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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288 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 ...
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2answers
748 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 ...
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2answers
629 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
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1answer
785 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
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1answer
902 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
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1answer
677 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 ...
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1answer
295 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 ...
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4answers
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 ...
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4answers
1k 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 ...
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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 ...
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2answers
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 ...
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6answers
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
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4answers
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 ...
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3answers
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
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1answer
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- ...
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0answers
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
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1answer
248 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 ...
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1answer
487 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 ...
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2answers
532 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
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2answers
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 ...
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2answers
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
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1answer
375 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 ...
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1answer
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 ...
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2answers
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
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3answers
893 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
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1answer
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
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0answers
585 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 ...
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2answers
430 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
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2answers
417 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 $A(z) = A(z + T)$, ...
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0answers
464 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 ...
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2answers
658 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 ...
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2answers
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 ...
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1answer
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
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2answers
688 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
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3answers
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) ...
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1answer
622 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
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3answers
930 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
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1answer
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
3answers
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
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2answers
638 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
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1answer
454 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 ...
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1answer
566 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
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1answer
364 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 ...
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3answers
1k 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, ...
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3answers
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
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0answers
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
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3answers
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 ...
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0answers
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 ...
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0answers
825 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
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1answer
940 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 ...