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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.

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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 $k=0:\quad g(n,k) = 1$ $...
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0answers
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(...
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1answer
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 ...
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0answers
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 ...
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0answers
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
2answers
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
1answer
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,...
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votes
1answer
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\...
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1answer
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
2answers
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 ...
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0answers
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)=...
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votes
3answers
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
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0answers
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
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1answer
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 $\alpha \ge 1$...
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1answer
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: $...
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1answer
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 ...
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1answer
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 ...
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vote
0answers
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....
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0answers
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+\...
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0answers
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):...
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0answers
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
1answer
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-...
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0answers
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....
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0answers
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/...
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0answers
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 ...
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0answers
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{\...
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0answers
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-...
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0answers
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 ...
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0answers
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}...
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0answers
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 ...
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2answers
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 ...
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1answer
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 ...
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0answers
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 $i \in 1,2..,d$. ...
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1answer
220 views

Asymptotic behaviour of function from integral representation

In Appendix A of this paper, it is claimed that the asymptotic behaviour of $$\phi_1(y,\lambda)=\frac{1}{\Gamma(\frac{1-\lambda}{2})}\int_0^\infty dt~e^{-t}\cos(2y\sqrt{t})t^{-\frac{1+\lambda}{2}},$$ ...
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0answers
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 ...
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1answer
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{...
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0answers
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 \...
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1answer
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 $$F\left (n+\frac{m}{2},n+\frac{m+1}{2};k+\frac{3}{2};-\tan^2{\phi}\right) \tag{1}$$ ...
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0answers
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 $\sigma$ ...
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2answers
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 $I_n(u)$...
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votes
0answers
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 ...
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vote
1answer
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 ...
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vote
1answer
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(...
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votes
0answers
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
1answer
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{\...
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vote
0answers
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)=...
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vote
1answer
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
1answer
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
1answer
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
2answers
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 ...