26th-power Dedekind eta function identity?
Given the Dedekind eta function $\eta(\tau)$. Define $m = (p-1)/2$ and a $24$th root of unity $\zeta = e^{2\pi i/24}$.
Let p be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$:
$$\sum_{k=0}^{p-...
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votes
0
answers
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Trigonometry related to Rogers--Ramanujan identities
For integers $n\ge2$ and $k\ge2$, fix the notation
$$
[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad
[m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.
$$
Consider the following trigonometric numbers:...
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votes
2
answers
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Is there a "right" proof of Riemann's Theta Relation?
Let $\theta$ denote the usual Jacobi Theta function (with auxiliary parameter $\tau = i$, for simplicity), i.e.
$$
\theta(z) = \sum_{n \in \mathbb{Z}} \exp(-\pi (a + n)^2 + 2 \pi i n z) \ .
$$
I'm ...
15
votes
2
answers
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The complete list of continued fractions like the Rogers-Ramanujan?
I have two questions about q-continued fractions, but a little intro first. Given Ramanujan's theta function,
$$f(a,b) = \sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2}$$
then the following,
$$A(q) =...
15
votes
0
answers
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views
Prove $\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge {\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$
I would like to prove that
$$\int_{0}^{\infty} \cos(\omega x) \exp(-x^{\alpha}) \, {\rm d} x \ge
{\alpha^2 \sqrt{\pi} \over 8} \exp \left( -\frac{\omega^2}{4} \right)$$
for any $\omega > 0$ and $...
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votes
1
answer
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Geometric meaning of a trigonometric identity
It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then
$$
a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta +...
14
votes
6
answers
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Closed form of an infinite series
Does the following infinite series have a closed form?
$$
\sum_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}
$$
13
votes
3
answers
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views
Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?
For naturals $n\ge m$, define
$$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$
with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $.
Is it ...
13
votes
1
answer
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Connection between Infinite continued fractions, elliptic integrals and AGM
It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean.
$$C(x) = x + \frac{1^{2}}{2x + \frac{3^{...
11
votes
0
answers
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What properties characterize the function $L(x) = x+\exp(x) \log(x)$?
As might be known, the function $L(x) = x+\exp(x)\log(x)$ plays a prominent role in the Lagarias formulation of the Riemann hypothesis:
$$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$
My question is, ...
11
votes
1
answer
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What's the difference between a Riemann theta and a Siegel theta function?
One of the things I'm working on has required me to look into the literature of multidimensional theta functions, and I've gotten a bit confused on a few naming details.
A look at the DLMF says that "...
11
votes
1
answer
343
views
A problem involving the Error Function
I am looking at the following function on the domain $x\geq 0$:
$$F(x)=(x+a)e^{x^2}(1-\mathrm{erf}(x))-\frac{b}{\sqrt\pi},$$
where $a>0$, $0<b<1$ are parameters. From plotting this function ...
11
votes
0
answers
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Linear eta product identities - how many are there?
For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n}) $, let for brevity $e_k:=\eta(q^k)$. With this notation, a blog entry of Michael ...
10
votes
1
answer
433
views
On Ramanujan's pi formula $\frac 1\pi=\sum_{k=0}^\infty\frac {(4k)!}{k!^4}\frac {Ak+B}{396^{4k}}$ and the solvable quintic $z^5-5z-396 = 0$?
I. Four quintics?
The general quintic can be transformed in radicals to at least three one-parameter forms. For simplicity, assume this free parameter to be some generic "alpha". Hence,
$$x^...
9
votes
0
answers
318
views
When exactly is the principal AGM equal to the optimal AGM?
Definitions
Following the terminology of SageMath, let the principal arithmetic-geometric mean, $\operatorname{AGP}$, of $(a,b)\in\mathbb{C}^2$ for $a\ne 0$, $b\ne 0$, $a\ne\pm b$ be defined as ...
8
votes
1
answer
651
views
lower bound for absolute value of a hypergeometric function
I am working with a certain Gauss hypergeometric function, $_{2}F_{1}(a,a-b;2a;1-z)$, where $a, b \in {\mathbb R}$ with $a, a-b > 0$ and $0<b<1$.
It appears that $\left| _{2}F_{1}(a,a-b;2a;1-...
8
votes
1
answer
565
views
Rotation invariance of an integral
Consider the integral depending on 2 parameters
$$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$
where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
8
votes
0
answers
305
views
How to prove that $ \sum_{m=0}^{\infty} { \Gamma\{(1+2m)/\alpha\}\over \Gamma(1/2+m)} { (-t^2/4)^{m}\over m !} \ge (\alpha/2)^{3}\exp(-t^{2}/4) $
I would love to prove the following inequality
$$
{1\over \sqrt{\pi} } \sum_{m=0}^{\infty}
\Gamma\{(1+2m)/\alpha\}
{ (-t^2)^{m}\over (2m) !}=$$
$$
\sum_{m=0}^{\infty}
{ \Gamma\{(1+2m)/\alpha\}\over \...
8
votes
1
answer
386
views
Lower bound for $\frac{\sum_{i,j}\min((f_i-f_j)^2,(g_i-g_j)^2)}{\sum_{i,j}\max((f_i-f_j)^2,(g_i-g_j)^2)}$
Let $f\in\mathbb{R}^n$ and $g\in\mathbb{R}^n$ be two orthogonal unit vectors such that $\sum_{i}{f_i}=\sum_{i}{g_i}=0$.
Question. Can we prove this?
$$\frac{\sum_{\{i,j\}}\min((f_i-f_j)^2,(g_i-...
8
votes
0
answers
466
views
Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?
Define the Ramanujan theta function $f(a,b)$ as,
$$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$
and the Dedekind eta function,
$$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} \left(1-...
7
votes
5
answers
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views
How to calculate one Cauchy type determinant
As we know, a Cauchy determinant of size n admits the following explicit formula:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y _i)}{\...
7
votes
2
answers
2k
views
expression for infinite series with powers of factorial in denominator
The series
$$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$
has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...
7
votes
2
answers
1k
views
Characterizing the real analytic Eisenstein series
Consider the classical real analytic Eisenstein series
$$
E(z,s)=\left(\pi^{-s}\Gamma(s)\frac{1}{2}\right)\sum_{(m,n)\neq(0,0)}\frac{y^s}{|mz+n|^{2s}},
$$
where $z=x+iy$. We think of $E(z,s)$ as a ...
7
votes
1
answer
1k
views
Trigonometric identities
In the rant I wrote at
http://ncatlab.org/nlab/show/trigonometric+identities+and+the+irrationality+of+pi
I asked: Are these four identities the first four terms in a sequence that continues?
This ...