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.
831
questions
4
votes
2
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Analytic continuation of $_4F_3(1)$
The Gauss theorem
$${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$
allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series ...
- 41
4
votes
1
answer
449
views
Summations in $\tan^2$
Hey all,
I was just wondering if anyone had come across the following identities, valid for $m\in\mathbb{N}$. I've used Abramowitz and Stegun, Maple, Mathematica etc but can't find them anywhere. I ...
- 211
4
votes
1
answer
370
views
Inequality for Fourier transform of a power exponential function
Let
$$
f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }},
x \in \mathbb{R}, 0<\alpha<2,
$$
where
$\phi_1(\alpha)=\frac{\alpha}{2}
\left\{{\{\Gamma(3/\alpha)\}^{1/...
- 168
4
votes
4
answers
640
views
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+\...
- 838
4
votes
1
answer
303
views
Integral operator with Bessel kernel
For $x,y\ge 0$, let
$$
k(x,y)= \frac {J_1(2\sqrt{xy})}{\sqrt{xy}},
$$
where $J_1$ is the the Bessel function of the first kind
$$
J_{1}(z)=\sum_{k=0}^{\infty}(-1)^{k} \frac{\left(\frac{z}{2}\right)^{2 ...
- 650
4
votes
1
answer
202
views
Perform an integration over the unit interval of a two-parameter expression involving a Gauss hypergeometric function
In a quantum-information-theoretic context, I've encountered the problem
of integrating over $r \in [0,1]$, the function
\begin{equation}
r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2}...
- 723
4
votes
1
answer
1k
views
Infinite summation formula of Bessel functions
I would like to find a closed form for the following series involving the Bessel function $J_k(z)$:
$$
\sum_{k=0}^{+\infty}\frac{(\mu)_{k}}{k!(\lambda)_{k}}t^k\left(\frac{z}{2}\right)^{k}J_{k+\nu}(z),...
- 135
4
votes
1
answer
166
views
Monotonicity of integral of Bessel functions
Is it known, and if yes how does one show, that the function
$$
\psi(n):=n\int_0^{+\infty} e^{-x}I_0\left(\frac{x}{n}\right)^{n-1}I_1\left(\frac{x}{n}\right)\mathrm{d}x$$
is decreasing for all $n\ge ...
- 41
4
votes
1
answer
137
views
Jacobi elliptic functions with modulus on the unit circle
I am gathering some available informations on Jacobi elliptic functions $sn(z,k)$, $cn(z,k)$, $dn(z,k)$ with $k\in\mathbb{C}$, $|k|=1$. I can not find much on them in standard references (Abramowitz&...
- 2,188
4
votes
1
answer
201
views
Homeomorphisms that admit a decomposition
Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$.
If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ \text{...
- 183
4
votes
1
answer
304
views
Generalization of Frobenius formula involving Macdonald polynomials
Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as
\begin{equation}
p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~,
\end{...
- 2,273
4
votes
1
answer
1k
views
Quadratic Transformation of the Hypergeometric Function 2F1
The function ${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)$ can be transformed (as reported by A. Erdélyi) by the following formula
${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)=
(1-z)^{...
- 447
4
votes
1
answer
2k
views
minimal polynomials of trig functions of ($k \pi/p$) and divisibility of coefficients by p
Take an odd prime $p$ and put $x_0:=\sum\limits_{j=0}^{p-1}\left(a_{j}\sqrt{p}\cos\dfrac{j\pi}p+b_{j}\sin\dfrac{j\pi}p +c_{j}\tan\dfrac{j\pi}p\right)$, where the $a_{ij}$ are integers. If $f$ denotes ...
- 13.2k
4
votes
0
answers
133
views
Uniform bound for Bessel functions
In NIST the following bound is claimed: $|J_\nu(x)|\le 1$ for all $x,\nu\ge0$. This is trivial for integer $\nu$, and it is pretty easy to prove a bound with 1 replaced, say, by 2. Does anyone have a ...
- 8,775
4
votes
0
answers
160
views
Asymptotic analysis for a double integral related to Airy functions
Let $Ai(x,y)$ be the Airy kernel which is given by
\begin{equation}\label{equ2.12}
Ai(x,y)=
\begin{cases}
\dfrac{Ai(x)Ai'(y)-Ai(y)Ai'(x)}{x-y}, & x\ne y, \\
Ai'(x)^2-xAi(x)^2 & x=y. \\
\end{...
- 869
4
votes
0
answers
191
views
Two questions on asymptotic expansion of confluent hypergeometric functions for real variable $x, |x| \to \infty$
I'm looking into the asymptotic expansion for confluent hypergeometric function $_1F_1(a;b;z) \equiv M(a;b;z)$ and I've two quick questions regarding its asymptotic behavior for real values $x,$ i.e. ...
- 223
4
votes
0
answers
184
views
Inverse mellin transform
Let $K_1(t)$ be the K-Bessel function, then we have
$\int_{0}^{\infty}K_v(y)y^s\frac{dy}{y}=2^{s-2}\Gamma(\frac{s+v}{2})\Gamma(\frac{s-v}{2})$ See page 106 of Bump's book Automorphic forms and ...
- 609
4
votes
0
answers
681
views
Why are there elementary equations that are not solvable in closed form?
Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships.
$\log\colon x\mapsto\log(x)$; $x\...
- 1,063
4
votes
0
answers
95
views
Bessel in matrix?
Let $M_n$ be the matrix
$$M_n=\begin{pmatrix}
1&\binom{1}{1}\binom{1-1}{1-1} &0 &0\qquad \qquad \dots &0\\
1&\binom{2}{1}\binom{2-1}{1-1} &\binom{2}{2}\binom{2-1}{2-1} &0 \...
- 41.8k
4
votes
0
answers
238
views
Generalize $\frac1{48^{1/4}\,K(k_3)}\,\int_0^1 \frac{dx}{\sqrt{1-x}\,\sqrt[3]{x^2+27x^3}}=\,_2F_1\big(\tfrac13,\tfrac13;\tfrac56;-27\big)=\frac47\,$?
These involve separate cases $a=\tfrac13,\,a=\tfrac14,\,a=\tfrac16$ of,
$${_2F_1\left(a ,a ;a +\tfrac12;-u\right)}=2^{a}\frac{\Gamma\big(a+\tfrac12\big)}{\sqrt\pi\,\Gamma(a)}\int_0^\infty\frac{dx}{(1+...
- 12.1k
4
votes
0
answers
348
views
Christoffel-Darboux type identity
The classical Christoffel-Darboux identity for Hermite polynomials reads
$$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^{n+1} n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y}.$$
I am ...
- 613
4
votes
0
answers
161
views
Functional equations about Conway's box function
Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence).
The ...
- 203
4
votes
0
answers
235
views
A connection between basic hypergeometric series and number theory
I am studying functions given by the power series:
$$f(z)=1+\sum_{n=1}^{\infty}\frac{z^n}{(1-q)(1-q^2)\cdots(1-q^{n})}.$$
The parameter $q$ is usually assumed to be such that $|q|<1$. Then it is ...
- 2,188
4
votes
0
answers
176
views
numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums
I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for $e^{-x}$, because a computation directly from the polynomial ...
- 653
4
votes
0
answers
207
views
Asymptotic expansion of Mellin transform of products of modified Bessel function K
Let $n\ge 1$ be an integer, let
$$F(x,y)=\int_0^\infty u^{n(x+y)} (K_{x-y}(u))^n du$$
for $x,y\ge 0$.
When $n=1$, this is just Mellin transform of the Bessel K function. When $n=2$, $F(x,y)$ has ...
- 443
4
votes
0
answers
107
views
Is there a nice way to invert this expression?
Let us first define the Euler polynomials to be the polynomials $P_n(q)$ that satisfy
$$
\frac{qP_n(q)}{(1 - q)^{n+1}} = \Big(q\frac{d}{dq}\Big)^n\frac{q}{1 - q}.
$$
For example, $P_0(q) = P_1(q) = 1$...
- 6,240
4
votes
0
answers
676
views
What is the status on questions related to Bhargava's factorial function?
In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like:
For $k, l \in \mathbb{Z}$, we have $k! \times l!$...
- 7,646
4
votes
0
answers
202
views
System of linear ODEs with hypergeometric coefficients
For quite some time I have been trying to solve the following system of differential equations for the two functions $G$ and $H$ defined on the interval $[0,1]$:
$$
\begin{align}x G''(x)=&\mathscr{...
- 656
4
votes
0
answers
98
views
Boersma and Glasser formula
In http://iopscience.iop.org/0305-4470/38/8/005 (A differentiation formula for spherical Bessel functions) Boersma and Glasser proved the following interesting formula $$\left(1-\frac{\sqrt{z^2+a^2}}{...
- 16.4k
4
votes
0
answers
432
views
Solvable parametric $7$th and $13$th degree equations using $\eta(q)/\eta(q^p)$?
Q: Why is that some polynomial relations between eta quotients have a solvable Galois group, even if the deg is $n>4$?
For example, we have the well-known modular equation,
$$u^6 - v^6 + 5u^2v^2(...
- 12.1k
4
votes
0
answers
403
views
Why does the plastic constant appear in the snub icosidodecadodecahedron?
The golden ratio,
$$\phi =\frac{1+\sqrt{5}}{2}$$
appears (among other polyhedra) in the Platonic solids icosahedron and dodecahedron, and it's quite easy to see the significance of the discriminant $...
- 12.1k
4
votes
0
answers
218
views
A second polylogarithm ladder for the tribonacci and n-nacci constants
In "A seventeenth-order polylogarithm ladder", on page 6 (eq 25), Bailey et al give a dilogarithmic ladder that begins with,
$$0 = \operatorname{Li_2(\alpha_1^{-630})}-2\operatorname{Li_2(\alpha_1^{-...
- 12.1k
4
votes
0
answers
319
views
A coincidence concerning Fermat primes, binomial sums, and eta quotients?
Let $w_k$ be a primitive k th root of unity, where k is a power of 2. In response to a question, Robert Israel gave the solution,
$$\sum_{n=0}^\infty \frac{(-1)^n}{\binom{kn}{kn/2}} = \frac{2^k}{2^...
- 12.1k
4
votes
0
answers
1k
views
Cubic polynomials with "nice" roots, which can be expressed by trig functions of rational angles
Consider the cubic polynomial $x^3-ax+b$ for $a,b\in\mathbb N$.
It has three real roots which, by Cardano's formula, can of course be written in closed form using thirds of angles or cube roots of ...
- 13.2k
4
votes
0
answers
435
views
Why is Mellin-inverse of Gamma periodic?
Specific Case
The periodicity is obvious from computation:
$$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$
However, is there a way to see directly from the integral ...
- 1,243
3
votes
3
answers
544
views
Differential equation with some constraints
I posted this to stackexchange, and after some hours got a comment that was so pessimistic about finding some neat orderly solution, that I'm posting it here too. (In case anyone cares, this is ...
- 11.9k
3
votes
2
answers
752
views
An integral identity evaluating the gamma function
While reading a number theory paper I encountered the identity
$$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$
...
- 7,199
3
votes
2
answers
386
views
Is this infinite series related to some well-known special functions?
Please allow me to resort once again to the expertise of the MathOverflow community :
During research I encoutered the following infinite series :
$$\sum_{n=-\infty}^{+\infty} \frac{u^{2n}}{1+\rho^{...
- 1,942
3
votes
3
answers
607
views
A "known" tangent half-angle formula?
In another posting I wrote about a trigonometric relation I had derived, but that ended up not being the main point of the posting:
Strange pattern in rounding errors?
So as long as we're here, let'...
- 11.9k
3
votes
2
answers
1k
views
Dilogarithm, tetrahedrons, and hyperbolic space
The Bloch-Wigner function $D(z)$ gives the volume of an ideal tetrahedron in the hyperbolic space $\mathbb{H}^3$. Here $z$ is the cross-ratio $(z_1,z_2,z_3,z_4)$ parametrizing the tetrahedron in $\...
- 265
3
votes
3
answers
8k
views
Finding the length of a cubic B-spline
Can I find an analytical solution to the the length of an 2-dimensional cubic B-spline? All I can find are chorded approximations and the opinion that the analytic solution is "unbearably ...
- 133
3
votes
2
answers
559
views
Investigation of $\sum \limits_{k=-\infty}^\infty \frac{x^{k+n}}{ \Gamma(k+n+1)}$ where $n \in C$? [closed]
$$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}$$
because $x!=\Gamma(x+1)$ where $x$ is non-negative ...
- 302
3
votes
3
answers
629
views
Asymptotic forms of Legendre functions for large degree
Does anyone know where to find (or how to obtain) expressions for the Legendre functions for large degree, to second order? For example, to first order the expressions are
$$
P_n(\cosh(x)) ~
\...
- 573
3
votes
2
answers
330
views
Where to find or how to prove that the ratio of two Bernoulli polynomials is increasing?
It is well known that the classical Bernoulli polynomials $B_j(t)$ are generated by
\begin{equation*}
\frac{s\operatorname{e}^{ts}}{\operatorname{e}^s-1}=\sum_{j=0}^{\infty}B_j(t)\frac{s^j}{j!}, \quad ...
- 838
3
votes
2
answers
585
views
Non-asymptotic upper bound of right tail of Gamma function
I'm wondering if there is any non-asymptotic upper bound for the following Gamma function:
$$f_a(x)=\int_{x}^{\infty}t^a\exp(-t)dt$$
for $x>0,a>0$? Something like $x^a\exp(-x)$?
- 1,475
3
votes
2
answers
168
views
A calculation involving Lerch Transcendents
The Lerch Transcendent is defined here as
$$\Phi(z,s,a):=\sum_{k=0}^\infty \frac{z^k}{(k+a)^s}.$$
I am interested in the case $z=\frac 12,$ $s=1.$ The following limit showed up in estimating uniform ...
- 1,269
3
votes
2
answers
511
views
What is known about this power series?
In the course of some calculations, I came across the following powers series.
For fixed $C>1$ let
$$
f_C(u)=\sum_{k=0}^\infty\frac{u^k}{C^{k^2}}.
$$
This series converges for all $u\in\mathbb C$, ...
user1688
3
votes
2
answers
275
views
$\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} z^{2k}$ is an elementary function
I try to calculate the following series
\begin{align*}
S_{n,m}(z)=\sum_{k=0}^{\infty} {\frac{(-1)^{k} \Gamma(\frac{2k+n+m}{2})\,\Gamma(\frac{2k+m-1}{2})}{k!\Gamma(k+\frac{m}{2})}} \, z^{2k},
\end{...
- 588
3
votes
1
answer
236
views
How to compute $\int_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega)$
I would like compute the following
$$I_{t,x,y} = \int_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega); $$
where $\mathbb S^2$ is the two-...
- 588
3
votes
1
answer
452
views
An inequality involving Bessel functions of imaginary order
The following inequality: $$\frac{\pi k}{\sinh{(\pi k)}}\;|J_{ik}(\tau)|^2\le 1,\;\;\;k,\tau\ge 0,$$ for Bessel function $J_{ik}(\tau)$, I found in http://link.springer.com/article/10.1134%2F1.558677 (...
- 16.4k