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.
920 questions
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Which functions $f$ produce zeta values via the infinite summation of the form $\sum_{n \ge 0} f^{(k)}(n)$?
I have observed in recent posts by Zhi-Wei Sun (More conjectural formulas for Riemann's zeta function (IV)) and Deyi Chen (Some series related to $\zeta(3),\zeta(4),\zeta(5),\zeta(6),\zeta(7)$) ...
4
votes
3
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Closed form for the integral $\mathcal{J}_{\Delta'}$ over the complementary simplex
Let
$$
\Delta={(x,y)\in[0,1]^2 : x+y\le 1}
$$
and let
$$
\Delta'={(x,y)\in[0,1]^2 : x+y>1}.
$$
For positive integers $k,N$ consider
$$
\mathcal{J}_{\Delta'}=
\iint_{\Delta'}
x^{k-1}y^{k-1}(1-x-y)^{...
- 61
0
votes
0
answers
51
views
Monotonicity of a unique root curve arising from coupled cumulative entropy functionals
I would like to ask about the following monotonicity problem.
For $g>1$ and $\rho>0$, define
$
\bar b(g,\rho):=\frac{\rho g}{1+\rho g}
$
and
$$
\Phi_\ell(b;g,\rho):=
\begin{cases}
\left(\dfrac{1}...
3
votes
1
answer
241
views
Conjectural series involving the hyperbolic cosine function
Motivated by Questions 508872 and 508945， I have found some new conjectural series involving the hyperbolic cosine function $\cosh(x)=(e^x+e^{-x})/2$.
Conjecture 1. Let $m$ be any nonnegative integer, ...
- 19.2k
2
votes
1
answer
258
views
Admissible 2x2 matrices for dilogarithm
Here are the relevant papers:
Searching for modular companions, by Shashank Kanade (on arXiv, 2019)
The dilogarithm function, by Don Zagier link at Zagier's page (page 46)
The idea is, if we have a ...
- 633
1
vote
0
answers
73
views
Simplicity of complex zeros of a $\Gamma$-ratio / Bernstein function
Question. Fix $\beta>0$ and define
$$\phi_\beta(z)=\int_{0}^{\infty}(1-e^{-zx})(1-e^{-x})^{\beta}e^{-(\beta+2)x}\,dx,$$
interpreted by analytic continuation. With $t=e^{-x}$,
$$\phi_\beta(z)=\int_{...
- 11
6
votes
1
answer
355
views
Fourth power Airy integral with Fourier representation
How can I prove that, by using the Fourier integral representation of the Airy function, that
$$\Omega = \int_{0}^{\infty} \text{Ai}^4(x) \, dx = \frac{\ln(3)}{24 \pi^2}$$
where $\text{Ai}(x)$ is the ...
- 1,374
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votes
0
answers
274
views
Finding all cubic dilogarithm ladders
Don Zaiger gives a procedure for generating dilogarithm ladder relations in the below paper:
https://people.mpim-bonn.mpg.de/zagier/files/tex/LewinPolylogarithms/fulltext.pdf
Specially, page 12, ...
- 633
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1
answer
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views
Reference for formula for Gamma function on Wikipedia
I am looking for a reference to an identity related to Stirling's formula for the Gamma function. The Wikipedia page for the Gamma function states the following: if $\text{Re}(z) > 0$, then
$$\tag{...
- 2,202
0
votes
1
answer
82
views
Reference for sums of rational functions
Wikipedia explains how to evaluate sums of rational functions in terms of the Digamma function.
Unfortunately, I failed to find a quotable reference for this formula; the best I've found is at NIST, ...
- 607
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vote
2
answers
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views
Are shifted Dirichlet L-functions $\sum_{n=1}^\infty \chi(n)(n+a)^{-s}$ studied in the literature?
I am studying generalizations of Dirichlet L-functions through the framework of LC-functions (a generalization of the Hurwitz zeta function). In this context, I have encountered naturally the ...
- 473
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1
answer
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views
A system of T-functions on $(0,\infty)$
Let $r>1$ be real and
\begin{align*}
f_1(x) &= 1,\\
f_2(x) &= (x+4)^r,\\
f_3(x) &=(x+4)^r(x+3)^r,\\
f_4(x) &= (x+4)^r(x+3)^r(x+2)^r,\\
f_5(x) &=(x+4)^r(x+3)^r(x+2)^r(x+1)^r.
\...
- 258
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votes
1
answer
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views
Show the monotonicity of a function involving the difference of the digamma function
Denoting the digamma function by $\psi$, define
\begin{equation*}
g(n,s) = \psi\left(\frac{n+2}{s}\right)+\psi\left(\frac{n}{s}\right)-2\psi\left(\frac{n+1}{s}\right)+\frac{s}{n(n+1)}
\end{...
- 123
1
vote
0
answers
80
views
Integral representation of $\text{Li}^{(n,0)}_{-\nu}(z)$ for $\nu\in\mathbb{N}$ and $n\in\mathbb{N}^{+}$ [closed]
This is not a question since It's OK to Ask and Answer Your Own Questions.
The purpose of this post is just to appear in searches in case someone needs this formula.
Motivation
Let $\text{Li}_{\nu}(z)...
- 137
5
votes
1
answer
713
views
The Bring quintic and the Baby Monster?
I. Quintic
The general quintic was reduced to the Bring form $x^5+ax+b=0$ in the 1790s, while the Baby Monster was found in the 1970s.
We combine the two together using the McKay-Thompson series (...
- 13.3k