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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About a Christoffel-Darboux-type sum

Hi! I've been using the Christoffel-Darboux identity for the Hermite polynomials, $$\sum_{k=0}^n\frac{H_k(x)H_k(y)}{2^k k!}=\frac{1}{2^n n!}\frac{H_{n+1}(x)H_n(y)-H_n(x)H_{n+1}(y)}{x-y},$$ for some ...
3
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1answer
246 views

The relationship between Stirling number of the second kind and the polylogarithm

It is shown here on Mathworld's page on Stirling number of the second kind that $$ \sum_{k=1}^n S(n,k) (k-1)! z^k = (-1)^n \text{Li}_{1-n}(1+\frac{1}{z}) $$ where $S(n,k)$ is Stirling number of the ...
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0answers
177 views

Convert 2F1 to polynomial

Is there any transformation to convert each of the following versions of ${}_2F_1$ to a polynomial? The first one is $${}_2F_1\left(\frac{1-a}{2}, -\frac{a}{2}; b;\frac{4z}{(1+z)^2} \right), \quad ...
5
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1answer
435 views

Contour Integral with Gamma functions and 2F1

Given the following contour integral $$\frac{1}{2\pi j}\int^{c+j\infty}_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\, {}_2F_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: ...
2
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1answer
424 views

Quadratic Transformation of the Hypergeometric Function 2F1

The function ${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)$ can be transformed (as reported by A. Erdélyi) by the following formula ${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)= ...
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1answer
178 views

About one series. Are there some related special functions?

Hello, I have the following series: $$ \sum_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0, $$ where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum ...
4
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1answer
830 views

minimal polynomials of trig functions of ($k \pi/p$) and divisibility of coefficients by p

Take an odd prime $p$ and put $x_0:=\sum\limits_{j=0}^{p-1}\left(a_{j}\sqrt{p}\cos\dfrac{j\pi}p+b_{j}\sin\dfrac{j\pi}p +c_{j}\tan\dfrac{j\pi}p\right)$, where the $a_{ij}$ are integers. If $f$ denotes ...
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631 views

Cubic polynomials with “nice” roots, which can be expressed by trig functions of rational angles

Consider the cubic polynomial $x^3-ax+b$ for $a,b\in\mathbb N$. It has three real roots which, by Cardano's formula, can of course be written in closed form using thirds of angles or cube roots of ...
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807 views

Which trigonometric identities involve trigonometric functions?

Another question that's getting no answers on stackexchange: Once upon a time, when Wikipedia was only three-and-a-half years old and most people didn't know what it was, the article titled ...
4
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2answers
836 views

a dilogarithm identity: known or new?

I was playing around with dilogarithms and numerically found the following dilogarithm identity: $$\text{Li}_2\left(\frac{2 m}{m^2+m-\sqrt{((m-3) m+1) \left(m^2+m+1\right)}-1}\right)$$ ...
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0answers
257 views

Finite sum of modified bessel function of the second type

I have a sum of the form $\sum^n_{i=0} \frac{{t}^{i} z^{i}}{i!}K_{a+i}(z), \quad\quad z\in\mathbb{R},z>0$ where $K$ is the modified Bessel function of the second type and $a$ is an integer, ...
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2answers
604 views

Inverse Hankel Transform

I was reading through Akhiezer's book Lectures on Integral Transforms and in chapter nine, he states that the Hankel transform is unitary for $\nu > -1$, so that for a suitable function, $f$, ...
2
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1answer
228 views

is this bound on a kummer function known?

Is it already known that ${}_1F_1(a;b;x) \leq \Gamma(b)(1+|x|)^{-a}$ when $a$ is an integer, $a <0,$ and $b>0?$ If it is, what is a reference? my proof: Since the Kummer function can be ...
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1answer
295 views

Upper bounds on generalized Laguerre polynomials

I evaluated an integral and obtained an expression with a Laguerre polynomial. I'd like something more explicit and useable. Are there any known simple (e.g. exponential) upper bounds on the ...
3
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1answer
495 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 ...
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1answer
509 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
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1answer
963 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
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2answers
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 $f(x)$ ...
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0answers
701 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 ...
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3answers
533 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 ...
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3answers
706 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 ...
14
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2answers
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 $n=2m$, ...
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387 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
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0answers
421 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 ...
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1answer
411 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 ...
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1answer
612 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
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3answers
467 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 ...
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0answers
249 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)} ...
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2answers
431 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} ...
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1answer
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 ...
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3answers
434 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 ...
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1answer
1k 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 ...
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2answers
349 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 $$ ...
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216 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 ...
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0answers
442 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
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1answer
257 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)}} $
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1answer
185 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 ...
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0answers
290 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 ...
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1answer
286 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
744 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
618 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 ...
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1answer
775 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
898 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
672 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
290 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 ...
3
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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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5answers
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 ...