**13**

votes

**5**answers

2k 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 ...

**29**

votes

**4**answers

2k 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

387 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

**0**answers

264 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

244 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

477 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

531 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

369 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

374 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

472 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

882 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

775 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 ...

**7**

votes

**0**answers

582 views

### On Stark's conjecture for imaginary quadratic fields

In the famous paper "L-Functions at s = 1. IV. First Derivatives at s = 0" of Stark from 1980, it is shown that in the case of an imaginary quadratic field $K$ certain numbers of the form ...

**1**

vote

**2**answers

425 views

### Bessel functions in an Hermite-Gauss basis

¿Could somebody tell me how can i write a zero order bessel function in an Hermite-Gauss basis?
Thanks

**4**

votes

**2**answers

413 views

### Is there a generalization of Floquet theory to elliptic functions?

Hi,
Consider a system of linear differential equations
$$
{d f \over dz} = A(z) f,
$$
where $A(z)$ is a matrix-function. If $z \in \mathbb{R}$ and the function is periodic $A(z) = A(z + T)$, ...

**2**

votes

**0**answers

460 views

### Generalization of repeated error function integral

Is there a name for the following integral?
$f(x, y, n) = \int_y^\infty (t - x)^n \exp(-t^2) dt$
The parameter $n$ is positive. The first priority is integer $n$, but more generally the case of ...

**0**

votes

**2**answers

656 views

### Modified and unmodified Bessel functions of the second kind

There's a simple relationship between $J_\nu$, Bessel functions of the first kind, and $I_\nu$, modified Bessel functions of the first kind, namely $I_\nu(z) = i^{-\nu} J_\nu(iz)$. However, there ...

**4**

votes

**2**answers

1k views

### Are there any uses for complex sine? [sin z]

The sine function can take a complex argument. e.g. sin(x + iy)
But does it get used that way in any field? Either practical (e.g. electrical engineering) or in other fields of math? Naturally, I am ...

**1**

vote

**1**answer

211 views

### evaluating an integral related to the volume of Hessenberg orthogonal matrices

Consider the following integral,
$$
{1 \over 4\pi^{2}}\int_{0}^{2\pi}\int_{0}^{2\pi}
\sqrt{\, 9 -\sin^{2}\left(\theta_{1} \over 2\right)
\sin^{2}\left(\theta_{2} \over 2\right)\,}
\,{\rm ...

**5**

votes

**2**answers

670 views

### Proving a hypergeometric function identity

While playing around with the fractional calculus, I got stuck trying to show that two different ways of differintegrating the cosine give the same result. DLMF and the Wolfram Functions site don't ...

**14**

votes

**3**answers

1k views

### nth-order generalizations of the arithmetic-geometric mean

The arithmetic-geometric mean,
$a_{k+1}=\frac{a_k+b_k}{2} \quad b_{k+1}=\sqrt{a_k b_k}$
is one of the celebrated discoveries of Gauss, who found out that it is equivalent to computing a (complete) ...

**-3**

votes

**1**answer

620 views

### Gamma(e)=Pi/2,Zeta(e)=4/Pi ? [closed]

I find that Gamma(e) is close to Pi/2 and Zeta(e) is close to 4/Pi. So I have a question:
$\Gamma (e) = \pi /2$
$\zeta (e) = 4/\pi $
Is it true in fact?

**5**

votes

**3**answers

918 views

### Differential equation for a ratio of consecutive Bessel functions

My attempts to search via Google seem to be failing, so I thought of asking here.
All the derivatives of the function
$r_n(z):=\frac{J_n(z)}{J_{n-1}(z)}$
where $J_n(z)$ is the Bessel function of ...

**5**

votes

**1**answer

1k views

### When is ArcTan a rational multiple of pi?

Is there a characterisation for which $x\in\mathbb{R}$ the value $\arctan(x)$ is a rational multiple of $\pi$?
Or reformulated: What is the "structure" of the subset $A\subseteq\mathbb{R}$ which ...

**4**

votes

**3**answers

2k views

### Finding a recursion for a sum of Legendre polynomials

The polynomial
$a_n(x):=P_n(x)-\frac{n-1}{n}P_{n-2}(x)$
where $P_n(x)$ is a Legendre polynomial came up while I was investigating methods for estimating the error in Gaussian quadrature.
I am ...

**4**

votes

**2**answers

635 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 ...

**0**

votes

**1**answer

452 views

### modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder
$R_{k,m} \equiv H_{k} ~ \mod H_m$
for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial ...

**1**

vote

**1**answer

562 views

### Asymptotics of Hermite and hypergeometric function

I am looking for the asymptotics of the following integral
$\int_{\mathbb{R}} H_m^2(x) {\rm e}^{-2 \alpha^2 x^2} {\rm d} x = 2^{m-1/2} \alpha^{-2m -1} (1-2\alpha^2)^m \ \Gamma(m+1/2) ~ ...

**2**

votes

**1**answer

355 views

### On the elliptic logarithm and elliptic exponential

Browsing the wealth of functions available in Mathematica, one encounters two not-so-common functions: the "elliptic logarithm", which is an elliptic integral in another garb, and the "elliptic ...

**5**

votes

**3**answers

995 views

### Simultaneously computing a complete elliptic integral and its complement

The complete elliptic integral of the first kind
$K(m)=\int_0^{\pi/2}\frac{\mathrm{d}t}{\sqrt{1-m\sin^2t}}$
is easily computed via the arithmetic-geometric mean iteration; to wit,
...

**4**

votes

**3**answers

2k views

### Spherical Harmonics - a bunch of questions about them

Hi there,
Please tell me if I should divide these into individual questions next time.
Short intro:
Spherical Harmonics are a nice collection of functions. They are orthogonal and allow you to take ...

**26**

votes

**0**answers

1k views

### Curious $q$-analogues

Consider the Fibonacci polynomials
$$F_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} \right\rfloor }\binom{n-j}{j} x^{n - 2j} $$
and the Lucas polynomials
$$L_n (x) = \sum_{j = 0}^{\left\lfloor {n/2} ...

**7**

votes

**3**answers

1k views

### Recent work on hypergeometric functions

Does anyone know of a monograph/survey on the modern history of (basic or elliptic) hypergeometric functions and their applications?
I haven't had much time to search the literature, and because it ...

**19**

votes

**0**answers

1k views

### Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation
$$
[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad
[m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.
$$
Consider the following trigonometric ...

**1**

vote

**0**answers

812 views

### Trigonometric identities and (several?) complex variables

I don't know anything about several complex variables nor whether that topic will answer my questions below, but in one complex variable one learns that since $\sin x$ and $\cos x$ are entire ...

**3**

votes

**1**answer

936 views

### Trigonometric identities

In the rant I wrote at
http://ncatlab.org/nlab/show/trigonometric+identities+and+the+irrationality+of+pi
I asked: Are these four identities the first four terms in a sequence that continues?
This ...

**7**

votes

**1**answer

1k views

### Errata for Emil Artin's 'The Gamma Function'?

In the English translation of The Gamma Function by Emil Artin (1964 - Holt, Rinehart and Winston) there appears to be a mistake in the formula given for the gamma function on page 24:
$$\Gamma(x) = ...

**1**

vote

**0**answers

673 views

### Bessel function in polar coordinates

I want to write the Bessel function of the first kind in polar coordinates
$J_\alpha(z)=|J_\alpha(z)|e^{i\varphi_\alpha(z)}$
Is anything known about $\varphi_\alpha(z)$?
In particular, I'm ...

**1**

vote

**3**answers

581 views

### A trigonometry problem

Let x = pi/(2k+1), for k>0.
Prove that
cosxcos2xcos3x...coskx = (1/2)^k
I've confirmed this numerically for n from 1 to 30.
I'm finding it surprisingly difficult using standard trig formula ...

**7**

votes

**2**answers

506 views

### Relation between full elliptic integrals of the first and third kind

I am working on a calculation involving the Ronkin function of a hyperplane in 3-space.
I get a horrible matrix with full elliptic integrals as entries. A priori I know that the matrix is symmetrical ...

**3**

votes

**3**answers

3k views

### Finding the length of a cubic B-spline

Can I find an analytical solution to the the length of an 2-dimensional cubic B-spline? All I can find are chorded approximations and the opinion that the analytic solution is "unbearably gruesome". ...

**22**

votes

**5**answers

10k views

### Inverse gamma function?

This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this.
We have the ...

**8**

votes

**3**answers

804 views

### No simple duplication formula for factorials?

Many special functions including the gamma function have a duplication formula of some sorts. In the case of the gamma function it reads:
Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2)
On ...