**3**

votes

**1**answer

519 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 ...

**0**

votes

**1**answer

563 views

### infinite series with Hypergeometric functions

Can we get a closed form for the series
$\sum^\infty_{k=0} \frac{ t^k}{k!} \Gamma(k+a)\Gamma(k+\frac{1}{2}){}_2F_1(k+a,k+\frac{1}{2};n+1,x)$
any hints or clues are welcomed.

**5**

votes

**1**answer

1k views

### Two-variable generating functions for Laguerre polynomials

Where can I find generating functions for orthogonal polynomials in two variables?
Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for
$$
\sum_{n=0}^\infty ...

**5**

votes

**2**answers

1k views

### How to do integrals involving two Bessel functions and another function?

I often encounter the integrals in the following form:
$\int_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$,
where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ ...

**2**

votes

**0**answers

803 views

### Bessel functions in wave propagation and scattering

Is there a way to scale Bessel J(n,.) (Bessel of first kind) and Bessel H(n,.) (Bessel of third kind or Hankel)? I am having computer problems with higher orders (higher vlaues of n) and small ...

**4**

votes

**3**answers

570 views

### Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$

I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression
$
\sum_{k=0}^{\infty} \frac{z^k}{k!k!} = F_{1}[;1;z]
$
This is not very helpful though, and I 'd ...

**0**

votes

**3**answers

708 views

### How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times ...

**15**

votes

**2**answers

2k views

### Sum of trigonometric functions

Do somebody know the closed form of the following sum (m is an integer)
$$f(\beta)=\sum _{k=1}^{2m+1} \sin^{2 m+1}\left[\frac{-\beta+k \pi
}{1+2 m}\right]$$
If instead of $n=1+2m$ we put $n=2m$, ...

**1**

vote

**0**answers

409 views

### An infinte series involving the Modified Bessel Function of the second kind

The following series has had me held up for the past one week:
$$
\sum_{n=0}^\infty\frac{(2m)_n m^n}{(2m+1/2)_n n!}A^{3n/2} t^n K_{2m+n-1/2}(2\sqrt{A}t)~~~~ A>0, ~t>0, ~m\geq1/2
$$
where ...

**0**

votes

**0**answers

451 views

### modified bessel fucntion of the third kind

Hi I'm doing a computation where the modified bessel function of the third kind is the main source of computational strain, we are using a 10,000 dimension's for our distribution, is there any easier ...

**1**

vote

**1**answer

464 views

### Legendre Polynomial Identity

I have encountered the following sum involving Legendre polynomials, which I hope to reduce to something involving a $\delta$-function:
$$
\frac{d^2}{dx^2} \sum_{\ell=0}^{\infty} \frac{2 \ell + 1}{2 ...

**5**

votes

**2**answers

720 views

### On a polynomial related to the Legendre function of the second kind

The Legendre function of the second kind, $Q_n(z)$, along with the usual Legendre polynomial $P_n(z)$, are the two linearly independent solutions of the Legendre differential equation.
$Q_n(z)$ can ...

**2**

votes

**3**answers

474 views

### Differential equation with some constraints

I posted this to stackexchange, and after some hours got a comment that was so pessimistic about finding some neat orderly solution, that I'm posting it here too. (In case anyone cares, this is ...

**0**

votes

**0**answers

250 views

### High dimensional beta integral (question following the previous post)

Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} ...

**1**

vote

**2**answers

452 views

### High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello,
When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake:
$$
\int_{R^n} |x-y|^{-n+\alpha} ...

**11**

votes

**1**answer

2k views

### Geometric meaning of a trigonometric identity

It follows from the law of cosines that if $a,b,c$ are the lengths of the sides of a triangle with respective opposite angles $\alpha,\beta,\gamma$, then
$$
a^2+b^2+c^2 = 2ab\cos\gamma + 2ac\cos\beta ...

**1**

vote

**3**answers

469 views

### Pairing function monotonic respect to product of arguments [closed]

Has anyone ever created a "pairing function" (possibly non-injective)
with the property to be nondecreasing wrt to product of arguments, integers n>=2, m>=2. (We can also assume that n and m are ...

**1**

vote

**1**answer

2k views

### looking for monotonically increasing functions with range in [0,1] [closed]

Ideally, the function f(x) would tend toward zero as x tends towards negative infinity, and f(x) would tend towards 1 as x tends towards infinity, all the while being monotonically increasing.
for ...

**7**

votes

**2**answers

363 views

### Asymptotics of the $q$-harmonic series as $q\to1$

The following (very simply looking!) problem occurs in regularization
of the harmonic series
which can be formally thought of as the limit as $q\to1$, $|q|<1$, of
$$
...

**0**

votes

**0**answers

221 views

### Non-standard addition theorem for Legendre function of the first kind

There is known link text addition theorem for the Legendre functions of the first kind $P_{\nu}^m(x)$, which establishes the connection between $P_{\nu}^m(x)P_{\nu}^{m}(y)$ and $P_{\nu}(F(x,y))$ for ...

**3**

votes

**0**answers

466 views

### Mathematica package for obtaining hypergeometric function

In my current research in electromagnetics I am encountering integrals of the form $$ \int_0^\infty dt J_0( r t) \frac{\exp(-h \sqrt{t^2 - a^2})}{\sqrt{t^2 - b^2}} t . $$ $a$ and $b$ are complex ...

**2**

votes

**1**answer

276 views

### antiderivative involving modified bessel function

This little integral has been holding me up for weeks. Has anyone come across a similar integral in their work.
$\int {\frac{d x}{c-I_0(x)}} $

**1**

vote

**1**answer

187 views

### Can you control the amplitudes of a finite collection of sine curves just by controlling the amplitude of their superposition?

Dear all,
I would like to know whether the following claim is true. In particular, if it is true, then I would like to know if there is some textbook that contains the statement and maybe even the ...

**3**

votes

**0**answers

318 views

### Why is Mellin-inverse of Gamma periodic?

Specific Case
The periodicity is obvious from computation:
$$\cal{M}^{-1}\{\Gamma\}(x) := \frac{1}{2\pi i}\int_{c}\Gamma(s)x^{-s}d s=e^{-x}$$
However, is there a way to see directly from the integral ...

**1**

vote

**1**answer

293 views

### Maximum of a series of integrals of Hermite functions

Given the function $$f(A) := \sum_{n=1}^{\infty}\left( \int_A \varphi_0\varphi_n \right)^2,$$ where $A$ is any measurable subset of $\mathbb{R}$, and $\varphi_n$ is the $n$th Hermite function, I want ...

**0**

votes

**2**answers

767 views

### Functions defined as infinite products

Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too!
The original motivation for this is the ...

**0**

votes

**2**answers

662 views

### Cosine of a Partial Sum

Does anyone know of a closed formula for $cos(\displaystyle\sum_{n=0}^m a_{n})$? I've seen formulas for $cos(\displaystyle\sum_{n=0}^\infty a_{n})$ and $tan(\displaystyle\sum_{n=0}^m a_{n})$, but the ...

**4**

votes

**1**answer

813 views

### What's the difference between a Riemann theta and a Siegel theta function?

One of the things I'm working on has required me to look into the literature of multidimensional theta functions, and I've gotten a bit confused on a few naming details.
A look at the DLMF says that ...

**5**

votes

**1**answer

907 views

### Geometric meaning of trigonometric relations

According to a paper by Zhiqin Lu in the Mathematical Gazette (the British publication, not the Boston-area newsletter, if that still exists (or even if it doesn't)) in 2007(?), if $u+v+w=\pi$ and ...

**5**

votes

**1**answer

691 views

### Integral identity for Legendre polynomials

How does one prove the following integral identity, where $P_n(x)$ is the $n$th Legendre polynomial?
$$
\int_0^1 P_n(2t^2-1) dt = \frac{(-1)^n}{2n+1}
$$
Notes & Background
A variant of this ...

**2**

votes

**1**answer

299 views

### Legendre Function “Types”

Legendre functions can be of first and second kinds; $P$, $Q$.
They can have order $\mu$ and degree $\nu$; $P^\mu_\nu$ $Q^\mu_\nu$
But do they also have Types? Some of the numerical software I am ...

**7**

votes

**4**answers

1k views

### Determining the asymptotic behavior of a series

I am trying to determine the behavior of the following series as $n\to\infty$. Let $0<\mu<1$ be fixed and for every positive integer $n\geq 1$, consider the function $f_n(t)$ of a real variable ...

**7**

votes

**4**answers

2k views

### Estimating the probability that one Poisson RV is larger than another

Let $X$ and $Y$ be Poisson random variables with means $\lambda$ and $1$, respectively. The difference of $X$ and $Y$ is a Skellam random variable, with probability density function
$$\mathbb P(X - Y ...

**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 ...

**3**

votes

**2**answers

1k views

### Product of hypergeometric functions/Jacobi Polynomials

Are there any theorems related to the product of Jacobi/Legendre Polynomials and/or Hypergeometric functions? Specifically, I'm interested in the product of ${}\_{2}F_{1}[-n,-n+1;2;x]$ and ...

**24**

votes

**6**answers

3k views

### Why are hypergeometric series important and do they have a geometric or heuristic motivation?

Apart from telling that the hypergeometric functions (or series) are the solutions to the (essentially unique?) fuchsian equation on the Riemann sphere with 3 "regular singular points", the wikipedia ...

**31**

votes

**5**answers

3k views

### Groups, quantum groups and (fill in the blank)

In the study of special functions there are three levels of objects, classical, basic and elliptic. These correspond to classical hypergeometric functions, basic (q-) hypergeometric functions, and ...

**2**

votes

**3**answers

390 views

### About Turan`s problem(inequality) in multivariable

Hi. I have a question related to Turan`s problem, that is
Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$.
I am considering the generalized question for ...

**4**

votes

**1**answer

217 views

### Positivity of a rational function

A rational function is called positive if all its Taylor coefficients are positive.
Friedrichs-Lewy conjecture states the positivity of the rational function
\begin{eqnarray*}\frac{1}{
(1-x)(1- ...

**1**

vote

**1**answer

284 views

### monotonicity of functions related to modified Bessel function

Dear colleagues,
I recently met some problems related to the modified Bessel funtions of the first kind and the second kind. I want to know if there exist some results on the monotonicity of
...

**2**

votes

**1**answer

250 views

### Estimating associated Legendre function

For past months I've been trying to estimate associated Legendre function $P_{-\frac{1}{2}+it}^m (\cosh r)$ in order to study Laplacian eigenfunctions on a hyperbolic surface. I found reasonably sharp ...

**0**

votes

**1**answer

542 views

### Results on derivatives with respect to the parameter of Modified Bessel Function [duplicate]

Possible Duplicate:
Derivate Bessel Function with respect to order
Dear colleagues,
I have a question about the modified Bessel function of the second kind, $I_\nu(x)$ and $K_\nu(x)$. I ...

**5**

votes

**2**answers

537 views

### generalization of (Rogers) dilogarithm

Let $C$ and $S$ be abbreviations for $\cosh$ and $\sinh$, and consider the following
function:
$$f(x,y) = \int_{-y\le r+l \le y} \frac{ (C(x)S(l)C(r) - C(l)S(r))(C(x)C(l)S(r)-S(l)C(r)) }
...

**4**

votes

**2**answers

377 views

### Analytic continuation of $_4F_3(1)$

The Gauss theorem
$${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$
allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series ...

**6**

votes

**2**answers

2k views

### Sums of arctangents

$$
\begin{align}
\arctan(x) & = \arctan(1) + \arctan\left(\frac{x-1}{2}\right) \\
& {} - \arctan\left(\frac{(x-1)^2}{4} \right) + \arctan\left(\frac{(x-1)^3}{8}\right) - \cdots
\end{align}
$$
...

**3**

votes

**1**answer

377 views

### Summations in $\tan^2$

Hey all,
I was just wondering if anyone had come across the following identities, valid for $m\in\mathbb{N}$. I've used Abramowitz and Stegun, Maple, Mathematica etc but can't find them anywhere. I ...

**0**

votes

**1**answer

1k views

### invert complete elliptic integral of first kind K(k)

Hi,
I am not a mathematician although I use it in hydrodynamics research. I have a question regarding elliptic integrals for my research in wave theory
Given the value of the complete elliptic ...

**5**

votes

**2**answers

482 views

### A problem on sums of arctangents of rationals

Let $S$ be a set of rational numbers.
For "special" sets $S$, we can ask if $\pi$ can be written as a $\mathbb{Q}$-linear or $\mathbb{Z}$-linear combination of elements from '$\{\tan^{-1}(x): x \in ...

**7**

votes

**3**answers

951 views

### Conformal Mappings for hyperbolic polygon

I am searching for a conformal mapping from the upper halfplane onto a hyperbolic polygon, i.e. the sides of the polygon have to be geodesics.
The classical Schwarz Christoffel theorem does the job ...

**8**

votes

**1**answer

783 views

### Is this sequence of polynomials well-known?

While working on a problem in p-adic Hodge theory, and needing to write down a solution to a certain equation involving p-adic power series, I stumbled across a certain sequence of polynomials. Define ...