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.

learn more… | top users | synonyms

0
votes
1answer
89 views

Existence of bounded $n-$th derivative of the solution of differential equation

This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, ...
10
votes
0answers
140 views

A hypergeometric puzzle

$$ 143\,\sqrt {3}\;{\mbox{$_2$F$_1$}\left(\frac{1}{2},\frac{1}{2};\,1;\,{\frac {3087}{8000}}\right)}= 40\,\sqrt {5}\; {\mbox{$_2$F$_1$}\left(\frac{1}{3},\frac{2}{3};\,1;\,{\frac ...
4
votes
1answer
68 views

Legendre Q(n,x) function coefficients in terms of P(n,x) coefficients

Empirically, the Legendre functions of second kind, $Q_n(x)$, appear to be of form $$ Q_n(x)=\frac{P_n(x)}{2} \cdot\ln(\frac{1+x}{1-x})+p_n(x), $$ with $P_n(x)$ the Legendre polynomials of first kind ...
11
votes
2answers
995 views

How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...
1
vote
0answers
49 views

L2 norm of a M-Whittaker function

Let $M_{\kappa,\mu}(z)$ be the Whittaker function, as defined here http://en.wikipedia.org/wiki/Whittaker_function. Does any one know the evaluation of the following integral? ...
2
votes
0answers
69 views

Inequality related to regularized incomplete beta function

I am trying to analyze the following inequality related to regularized incomplete beta function $I$: $$p^{c}/2\geq I_{\frac{p}{1+p}}(\alpha,\alpha)= ...
4
votes
1answer
77 views

Sharp upper bounds on hypergeometric function ${}_2F_1[a,b,c;z]$ when $|z|\geq1$

Generally, hypergeometric function ${}_2F_1[a,b,c;z]$ is defined using Gauss series ${}_2F_1[a,b,c;z]=\sum_{n=0}^\infty\frac{(a)_n(b)_n}{(c)_nn!}z^n$ with $|z|<1$, and there seems to be a lot of ...
0
votes
0answers
55 views

A solution of a q-difference equation

Is it possible to find a solution of the $q$-difference equation $$f(q^{-1}x)-f(qx)=x(a-x)f(x),$$ with $f(0)=1$, (perhaps) in terms of basic hypergeometric series? Or in another rather explicit form? ...
3
votes
2answers
119 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 ...
2
votes
1answer
79 views

M-Wright function asymptotics

Let $M(z;\nu):= \frac{1}{\pi}\sum_{n=1}^{\infty} \frac{(-z)^{n-1}}{(n-1)!}\Gamma(\nu n)\sin(\nu n\pi)=\frac{1}{2\pi i}\int_{\text{H}_a}\exp(\sigma -z\sigma^{\nu})/\sigma^{1-\nu} d\sigma$, ...
1
vote
2answers
116 views

Evaluate an integral or Fourier coefficients

Consider an integral $$ \int_0^\pi \frac{\cos(kx)}{\cosh(ax)}\ dx $$ there $k\in \mathbb{Z}, a\in \mathbb{R}.$ Of course that is Fourier coefficient for the function $f(x)=\frac{1}{\cosh(ax)}.$ ...
0
votes
0answers
70 views

Integral of Bessel function of 1st kind with complex exponential

Does someone know the solution (simple closed form) of one of theses integrals: $$\int_0^t J_l(s) e^{-iA(t-s)}ds$$ $$\int_0^t \frac{J_l(s)}{s} e^{-iA(t-s)}ds$$ with $l>0$, $t>0$, $\Re(A)>0$, ...
6
votes
1answer
159 views

Abel's five terms relation from Yang-Baxter equation?

Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations? If yes, how? Thanks for any help.
3
votes
1answer
162 views

Computing Reciprocal Gamma

Reciprocal Gamma $1/\Gamma(z)$ is an entire function and so it has a convergent Taylor series expansion which was given in its wikipedia article. ...
0
votes
1answer
120 views

Legendre differential equation with additional term

In an application I encountered the ODE $$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right) \right) \left( ...
9
votes
1answer
246 views

Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $x_n$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$. My ...
2
votes
1answer
323 views

A functional inequality

$g:[0,1]\to[0,1]$ continuously differentiable and increasing such that for all integers $t>0$ and for all $r\in(0,1)$, $g(r^{t+1})>g(r)\cdot g(r^t)$. Does this imply that for all ...
3
votes
1answer
160 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 ...
1
vote
3answers
216 views

State of the Art in Approximating Fresnel Integrals

Background of my question is, that I need to calculate Clothoids and I found an AMS article "Chebyhev Approximations for Fresnel Integrals" by W.J. Cody from 1968 ...
3
votes
1answer
134 views

Numerical Evaluation of Some Triple Integral involving Negative Powers

Let $\beta_i\in (-1/2,0)$, $i=1,2,3,4$. I'm interested in obtaining numerical value of the following integrals: $$ \int_{0<u_1<u_2<u_3<1} (1-u_1)^{\beta_1}(1-u_2)^{\beta_2} ...
3
votes
2answers
233 views

Recognize this sum

I have strong feeling that the above function $$f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)}$$ is a known special function but I can't seem to recognize it. Here $\Gamma(x)$ ...
8
votes
2answers
287 views

On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation ...
4
votes
1answer
156 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 \ ...
1
vote
2answers
142 views

Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero?

I want to know under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where ${E_{\alpha ,1}}(z) = \sum\limits_{k = 0}^\infty ...
1
vote
2answers
212 views

Looking for a limit related to the series in a previous post

Can any one show that the following limit? $$ \lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1. $$ If one uses the ...
1
vote
0answers
43 views

Inversion of incomplete elliptic integral of third kind?

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)? That means given $\Pi(n,u,m) = f(x)$, I would like to obtain $u$ as ...
2
votes
0answers
50 views

Jacobi triple product for multidimensional lattices

The Jacobi triple product identity gives as a special case a product formula for the theta function of a 1-dimensional lattice. Is there a more general product formula for the theta function of an ...
9
votes
1answer
825 views

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
1answer
158 views

Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by $\quad L_n (x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad $ I want to prove that $$ \forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...
1
vote
2answers
185 views

Is there a class of functions closed against differentiation besides elementary? [closed]

Is there a finite set $P$ of non-elementary functions $f_n$ such that the derivative of any function $f$ from that set is not elementary, but expressible with functions from the same set $P$ plus ...
1
vote
1answer
89 views

The asymptotic behavior of hypergeometric function around -1

Recently, in studing some specific orthogonal polynomials on unit circle, I was lead to study the asymptotic behavior of the following hypergeometric function at the neighberhood of $-1$: $$ ...
3
votes
1answer
201 views

Asymptotic expansion of modified Bessel function $K_\alpha$

An integral representation for the Bessel function $K_\alpha$ for real $x>0$ is given by $$K_\alpha(x)=\frac{1}{2}\int_{-\infty}^{\infty}e^{\alpha h(t)}dt$$ where ...
7
votes
4answers
210 views

Is there an oscillating analog of the Gaussian distribution?

It frequently happens that, in some famillies of polynomials with positive coefficients, the coefficients of large polynomials look like a bell curve and tend to the distribution function of the ...
4
votes
2answers
475 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 $u\in\mathbb C$, ...
6
votes
1answer
237 views

Under which constraints are there only finite numbers of irreducible eta product identities?

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 $e_k:=\eta(q^k)$. An eta product identity (or eta identity for ...
15
votes
2answers
685 views

Is there a known solution to $f(x) = (1-x)f(x^2)$?

The functional equation $f(x) = (1-x)f(x^2)$ (with $f(0)=1$) has a simple solution that can be expressed as a rapidly converging infinite product $$f(x) = \prod_{n=0}^\infty (1 - x^{2^n}) = ...
0
votes
0answers
63 views

What function is “$U_{\nu}(\cdot, \cdot)$”?

I was searching in the Prudnikov (vol. 2) how to solve an integral and I finally found it. However, I didn't recognized a function that appears in the answer. Integral 1.8.2.4: $$ \int_0^x x^{\nu+1} ...
1
vote
2answers
129 views

Quantifying simplicity, in the case of trigonometric and exponential functions

The pair of identities the sine and cosine of a sum of two terms as functions of the sines and cosines of the terms separately is not as simple as the identity that expresses the exponential of a sum ...
0
votes
0answers
36 views

Inverse Barnes G(n) function?

It is known that Barnes $G$-function is an analytic continuation of the $G$-function defined in the construction of the Glaisher-Kinkelin constant. http://mathworld.wolfram.com/BarnesG-Function.html ...
3
votes
0answers
160 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 + ...
35
votes
1answer
1k views

There's something strange about $\sqrt{d\big(j(\tau)-1728\big)}$

Given the j-function $j(\tau)$, I was looking at, $$F(\tau) = \sqrt{d\big(j(\tau)-1728\big)}$$ which appears in Ramanujan-type pi formulas. Let $C_d$ be the prime factors of the constant term of the ...
3
votes
1answer
132 views

On Continuous Replicative Functions

I asked this question on math.stackexchange here, but it did not receive much attention. Thus, I was suggested to post it here. Knuth, in The Art of Computer Programming Vol. 1, defines a replicative ...
2
votes
0answers
60 views

Associated Legendre/Gegenbauer functions with complex degree at larger order

I am interested in approximating the associated Legendre function (also known as conical function) \begin{equation} P_{-1/2 + i p}^{\frac{2-N}{2}}(x) \end{equation} when $N \to \infty$. The real ...
1
vote
3answers
172 views

A definite integral of hypergeometric function 2F1

I am wondering whether there exists a closed form for the definite integral $$F(x)=\int_0^1t^{-a}(1-t)^{N}(1-xt)^{-a}{}_2F_1(-a,k-a-1/2,k-a;4xt(1-xt))dt,$$ where $a\in(0,1)$ and $N,k$ are positive ...
1
vote
1answer
127 views

Accurate bounds for derivatives of Legendre polynomials

Let $P_n(x)$ denote the $n$th Legendre polynomial. What bounds can one give for $d_{n,m}(x) = |\frac{d^m}{dt^m}P_n(t)|_{t=x}$ assuming that $|x| \le 1$? Clearly $$d_{n,m}(x) \le d_{n,m}(1) = ...
1
vote
0answers
75 views

Quadratic transformation of hypergeometric function 2F1

I want to know whether there is some transformation between $_2F_1(a,b;c;x)$ and $_2F_1(a',b';c';x(1-x))$. Here is an example called the Kummer quadratic transformation, which may be known to most of ...
2
votes
0answers
74 views

From Selberg integral to Dyson integral

My question is from the drivation from Slberg integral to Dyson integral in this paper: Selberg integral : $$ S_n(\alpha,\beta,\gamma) = \int_0 ^1 \cdots \int_0 ^1 \prod_{i=1}^n ...
3
votes
0answers
132 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)~, ...
4
votes
1answer
299 views

Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...
2
votes
1answer
186 views

About large z behavior of hypergeometric function $_2F_1(1/2,1/2,1;z)$

The hypergeometric function $_2F_1(\large \frac{1}{2},\frac{1}{2},1;\frac{1-\frac{u}{\Lambda^2}} {2} \large)$ at large $\mid u\mid$ can be approximated by $$ -\frac{\Lambda}{\pi} \sqrt{\frac{2}{u}} ...