**-6**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

vote

**0**answers

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

**2**answers

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

**0**answers

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**

votes

**1**answer

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

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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

**1**answer

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**

votes

**3**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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 ...

**1**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**3**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**1**answer

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**

votes

**3**answers

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**

votes

**0**answers

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**

vote

**2**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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

**2**answers

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

**4**answers

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

**0**answers

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 ...