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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2
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86 views

A question on Ramanujan's $1/\pi$ formulas

It is known that Ramanujan discovered a number of formulas fo $1/\pi$. All of these formulas are of the form $$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(1/2)_n(s)_n(1-s)_n}{(1)_n^3}(a+bn)z^n,$$where ...
3
votes
1answer
597 views

Pochhammer symbol of a differential, and hypergeometric polynomials

I have a minor result which I'm sure has come up somewhere before but I can't seem to find it. Consider a confluent hypergeometric function of the form $$\newcommand{\ff}{{}_1F_1} ...
1
vote
0answers
29 views

Proving injectivity of a multivariable function

Let $I$ denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by, $$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over ...
2
votes
1answer
56 views

Closed Form Expression for Nested Series Summation?

Just wandering if there are any criteria that can decide whether a finite series summation has closed form or not. for example, In the following nested summation, $n$ is some even integer that will be ...
11
votes
2answers
420 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 ...
6
votes
1answer
956 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 ...
34
votes
1answer
2k 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 ...
5
votes
1answer
370 views

Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...
1
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0answers
28 views

Rearrangement of summation expression

Referring to the spherical harmonics expansion in this article: Méléard, P., Pott, T., Bouvrais, H., & Ipsen, J. H. (2011). Advantages of statistical analysis of giant vesicle flickering for ...
1
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0answers
54 views

Rearrangement of a spherical harmonics expansion

Referring to this article: http://i.stack.imgur.com/sfQ1C.png and http://i.stack.imgur.com/LelKb.png How is it that they get from equation 2 to equation 3? Whenever I do it, I get only cosine ...
1
vote
0answers
172 views

system with solutions $\{x-a:0\leqslant a\leqslant z-1\}$ [closed]

What must be $F$ there where $0=F(1,x,0)=F(x-0,x,z)=F(x-1,x,z)=F(x-2,x,z)=F(x-3,x,z)=$ $\dots$ $=f(x-z-1,x,z)=0$? Define $F$ in the domain where a continuous function exists that behaves so for ...
6
votes
1answer
100 views

An identity of complicated combinations of gamma functions (related to hypergeometric functions)

Can somebody help me in proving the following equation? \begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) ...
3
votes
0answers
817 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 ...
4
votes
4answers
170 views

Integrals involving the Tricomi hypergeometric function

I am looking for a reference for the two following equalities involving the Tricomi function $U$ and the Meijer function $G$. I have found these formulas on the website http://functions.wolfram.com/, ...
-1
votes
2answers
322 views

What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [closed]

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...
1
vote
0answers
139 views

Conway's box function iterated to produce a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
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 ...
0
votes
0answers
66 views

Energy Oscillations in a One Dimensional Crystal

Good day! Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)? article, that I have Especially ...
4
votes
1answer
164 views

Special Function, Series Expansion, or Simpler Form of a Certain Infinite Product?

$\prod _{n=1}^{\infty } \left(1+a (c+n)^b\right)$ where a > 0, b < -1, and c >= 0 Is there a special function, series expansion, or other simpler (or maybe just interesting) representation of ...
8
votes
2answers
3k views

Sums of arctangents

$$ \begin{align} \arctan(x) & = \arctan(1) + \arctan\left(\frac{x-1}{2}\right) \\ & {} - \arctan\left(\frac{(x-1)^2}{4} \right) + \arctan\left(\frac{(x-1)^3}{8}\right) - \cdots \end{align} $$ ...
1
vote
0answers
105 views

Four kinds of generalized hypergeometric formulas for $\pi$

Given, $$\begin{array}{|c|c|c|c|} \hline n&a_n&b_n&c_n\\ \hline 1 & 6541681608 & 163096908 & -640320^3\\ \hline 2 & 85840 & 4492 & -14112^2\\ \hline 3 & 28302 ...
5
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0answers
459 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} ...
10
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0answers
441 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 + ...
6
votes
2answers
334 views

expression for infinite series with powers of factorial in denominator

The series $$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$ has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...
5
votes
2answers
418 views

A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where $$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$ is the Lerch ...
5
votes
2answers
269 views

What is a “generalized zeta function”?

Out of procrastination I computed $$\sum_{k=1}^\infty k^{-k^2}\sim 1.06255080549625593786944593879.$$ The inverse symbolic calculator identified this number as "From generalized Zeta function". I do ...
10
votes
1answer
336 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 $j'_{n+1/2,1}$ denote the first zero of its derivative, which is also the location of the first maximum of $\mathcal{J}_{n+1/2}$. ...
5
votes
1answer
522 views

Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...
2
votes
0answers
56 views

how understand periodicity in a combination of power, gamma and zeta functions?

Riemann's functional equation may be written: $$ \frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1} $$ and so by symmetry: $$ \frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} ...
-2
votes
2answers
152 views

On Bohr-MollerupTheorem [closed]

In http://mathworld.wolfram.com/Bohr-MollerupTheorem.html, Bohr-Mollerup Theorem is given where it is stated that $\Gamma$ function is the unique log convex function that satisfies ...
4
votes
1answer
468 views

What is the advantage of inverting elliptic integrals?

In the case of the circle I can hardly make any conclusions from the integral $(1)$, most of the theorems come from geometrical considerations. It's not clear how to prove periodicity from this ...
-2
votes
3answers
2k views

Factorial of 3/2? [closed]

How do you compute the factorial of something like $3/2$ or $-2$? Wolfram Alpha gives an answer, but how does it arrive at that point?
1
vote
1answer
109 views

Approximation of $ _2F_1((b-1)a,b;ba;x) $

Is there any (simple) approximation of this Hypergeometric function: $ _2F_1((b-1)a,b;ba;x) $, where $0<x<1$ and $b>a>1$. Thanks!
5
votes
0answers
122 views

Elementary function relative to erf

The modified Bessel function of the 1st kind $I_0$ is defined by $$ I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta $$ and arises, among other places, in the probability density function of a ...
6
votes
0answers
172 views

Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$ ...
2
votes
1answer
301 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 ...
0
votes
1answer
77 views

Upper bound for a ratio of modified Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...
1
vote
0answers
202 views

What is the status on questions related to Bhargava's factorial function?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like: For $k, l \in \mathbb{Z}$, we have $k! \times ...
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2answers
134 views

Is there any simpler form of this function

Assume that $n$ is a positive integer. Is there any simple form of this hypergeometric value $$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$
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0answers
94 views

What is $\int (1-e^{-x})^n dx$? [closed]

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is $(-1)^n B(e^x; -n, n+1)$ I ...
1
vote
1answer
150 views

How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and ...
1
vote
0answers
105 views

A hypergeometric identity [closed]

Is there any simple proof of this identity $${_4F_3}[\{\frac{1}{2}+\frac{n}{4},1+\frac{n}{4},-p,p\},\{\frac{1}{2},\frac{3}{2},\frac{1}{2}+\frac{n}{2}\},1]=\frac{\Gamma[\frac{1+n}{2}] ...
1
vote
0answers
74 views

q-Hermite polynomials

It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$ are orthogonal in $\theta \in [0, \pi]$ with ...
2
votes
2answers
165 views

How to calculate one Cauchy type determinant

As we know, a Cauchy determinant of size n admits the following explicit formula: $$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y ...
4
votes
1answer
559 views

How to prove this identity on double summation series?

I suspect the following identity is valid, but I can not prove it. I just calculate it numerically. ...
7
votes
3answers
1k views

a dilogarithm identity: known or new?

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

Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared: $$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$ where $\rho,s>1$, ...
6
votes
0answers
333 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
201 views

Legendre polynomials and formal groups

Let $P_n(x)$ be Legendre polynomials: $$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$ Usual arguments from the theory of formal groups allow to prove that for any $n$ ...
9
votes
8answers
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 ...