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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35
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1k views

There's something strange about $\sqrt{d\big(j(\tau)-1728\big)}$

Given the j-function $j(\tau)$, I was looking at, $$F(\tau) = \sqrt{d\big(j(\tau)-1728\big)}$$ which appears in Ramanujan-type pi formulas. Let $C_d$ be the prime factors of the constant term of the ...
29
votes
4answers
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 ...
28
votes
7answers
2k views

How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?) Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
27
votes
1answer
1k views

Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...
26
votes
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} ...
24
votes
5answers
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 ...
23
votes
2answers
887 views

A 14th and 26th-power Dedekind eta function identity?

Given the Dedekind eta function $\eta(\tau)$. Let p be a prime and define $m = (p-1)/2$. Let p be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$: $$\sum_{k=0}^{p-1} \Big(e^{\pi i m k/12} ...
21
votes
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 ...
19
votes
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 ...
15
votes
2answers
685 views

Is there a known solution to $f(x) = (1-x)f(x^2)$?

The functional equation $f(x) = (1-x)f(x^2)$ (with $f(0)=1$) has a simple solution that can be expressed as a rapidly converging infinite product $$f(x) = \prod_{n=0}^\infty (1 - x^{2^n}) = ...
14
votes
2answers
1k views

The function $\sum_{0}^{\infty} x^n/n^n$

The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...
14
votes
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) ...
14
votes
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 $n=2m$, ...
14
votes
1answer
321 views

Is this combination of generalized polygamma and dilogarithm actually zero? $\Im\;\psi^{(-2)}(1+i)+\frac1{4\pi}\text{Li}_2(e^{-2\pi})-\log\sqrt{2\pi}+\frac{5\pi}{24}+\frac12$

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^{-4\times10^3}$ in ...
11
votes
2answers
983 views

How much can one say about this differential equation?

Consider the ODE $y^{\prime \prime}(x) = \cos(x) y(x)$ with boundary value conditions $y(0)=1$, $y(1)=2$. Solving it results in a linear combination of Mathieu functions, but what I find more ...
11
votes
1answer
253 views

Can a harmonic number be a rational number for non-integer rational argument?

Define harmonic numbers for a complex argument $z$ as $H_z=\frac{\Gamma'(z+1)}{\Gamma(z+1)}-\Gamma'(1)$. For $n\in\mathbb{N}$, $H_n$ are usual harmonic numbers $\sum^n_{k=1} k^{-1}$ . They are ...
11
votes
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 ...
11
votes
1answer
318 views

Is there something wrong with Hörmander's theorem on stationary phase method

It is well-know that the Bessel function has the asymptotic expansion $J_n(\omega) \sim \left( \frac 2 {\pi \omega} \right)^{1/2} \left( \cos \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) - ...
11
votes
1answer
379 views

Schur functors generalization to “Jack”, “Hall-Littlewood”, “Macdonald” functors ?

Schur functors are functors from the category of vector spaces to itself. If we take an operator $M: V->V$ and apply a Schur functor to it and then calculate trace $Tr(M^{\Lambda})$ we will get ...
10
votes
2answers
728 views

The complete list of continued fractions like the Rogers-Ramanujan?

I have two questions about q-continued fractions, but a little intro first. Given Ramanujan's theta function, $$f(a,b) = \sum_{n=-\infty}^{\infty}a^{n(n+1)/2}b^{n(n-1)/2}$$ then the following, ...
10
votes
0answers
129 views

A hypergeometric puzzle

$$ 143\,\sqrt {3}\;{\mbox{$_2$F$_1$}\left(\frac{1}{2},\frac{1}{2};\,1;\,{\frac {3087}{8000}}\right)}= 40\,\sqrt {5}\; {\mbox{$_2$F$_1$}\left(\frac{1}{3},\frac{2}{3};\,1;\,{\frac ...
10
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0answers
366 views

Connection between Infinite continued fractions 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 + ...
9
votes
7answers
1k views

A good reference to grok hypergeometric functions?

When I was introduced during my degree to special functions, I made friends with a number of nice functions - Laguerre, Legendre, Hermite, Bessel, and whatnot - but I made only a passing acquaintance ...
9
votes
1answer
823 views

Has anyone seen this series?

I come across the following infinite series. $$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$ In particular, I am interested in the case where $a=1/4$. ...
9
votes
2answers
810 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 ...
9
votes
1answer
246 views

Is a Bessel function larger than all other Bessel functions when evaluated at its first maximum?

Let $\mathcal{J}_{n+1/2}$ be the Bessel function of order $n+1/2$. Let $x_n$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$. My ...
9
votes
2answers
484 views

The fraction of the sphere a fixed distance from a subspace

The following problem has a beautiful geometric interpretation in terms of the proportion of points on the Euclidean sphere in $\mathbb{R}^d$ that lie at least a certain distance away from a ...
9
votes
0answers
306 views

Linear eta product identities - how many are there?

For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n}) $, let for brevity $e_k:=\eta(q^k)$. With this notation, a blog entry of Michael ...
8
votes
3answers
808 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 ...
8
votes
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 ...
8
votes
2answers
766 views

Is there a “right” proof of Riemann's Theta Relation?

Let $\theta$ denote the usual Jacobi Theta function (with auxiliary parameter $\tau = i$, for simplicity), i.e. $$ \theta(z) = \sum_{n \in \mathbb{Z}} \exp(-\pi (a + n)^2 + 2 \pi i n z) \ . $$ I'm ...
8
votes
2answers
278 views

On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation ...
7
votes
2answers
355 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 $$ ...
7
votes
3answers
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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 ...
7
votes
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 ...
7
votes
4answers
449 views

Trig functions based on convex curves

Pardon my naivety, but I wonder if much use has been found for trigonometric functions defined in terms of a centrally symmetric convex curve $K$ replacing the circle $C$. For example, here is the ...
7
votes
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 ...
7
votes
1answer
158 views

Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by $\quad L_n (x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad $ I want to prove that $$ \forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...
7
votes
4answers
210 views

Is there an oscillating analog of the Gaussian distribution?

It frequently happens that, in some famillies of polynomials with positive coefficients, the coefficients of large polynomials look like a bell curve and tend to the distribution function of the ...
7
votes
3answers
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 ...
7
votes
2answers
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Duality of eta product identities: a new idea?

Looking at the collection of Eta Function Product Identities by Michael Somos, it seems like generally those identities come in pairs: let's call two eta product identities $\sum\limits_{i=1}^r ...
7
votes
1answer
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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) = ...
7
votes
2answers
509 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 ...
7
votes
0answers
250 views

Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson

Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites. Let the ...
7
votes
0answers
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 ...
6
votes
2answers
421 views

Strange pattern in rounding errors?

This will look at first like a posting about trigonometry, then maybe about statistics, then finally about peculiarities of either a certain random process; or the pseudorandom number generator that ...
6
votes
2answers
2k views

Sums of arctangents

$$ \begin{align} \arctan(x) & = \arctan(1) + \arctan\left(\frac{x-1}{2}\right) \\ & {} - \arctan\left(\frac{(x-1)^2}{4} \right) + \arctan\left(\frac{(x-1)^3}{8}\right) - \cdots \end{align} $$ ...
6
votes
2answers
815 views

Is there a known formula for fractional derivative of cot x?

I wonder if there any established formula for fractional derivative of a function $\pi \cot (\pi x)$. I derived the following expression: $(\pi \cot (\pi ...
6
votes
1answer
237 views

Under which constraints are there only finite numbers of irreducible eta product identities?

For the Dedekind eta function, defined as usual by $\eta(q) = q^{\frac1{24}} \prod\limits_{n=1}^{\infty} (1-q^{n})$, let for brevity $e_k:=\eta(q^k)$. An eta product identity (or eta identity for ...
6
votes
1answer
158 views

Abel's five terms relation from Yang-Baxter equation?

Can the famous Abel's five terms relation satisfied by the dilogarithm be derived from (a particular case of) the theory of Yang-Baxter equations? If yes, how? Thanks for any help.