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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26
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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} ...
23
votes
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745 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 ...
18
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 ...
10
votes
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355 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
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282 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 ...
7
votes
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227 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
566 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
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125 views

Evaluating an infinite product of q-exponentials

For the $q$-exponential $$e_q(u) = \sum_{n=0}^{\infty} \frac{u^n}{[n]_q!}$$ with $[k]_q=\frac{1-q^k}{1-q}$ and $[n]_q! = [n]_q [n-1]_q \cdots [1]_q$, we don't have the property $e_q(u) e_q(v) = e_q ...
5
votes
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176 views

Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?

Define the Ramanujan theta function $f(a,b)$ as, $$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$ and the Dedekind eta function, $$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} ...
5
votes
0answers
185 views

Identification of a curious function

The following question was asked on math.stackexchange, but there were no replies. During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k ...
5
votes
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416 views

Parabolic cylinder functions - explicit estimates?

I need estimates for the parabolic cylinder functions $U(a,z)$ (first studied by Whittaker). Most work in the literature focuses on $a$ real. As it happens, I am interested in $U(a,z)$ on a strip in ...
4
votes
0answers
211 views

Why does the plastic constant appear in the snub icosidodecadodecahedron?

The golden ratio, $$\phi =\frac{1+\sqrt{5}}{2}$$ appears (among other polyhedra) in the Platonic solids icosahedron and dodecahedron, and it's quite easy to see the significance of the discriminant ...
4
votes
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297 views

hypergeometric function $_2F_1(-n;-r;1;2)$

The hypergeometric function $_2F_1(-n;-r;1;2)$ appears in many different situations. For instance, it counts the number of integer points within a sphere in the $l_1$ norm, i.e., $$_2F_1(-n;-r;1;2) ...
4
votes
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232 views

A coincidence concerning Fermat primes, binomial sums, and eta quotients?

Let $w_k$ be a primitive k th root of unity, where k is a power of 2. In response to a question, Robert Israel gave the solution, $$\sum_{n=0}^\infty \frac{(-1)^n}{\binom{kn}{kn/2}} = ...
3
votes
0answers
139 views

Solvable parametric $7$th and $13$th degree equations using $\eta(q)/\eta(q^p)$?

Q: Why is that some polynomial relations between eta quotients have a solvable Galois group, even if the deg is $n>4$? For example, we have the well-known modular equation, $$u^6 - v^6 + ...
3
votes
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125 views

Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as \begin{equation} p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~, ...
3
votes
0answers
56 views

Closed form for a simple hypergeometric q series

I've run across an interesting hypergeometric q-series that I feel must have been found before: $\sum_{n=0}^{\infty}(-1)^n$$\frac{e^{n b y}}{\prod_{k=1}^{n}(e^{\pi k b^2}-e^{\pi k b^{-2}})} = ...
3
votes
0answers
368 views

An Expression for $\log\zeta(ns)$ derived from the Limit of the truncated Prime $\zeta$ Function

I think, here, I found $$ P_x(s)=\sum_{p < x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} \left[ {\rm li}(t^{\frac zn-s}) \right]^{x}_2 \tag{7} ...
3
votes
0answers
108 views

Error Function limes

How can i calculate $\prod_{n=1}^{\infty}{erf(n)} $ with $erf(z) = \frac{2}{\sqrt{\pi}}\int_0^z e^{-z^{2}} \mathrm{d}z$? I know it's something like 0,84. And i see that only the first terms are ...
3
votes
0answers
160 views

Can Bernoulli polynomials be extended to fractional orders without losing elementarity?

Can Bernoulli polynomials $B_s(x)$ be extended to fractional $s$ in such a way so that for any given $s$ the function $B_s(x)$ still could be expressed in elementary functions of $x$?
3
votes
0answers
413 views

Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions?

I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions ${}_1F_2(1;a,a+1;z)$. I was unable to find useful functional relations for this specific ...
3
votes
0answers
425 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 ...
3
votes
0answers
277 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 ...
2
votes
0answers
41 views

Jacobi triple product for multidimensional lattices

The Jacobi triple product identity gives as a special case a product formula for the theta function of a 1-dimensional lattice. Is there a more general product formula for the theta function of an ...
2
votes
0answers
45 views

Associated Legendre/Gegenbauer functions with complex degree at larger order

I am interested in approximating the associated Legendre function (also known as conical function) \begin{equation} P_{-1/2 + i p}^{\frac{2-N}{2}}(x) \end{equation} when $N \to \infty$. The real ...
2
votes
0answers
60 views

From Selberg integral to Dyson integral

My question is from the drivation from Slberg integral to Dyson integral in this paper: Selberg integral : $$ S_n(\alpha,\beta,\gamma) = \int_0 ^1 \cdots \int_0 ^1 \prod_{i=1}^n ...
2
votes
0answers
109 views

A generalization of Macdonald functions?

I am interested in finding a set of functions $f(z_1,\cdots ,z_k;q,\,t)$, conjecturally polynomials, which depend on two parameters $(q,t)$ and an integer $k$, and are orthogonal under the following ...
2
votes
0answers
88 views

A second polylogarithm ladder for the tribonacci and n-nacci constants

In "A seventeenth-order polylogarithm ladder", on page 6 (eq 25), Bailey et al give a dilogarithmic ladder that begins with, $$0 = ...
2
votes
0answers
280 views

The Riemann Zeta Function summing over the Gamma Function

Has anyone studied a function of the form $$\eta(s) = \sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^{s}} = \sum_{n=0}^{\infty}\frac{1}{k!^s}$$ This series is appearing in my research on the volumetric ...
2
votes
0answers
111 views

Are numbers $h_{r,s} = \sum_{k} P(r;s;k) \frac{1}{n^{2k}} \bigg(1-\frac{1}{n}\bigg)^{n-2k}$ irrational?

I asked this question on MSE and Mike Spivey gave an insightful answer. I decided to put it here nevertheless in case someone else gets interested. If this violates rules on MO, please let me know, ...
2
votes
0answers
248 views

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 ...
2
votes
0answers
662 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 ...
2
votes
0answers
449 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 ...
1
vote
0answers
26 views

Inversion of incomplete elliptic integral of third kind?

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)? That means given $\Pi(n,u,m) = f(x)$, I would like to obtain $u$ as ...
1
vote
0answers
66 views

Quadratic transformation of hypergeometric function 2F1

I want to know whether there is some transformation between $_2F_1(a,b;c;x)$ and $_2F_1(a',b';c';x(1-x))$. Here is an example called the Kummer quadratic transformation, which may be known to most of ...
1
vote
0answers
81 views

Simplify this expression with modified Bessel functions of the second kind

I'm interested in the function $$\frac{K^{(0,2)}(0,x)}{K(0,x)}$$ where the numerator is the modified Bessel function of the second kind twice differentiated with respect to $\alpha$ and taken at ...
1
vote
0answers
50 views

Sufficient conditions for sums of Laguerre polynomials to be non-negative

I am interested in sufficient conditions on non-negative sequences of coefficients $\{c_{2n}\}_{n\ge 0}$ guaranteeing that $$%\begin{equation}\label{cond} \sum_{n=0}^\infty c_{2n} L_{2n}^{(1)}(x)\ge ...
1
vote
0answers
152 views

complex contour integral calculation after Möbius transformation

Good day to everyone. In my scientific research I've got stuck with a contour integration problem. I would like to evaluate the following integral: $$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...
1
vote
0answers
278 views

Inequality for complex Hankel function

Let $x>1$ and $0<\varphi<\frac{\pi}{2}$ be fixed. I would like to show that for any $s>0$, the following inequality holds: $$ \left| H_{\frac{is}{e^{i\varphi } \cos \varphi}}^{\left( 1 ...
1
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0answers
151 views

Can the series $\sum\limits_{n=0}^\infty q^{F_n}$ be expressed in terms of theta functions?

Let $F_0=0,F_1=1,...$ be the Fibonacci numbers. Is there a known closed form for the sum $\sum\limits_{n=0}^\infty q^{F_n}$? By closed form, I mean in terms of well-known functions, the first ones to ...
1
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0answers
180 views

An integral with Gamma functions (Part 2)

I was wondering if there is a generalization of the integral discussed here to a case like, \begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ...
1
vote
0answers
110 views

an infinite series expansion in terms of the polylogarithm function

we have the complex valued function : $$f(z)=\sum_{n=0}^{\infty}a_{n}Li_{-n}(z)$$ we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function ...
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0answers
175 views

solving a sum of Hypergeometric function 2F3

I am trying to find a closed form solution for the two sums given by $$\sum^n_{k=0}\frac{y^k}{k!}(-a+n)_k \left(\frac{2}{z} \right)^k {}_2F_3\left({-\frac{k}{2}, \frac{1 - k}{2}}; {-k, -a+n,1+a-k-n}; ...
1
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0answers
313 views

P-Adic poly Bernoulli numbers

we can define p-adic Bernoulli polynomials by using q-integral on $Z_p$ and T.Kim's method, But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $Z_p$ ?
1
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0answers
707 views

Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta

The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation. In the discrete calculus there is ...
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0answers
584 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 ...
1
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0answers
250 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, ...
1
vote
0answers
365 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 ...
1
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0answers
260 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 ...
1
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0answers
806 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 ...