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.

**2**

votes

**2**answers

108 views

### Find formula for recurrence relation with two function and two variables

$f(n,k) = 2g(n-2,k-1)+f(n-1,k)$
$g(n,k) = g(n-1,k-1)+f(n,k)$
when $n\le0$ or $k\le0: \quad f(n,k) = 0$
when $n < k:\quad f(n,k) = 0$
when $n-k<-1:\quad g(n,k) = 0$
when $k=0:\quad g(n,k) = 1$
$...

**-1**

votes

**0**answers

121 views

### Integral of $C^\infty$ Analog of Unit Step Function

The function
$$
f(x)\ :=\ e^{1-\frac{1}{\sqrt{1-x^2}}}
$$
has the properties that
$\frac{d^nf}{dx^n}(\pm 1)=0$,
$\frac{d^n}{dx^n}\left(\sqrt{1-x^2}-f(x)\right) = 0$ at $x=0$,
and its integral $F(...

**0**

votes

**1**answer

92 views

### Is there special name for this function ?, $f(n)$ = smallest $k$, s.t. $\log^k(n) \leq 1$

$\log^k(n) = \log(\log(\log(...\log(n))));$
Let $f(n)$ = smallest $k$, s.t. $\log^k(n) \leq 1$
Is there known name for function $f$ ? Or it's an instance of some known function ?
Basically I want ...

**1**

vote

**0**answers

66 views

### Is the ratio of two distinct zeros of a Bessel function of the first kind an irrational number?

Let $J_1$ be the Bessel function of the first kind with parameter $\alpha=1$. Namely, $J_1$ satisfies the differential equation for $y$ given by $x^2y''+xy'+(x^2-1)y=0$ and $J_1(0)=0$.
Is it true ...

**1**

vote

**0**answers

52 views

### Linear dependence of solution?

Consider the function
$f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$
In the following we want to study the ...

**3**

votes

**2**answers

284 views

### Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$

The question is to prove:
$$
\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1.
$$
Numerically it seems to hold true. So I have made some attempts to ...

**2**

votes

**1**answer

190 views

### Schwartz space on $\bigcup_{n=1}^CR^n$

I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...

**-1**

votes

**1**answer

128 views

### How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]

It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$
Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...

**0**

votes

**1**answer

63 views

### Derive unique denominators from number [closed]

I'm wondering if it is possible to run a set of numbers ('target numbers') through a function and get out a number that when queried against in some way (with a 'target' number as all/part of the ...

**1**

vote

**2**answers

130 views

### Integral of product of Gaussian pdf and cdf [closed]

$\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ is the pdf of a standard normal distribution.
$\Phi(x) = \int_{- \infty}^x \phi(t) dt$ is the cdf of a standard normal distribution.
How does one ...

**0**

votes

**0**answers

34 views

### A two argument sepcial function related to Legendre polynomial and Meixner polynomial

This problem raised when I was trying to evaluate a complicated integral. A polynomial with 2 arguments emerged and I could not recognize it. Let's call it $F_n(k,x)$, what I know is that $F_n(0,x)=...

**2**

votes

**3**answers

99 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)) ~
\...

**3**

votes

**0**answers

147 views

### best-possible inequalities for hypergeometric functions

In what follows, let $n$ be a positive integer and $0<a<1/2$. I am interested in the Gauss hypergeometric functions, $_{2}F_{1}( -n, -n-a; 1-a; z)$. Notice that these are polynomials, if that is ...

**2**

votes

**1**answer

119 views

### $\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0 ?$ with $\alpha \ge 1$ and $n=1, 2,\cdots$

Could You give a poof, comment or reference for the inequality as follows:
$$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0$$
for all $n=1,2,3,\ldots$ and $0<x<\pi$ and $\alpha \ge 1$...

**4**

votes

**1**answer

239 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:
$...

**3**

votes

**1**answer

206 views

### Is $\frac{\pi}{4}L_0(z) = \sum\limits_{n=1}^{+\infty} (-1)^{n+1} \frac{I_{2n-1}(z)}{2n-1}$ between Bessel and Struve known?

Based of the detailed attempt to solve the integral $\int e^{\sin(x)} dx$ I stumbled upon a connection between modified Struve and modified Bessel function of the first kind. But, I cannot find a ...

**13**

votes

**1**answer

411 views

### A question on the sine function

The Fejer-Jackson-Gronwall inequality involving the sine function is as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
Here I ask the ...

**1**

vote

**0**answers

44 views

### Bound for truncation error of continued fraction for $E_1(z)$

Let $z \in \mathbb C \setminus(-\infty,0)$. It is known that
$$E_1(z) = \cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+\cfrac{2}{z+\cfrac{3}{1+\cdots}}}}}}.$$
For example, see http://functions....

**0**

votes

**0**answers

29 views

### Reduction of a Jacobi-type continued fraction

I am trying to reduce the following Jacobi-like continued fraction(or J-fraction):
$$f(z)=z+K_{n=1}^{\infty}\frac{R_{n}k^2}{z+Q_{n}}$$
where,
$$R_{n}=n\left(n-\frac{1}{2}\right),\; Q_n=n\left(n+m+\...

**0**

votes

**0**answers

101 views

### An upper bound for the Riemann zeta function

In our recent researches and by using "Limit summability of real functions"
(see this and this), we obtain the following upper bound for the (real) Riemann zeta functionn (as an example of the topic):...

**0**

votes

**0**answers

60 views

### Is $\binom{py-1/2}{py}-\binom{y-1/2}{y}$ decreasing for $y\geq 1$?

Fix $p\in(0,1)$ and consider the function $f:[1,+\infty]\to\mathbb{R}$
$f(y)=\binom{py-1/2}{py}-\binom{y-1/2}{y}$.
From numerical experimentation, it seems that this function is decreasing and ...

**7**

votes

**1**answer

250 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-...

**1**

vote

**0**answers

91 views

### Functional equation with Fourier transform

What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:
$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$
Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....

**0**

votes

**0**answers

93 views

### Barnes's double $\Gamma$-function, $\gamma_{h}(0) = -\frac{1}{12}$?

I apologize if I may have gotten the name of the function incorrectly. The function $\gamma_h(x)$ is defined in Appendix A of the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/...

**0**

votes

**0**answers

96 views

### On certain integrals of exponential functions with respect to Gaussian measures

I have questions about the integral
$$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$
for $a,b,c>0$.
What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...

**1**

vote

**0**answers

56 views

### Finite sum of spherical Bessel functions

In L.G. Afanasyev, A.V. Tarasov, Breakup of relativistic pi+pi- atoms in matter, Phys. At. Nucl. 59 (1996) 2130 the following identity is given for the spherical Bessel functions $j_n(z)=\sqrt{\frac{\...

**0**

votes

**0**answers

74 views

### Gaussian integral over logarithm of shifted error function

Suppose we have the following integral:
$$ \int^{\infty}_{-\infty} \frac{dz}{\sqrt{2\pi}}e^{\frac{-z^2}{2}
} \log\left( \text{erf} \, a (z-b) +1 \right), \ \ \ \ a,b \in \mathbb{R} $$
Does a closed-...

**4**

votes

**0**answers

92 views

### Nekrasov Partition function and the leading term of Prepotential

I've got a pretty basic question from the paper SEIBERG-WITTEN THEORY
AND RANDOM PARTITIONS, https://arxiv.org/pdf/hep-th/0306238.pdf.
In (4.25) the author expressed the partition function ...

**3**

votes

**0**answers

72 views

### Identity for the product of two different associated Legendre polynomials

In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated:
$$
\small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}...

**1**

vote

**0**answers

64 views

### Uniform Asymptotic Approximation of the Whittaker function

I would like to know if there exist a uniform asymptotic approximation of the Whittaker function $W_{\kappa,i\mu}(x)$ for $\kappa<0$, $x >0$, and with $\mu \to +\infty$. The case of $\kappa \ge ...

**4**

votes

**2**answers

157 views

### Are there noteworthy functional properties of the exponential integral?

I noticed that the following function
$$\mbox{Ei}(x) := - \int_{-x}^{\infty}\frac{e^{-t}}{t} \,\mathrm d t$$
occurs increasingly in different areas of physics and in mathematics.
I am wondering if ...

**1**

vote

**1**answer

137 views

### Alfred van der Poorten--rational functions paper

Does anybody has a copy of the following paper:
Alfred van der Poorten, Some facts that should be better known, especially about rational functions; Number Theory and Applications”, Richard A. Mollin ...

**0**

votes

**0**answers

26 views

### Functions of squares of gradients and the spectrum of the Hessian

This is a very ill-formed formed question but kindly indulge me!
Say one has function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ and a constant $a >0$. And let $g = \nabla f$ and $i \in 1,2..,d$. ...

**3**

votes

**1**answer

220 views

### Asymptotic behaviour of function from integral representation

In Appendix A of this paper, it is claimed that the asymptotic behaviour of
$$\phi_1(y,\lambda)=\frac{1}{\Gamma(\frac{1-\lambda}{2})}\int_0^\infty dt~e^{-t}\cos(2y\sqrt{t})t^{-\frac{1+\lambda}{2}},$$
...

**2**

votes

**0**answers

45 views

### The canonical form of the first Painlevé equation

The first Painlevé equation is traditionally written as
$$y''=6y^2+x. $$
Using scaling in both the dependent and independent variables, one can transform this equation into
$$Y''=aY^2+bx $$
for ...

**4**

votes

**1**answer

186 views

### Identities for Chebyshev polynomials of the second kind

While calculating an integral in a quantum mechanical problem by two different methods, I came across the following identity
$$\sum_{k=0}^n\sum_{m=0}^{2k}(-2)^m\binom{2(n-k)}{n-k}\binom{2k}{k}\binom{...

**1**

vote

**0**answers

55 views

### Vibration of point load on a halfspace

The amplitude of vibration of surface of halfspace at a distance r from a point harmonic load of amplitude Q is given by
$ w(r,0) = $
$ Q\over 2\pi G $
$ \int_0^\infty $
$ k^{2}\alpha pJ_0(pr)dp \...

**1**

vote

**1**answer

223 views

### Value of the hypergeometric function

Let $n$, $m$ and $k$ be some (positive) integers such that $(k+3/2)-(n+m/2)<0$. Can the hypergeometric function
$$F\left (n+\frac{m}{2},n+\frac{m+1}{2};k+\frac{3}{2};-\tan^2{\phi}\right) \tag{1}$$
...

**1**

vote

**0**answers

90 views

### Integral involving square of associated Laguerre polynomial and sperical bessel function

In a quantum mechanical problem I encountered the integral
$$I_k=\int_0^\infty x^{2(l+1)-k}j_k(\sigma x)e^{-x}[L_{n-l-1}^{2l+1}(x)]^2 dx,$$
where $j_k(x)$ is a spherical Bessel function, and $\sigma$ ...

**0**

votes

**2**answers

122 views

### Literature about the integral of Bessel $\int_0^x I_{0,1}(u) e^{-a u}du$?

Thanks to sound remarks here and here, and looking again at these equations, I noticed that my puzzle boils down to these 2 special functions:
$$\int_0^x I_i(u) e^{-a u}du, \quad i=0,1$$
where $I_n(u)$...

**0**

votes

**0**answers

21 views

### Show a Gaussian Hypergeometric Function is nondecreasing

For $n, m \in \mathbb{Z}$, $n > 4$, and $\gamma \le 1/n$, show that
$$\frac{n \, _2F_1\left(1,\frac{1}{2} (-m+n+2);\frac{n+4}{2};\frac{n \gamma }{1+(n-m)\gamma}\right)}{(n+2) (1+(n-m)\gamma)}$$
is ...

**1**

vote

**1**answer

159 views

### Root problem involving error function

I ran into this problem in my research:
Let $y_0$ be the root of
$$-(y+a)e^{y^2}\mathit{erfc}(y)+\frac{b}{\sqrt{\pi}}=0$$
on interval $[-a,\infty)$, while $a>0$ and $0<b<1$.
How can I ...

**1**

vote

**1**answer

104 views

### Seeking the derivation of the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}$

In this answer on math.stackexchange.com the Fourier Sine Transform of $x^{2\nu}(x^2+a^2)^{-\mu-1}$ is given in terms of the generalized hypergeometric function:
$$\frac{1}{2}a^{2\nu-2\mu}\frac{\Gamma(...

**6**

votes

**0**answers

122 views

### Reduction of a type of hyperelliptic integrals to elliptic integrals

(Was asked on Math.Stackexchange)
In [1] (hereinafter refered to as "the handbook"), it is said that
... the more general integral (Eq 575.16)
$$ \int R(\tau)\sqrt{(\tau-r_1)(\tau-r_2)(\tau-...

**2**

votes

**1**answer

266 views

### Clausen’s identity for associated Legendre polynomials

Clausen’s identity for Legendre polynomials has the form (see, for example,
A generating function of the squares of Legendre polynomials, by Wadim Zudilin: https://arxiv.org/abs/1210.2493)
$$P_n(\cos{\...

**1**

vote

**0**answers

63 views

### Orthogonal polynomials with quadratic recurrence coefficients

Consider the monic orthogonal polynomials determined by the recurrence
$$p_{n+1}(x)=(x-n(n+b))p_{n}(x)-n(n+a)p_{n-1}(x), \quad n\in\mathbb{N},$$
with the initial conditions $p_{-1}(x)=0$ and $p_{0}(x)=...

**1**

vote

**1**answer

449 views

### Series involving factorials

Playing around with this series for natural values of $a,b$, it appears that more generally for $c\in\mathbb N$, $$\sum_{k=0}^\infty \frac{ (a+k)! \ (b+k)!}{k!\ (a+b+c+ k+1)! }=\frac{a!\ b!\ (c-1)!}{...

**2**

votes

**1**answer

213 views

### Compute Confluent Hypergeometric Function 1F1

I am attempting to compute the (Kummer's) confluent hypergeometric function (see also here)
\begin{align}
M\left(\frac{n}{2}, n +\frac{3}{2}, -z\right) = {}_1F_1\left(\frac{n}{2};n +\frac{3}{2};-z\...

**5**

votes

**1**answer

108 views

### What are the most general methods for solving equations in closed form with Lambert W?

What are the most general methods for solving equations in closed form with help of Lambert W function or with a generalization of Lambert W function?
I gave a method in MSE here.
Which algorithms ...

**1**

vote

**2**answers

136 views

### Reference request for Stieltjes Transform

I am wondering to use Stieltjes transform to signal processing like Fourier Transform. The Fourier is known to give the frequencies of a signal but not sure what Stieltjes transform gives. I am only ...