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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27 views

applications that involves the Legendre Polynomials [on hold]

i am requested to make a basic research about the applications that involve the Legendre Polynomials.. thanks in advance
2
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1answer
130 views

Ask for a special function related to the error function

I am wondering whether anyone knows the following integration has a named special function or a reference $$ F_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y $$ for ...
-1
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0answers
38 views

Floquet solution to Mathieu equation in terms of Mathieu sin and cos [on hold]

In wikipedia https://en.wikipedia.org/wiki/Mathieu_function#Floquet_solution I want to know how the Floquet solution is plotted. One way I am thinking is to write Floquet solution in terms of the ...
2
votes
0answers
45 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 ...
2
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0answers
52 views

Fractional derivative of the Wright function

It is mentioned in some papers (Appendix in this paper, for example) that the (formal) solution of the fractional drift (or transport) equation $$ ...
0
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1answer
49 views

Request for references about computing or estimating Rademacher complexity

Is Rademacher complexity defined for any space of functions? Or are there restrictions on the function space over which this can be defined? For example is the Rademacher complexity defined or has ...
1
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0answers
42 views

The integral of $\Gamma\left(\zeta\right) \, W_{-\zeta,\mu}(z) $

Someone has a reference that addresses an integral of the followns type $$I_{a,b,x} = \int_{0}^{+\infty} \zeta^{-a} \, \Gamma\left(\zeta+b\right) \, W_{-\zeta-b,\tfrac{-1}{2}}(x) \, d\zeta$$ where ...
2
votes
2answers
156 views

Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
3
votes
0answers
59 views

Factoring Bessel functions into an amplitude and a phase

Take some $\nu>0$. Let $J_\nu(x)$ be the Bessel function of the first kind. Let's restrict its domain to $\mathbb R^+$. Is it possible to find a pair of functions $A_\nu(x), \phi_\nu(x):\mathbb ...
7
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1answer
252 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
2
votes
2answers
89 views

Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain: $$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$ where $m>0$ is fixed. Question. To ...
3
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1answer
77 views

Functional equations associated with addition theorems for elliptic functions

I'm trying to read the article "Functional equations associated with addition theorems for elliptic functions and two-valued algebraic groups" by Bukhshtaber,V. M. Russian Mathematical ...
3
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1answer
2k views

Is there any heartbeat like function? [closed]

I'm looking for a function, where the result is something like this: I tried to figure it out myself, but I have no idea how to manage it. f(x) = ... Thanks in ...
2
votes
1answer
70 views

Challenging problems concerning Jacobian elliptic functions with complex modulus

I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since ...
12
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3answers
412 views

Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?

For naturals $n\ge m$, define $$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$ with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $. Is it ...
2
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1answer
83 views

About eigen-functions of the Gaussian kernel

If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
5
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2answers
206 views

Alternating binomial Dirichlet series

I have come across the following deceptively simple expression: $$ H_n^s=\sum_{j=1}^n(-1)^{j-1}\left(\begin{array}{c}n\\j\end{array}\right)j^{-s} $$ We have (using eg mathematica, though probably ...
5
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1answer
213 views

Estimate of incomplete binomial integral

Let $0\le k \le n$. Prove that $$ n\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2. $$ As far as I know 1) it is proved for $\frac{k}{n+1}\le 1/2$ and 2) not proved for $1/2 ...
0
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1answer
53 views

What is the relationship between solutions for the parameterised second order differential equations

Let us consider the following parameterised complex-valued second order differential equations, and $u(x,\lambda)$ be the solution for $$ u''+u'-i\lambda V(x)u=0, \, x\in [0,1], $$ What is the ...
8
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133 views

Summation of series involving $\sinh$ of a square root

Consider the following series: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})} $$ From the physical ...
11
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521 views

Connection between Infinite continued fractions, elliptic integrals and AGM

It is known that at $x=1$, the following continued fraction represents $\frac{4}{\pi}$ and can be approximated rapidly using Gauss' Arithmetic Geometric mean. $$C(x) = x + \frac{1^{2}}{2x + ...
4
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103 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 ...
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1answer
70 views

Definite intergal with two K-Bessel functions and x

I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers): $$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$ I could not find it in ...
2
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0answers
44 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 ...
2
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0answers
44 views

Riemann theta function with asymptotically large Toeplitz Matrix

As a follow up to How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently Suppose that $M$ is a large Toeplitz matrix. With a suitable scaling $K^{-n}$ for some $K$, what will the Riemann ...
1
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1answer
50 views

Upper-bounding the value of a generalized Laguerre polynomial (using recurrence relation?)

I would like to produce an easily-interpretable explicit upper bound (i.e. no unspecified constants) for the function $$ f(n) := L_n^{\left(-n-\frac{d}{2}\right)}\left(-\frac{1}{2}\right), \quad n,d ...
3
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3answers
2k views

Integral over error function and normal distribution

Help me understand why $\int_{-\infty}^{\infty}\frac{1}{2}[1+\operatorname{erf}(\frac{\theta-x}{\sqrt{2q^2}})]\frac{1}{\sqrt{2\pi\sigma^2}}{\exp(-\frac{(x-\mu)^2}{2\sigma^2})}dx \approx ...
2
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0answers
95 views

Hypergeometric function asymptotics

I came across the following hypergeometric function recently: $$ _2F_1(1-n,p-2n+1;p-n+1;x) $$ where $p > 0$ is a non-integer constant, $n$ some large positive integer, and $x > 0$ a small ...
0
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0answers
48 views

Division of half-integer Legendre functions of the second kind with different arguments

I'm in search of a formula for: $\frac{Q_{n-\frac{1}{2}}(\chi_1)}{Q_{n-\frac{1}{2}}(\chi_2)}= ??$ where I am hoping the result to be a function of $\frac{\chi_1}{\chi_2}$. Does anyone know of such ...
3
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0answers
159 views

Combination of Generating Functions

Suppose I have the following generating functions: $$\frac{x^ke^{\left(z-\frac{1}{N}\right)x}}{N^{k-1}k!\sum_{j=0}^{N-1}w_N^{-jk}e^{\frac{w_N^jx}{N}}}=\sum_{j=0}^\infty H_{N,k,j}(z)\frac{x^j}{j!}$$ ...
7
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1answer
184 views

Asymptotics of a special function

In my research, I came up with a special function which I denote by $B(q)$ and is defined by the integral $$B(q)\equiv \int_{-\pi/2}^{\pi/2} ...
3
votes
1answer
88 views

How to show monotonocity and the limit? [closed]

Let me reformulate my recent question. Let $n, N$ denote density and cdf of Gaussian distribution. Let us consider its modification, given by density: $$\phi(x) = C\left\{ \begin{array}{lcc} ...
3
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0answers
164 views

A generalization of Rogers-Ramanujan identity

The generalized Rogers-Ramanujan identity has the following form $$\sum_{k_1\geq\cdots\geq k_r\geq 0}\frac{x^{k_1^2+\cdots +k_r^2+k_i+\cdots +k_r}}{(x)_{k_1-k_2}\cdots ...
3
votes
1answer
71 views

Inverse Laplace transform of a hypergeometric function

This is a repost from Math Stack-exchange where I did not manage to get an answer. http://math.stackexchange.com/questions/1491027/inverse-laplace-transform-of-a-hypergeometric-function I managed to ...
9
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3answers
429 views

Complex proof of $B(a,b)=\Gamma(a)\Gamma(b)/\Gamma(a+b)$

It is a question in spirit of this one. Is there a way to prove Euler's formula $$ \int_0^1 x^{a-1}(1-x)^{b-1}dx=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ using contour integration (and maybe ...
3
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0answers
87 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 ...
1
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2answers
354 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 ...
2
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1answer
121 views

An extreme of Jacobi elliptic function on an interval

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...
5
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2answers
643 views

Generalized trigonometric functions $Cos(n) v$ and $Sin(n) v$.

I just discovered a paper from 1948, Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen by R. Grammel which introduces functions he calls Cos(n) and Sin(n), representing a parameterization of the ...
3
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0answers
49 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 ...
4
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1answer
713 views

Pochhammer symbol of a differential, and hypergeometric polynomials

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it. Consider a confluent hypergeometric function of the form $$\newcommand{\ff}{{}_1F_1} ...
1
vote
1answer
74 views

Inverse error function in Hardy space?

Let $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}} \int_{-\infty}^x \exp(-t^2) \, dt$ be the error function $\mathrm{erf}: \mathbb{R} \to (-1,1)$. It is monotonously increasing and therefore has an inverse ...
5
votes
0answers
223 views

Recurrence Formula for Zernike polynomials

I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...
5
votes
2answers
200 views

Integrals involving trigonometric functions and polynomials

Can one describe all the real polynomials $P(x)$ such that the following integrals converge: $$ \int_0^{\infty} \sin(P(x))dx, \int_0^{\infty} \cos(P(x))dx ? $$ Among special cases are such ...
5
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1answer
111 views

A special function solution of a fourth-order ODE

I want to consider the solutions of the following fourth-order ODE: $$ f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0, \tag{$\ast$}$$ where $a,b$ are complex parameters. It turns out that with a Fourier ...
2
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0answers
34 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 ...
5
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1answer
145 views

On a Sum of Gamma Functions

I am working on a problem where the following sum appears: $$F(s, t)=\frac{1}{\Gamma(1+2\alpha)}\sum_{n=0}^{\infty}{\frac{s^{n} ...
6
votes
2answers
682 views

Bounds on remainder term of power series of elementary functions

This is mainly a question about the remainder term of power series for elementary functions. I'm very interested in aspects of calculating or computing elementary operations and functions, by which I ...
5
votes
4answers
732 views

An identity for the cosine function

Let $x = \pi/(2k+1)$, for $k>0$. Prove that $$ \cos(x)\cos(2x)\cos(3x)\dots\cos(kx) = \frac{1}{2^k} $$ I've confirmed this numerically for $n$ from $1$ to $30$. I'm finding it surprisingly ...
11
votes
0answers
259 views

Inverse Mellin of the exponential of the digamma function

I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying $$ \int_0^\infty f_p(x)x^{s-1}dx=e^{-p\psi(s)} $$ where $\psi(s)$ is the usual digamma function. The inverse Mellin formula ...