**19**

votes

**3**answers

630 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 ...

**0**

votes

**1**answer

126 views

### Find a generalized hypergeometric-based function yielding certain ratios of fifth-degree polynomials

Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials)
\begin{equation}
...

**11**

votes

**1**answer

270 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 ...

**1**

vote

**2**answers

213 views

### Hypergeometric identities

Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply
$$\sum_{j=0}^k\frac{(-1/2)_{k-j}(m+1)_j(-m)_j}{(1/2)_j(k-j)!j!}
...

**8**

votes

**2**answers

837 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 ...

**1**

vote

**1**answer

457 views

### Extension of the Jacobi triple product identity

The Jacobi triple product and the mathematical identity of it is:
$$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$
I would ...

**0**

votes

**1**answer

162 views

### This might be a trivial question on Hurwitz's zeta function.

In the book I am reading they write that for Hurwitz zeta function, $\zeta(x,s)=\sum_{n=0}^{\infty} \frac{1}{(x+n)^s}$, the next sum in the RHS converges for $\Re(s)>-1$, and I don't see how ...

**4**

votes

**1**answer

420 views

### Identity involving Fresnel integrals

In the paper E. Mehlum, Appell and the apple (nonlinear splines in space), Technical
Report No. 1676 (1981), Central institute for industrial research, Oslo (reproduced in the book Mathematical ...

**1**

vote

**0**answers

227 views

### Whether does the following equation have only one finite zero?

Dear MOs,
Here is a calculus problem which bored me for sometime. Let $a>0$ and $b<0$ be fixed.Define the following function (EDIT: Following the comment by Barry Cipra, you may only consider ...

**2**

votes

**2**answers

294 views

### Computing hypergeometric function of matrix argument

In the context of the Bingham probability distribution the ${ }_1F_1$ hypergeometric function of matrix argument naturally arises as a normalization constant of the probability distribution function. ...

**6**

votes

**0**answers

139 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 ...

**3**

votes

**3**answers

138 views

### Multivariate functions whose value is independent of the order of the arguments

Let $r_1, r_2, \ldots, r_k$ be positive integers with or without repetition such that $1\le r_i \le n$ for $i = 1, 2, \ldots, k$. Let $f$ be a continuous multivariate function with the property that ...

**9**

votes

**2**answers

487 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 ...

**1**

vote

**3**answers

298 views

### trigonometric identity needed for sums involving secants

I am looking for a closed-form formula for the following sum:
$\displaystyle \sum_{k=0}^{N}{\frac{\sin^{2}(\frac{k\pi}{N})}{a \cdot ...

**-1**

votes

**3**answers

386 views

### problem related to Airy functions [closed]

I have solved the Schrödinger equation for a triangular well potential and the solution comes in terms of Airy functions...now i am facing the following problems:
What are the normalization ...

**3**

votes

**1**answer

539 views

### Integral representation of the modified Bessel functions of the second kind and asymptotic expansion

The modified Bessel function (Macdonald function) $K_\alpha(z)$ is known to have the following asymptotic expansion for large positive $z$:
$$
K_\alpha(z)=\sqrt{\frac{\pi}{2z}}e^{-z}\sum_{k=0}^\infty ...

**3**

votes

**1**answer

1k views

### Is there any heartbeat like function? [closed]

I'm looking for a function, where the result is something like this:
image
I tried to figure it out myself, but I have no idea how to manage it.
f(x) = ...
...

**4**

votes

**0**answers

402 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

**0**answers

136 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, ...

**1**

vote

**1**answer

329 views

### Proof of generalized Cauchy formula

I would like to know if there is a proof for the identity used in the superconformal index of 4d ${\cal N}=2$ gauge theory. In the paper by Rastelli el al, it was discovered that Eq. (10) is equal to ...

**4**

votes

**1**answer

324 views

### “Known” Pythagorean identity? (reference request)

Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $\tan\theta_1,\tan\theta_2,\tan\theta_3,\ldots$ (and if the sequence of $\theta$s is finite remember that the $k$th-degree elementary ...

**2**

votes

**1**answer

357 views

### Integral of Modified Bessel Function of the Second Type

Given the identity
$$ \int^\infty_0 K_v\left(\alpha\sqrt{x^2+z^2}\right) \frac{x^{2\mu+1}}{\left(\sqrt{x^2+z^2}\right)^v}\:\mathrm{d}x = \frac{2^\mu \Gamma(\mu+1)}{\alpha^{\mu+1}z^{v-\mu-1}} ...

**2**

votes

**1**answer

604 views

### trigonometric non-identity

a <- (runif(2000) * (40-10) + 10) * pi/180
b <- (runif(2000) * (40-10) + 10) * pi/180
c <- (runif(2000) * (40-10) + 10) * pi/180
This chooses 2000 (pseudo-)random numbers in $(0,1)$ and ...

**10**

votes

**0**answers

379 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 + ...

**3**

votes

**3**answers

445 views

### A “known” tangent half-angle formula?

In another posting I wrote about a trigonometric relation I had derived, but that ended up not being the main point of the posting:
Strange pattern in rounding errors?
So as long as we're here, ...

**6**

votes

**2**answers

443 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 ...

**5**

votes

**0**answers

427 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 ...

**2**

votes

**1**answer

370 views

### Asymptotic expansion of integral (Bessel function really)

The integral
$$I = \int_{-\infty}^\infty \frac{e^{-\varepsilon x^2}} { \sqrt{1+x^2} } dx$$
is convergent for $\varepsilon > 0$ and can even be given in terms of the Bessel function $K_0$. As ...

**2**

votes

**1**answer

95 views

### Dertivative of a Special Function with respect to Order

The marcum Q-function is defined by
$$ Q_m(a,b) = \int^\infty_b x \left(\frac{x}{a}\right)^{m-1} \exp\left(-\frac{x^2+a^2}{2}\right)
I_m\left(a x\right)
\:\mathrm{d} x,$$
where $m\in\mathbb{N}$ , ...

**3**

votes

**0**answers

118 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 ...

**1**

vote

**0**answers

141 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 ...

**0**

votes

**2**answers

206 views

### Series representation of ratio of two Meijer G-functions

Let me use the notation from Maple http://www.maplesoft.com/support/help/Maple/view.aspx?path=MeijerG for the Meijer G-function. Then let me define,
$f_+(x) = MeijerG( [[+1/2],[]], [[0,0],[]], x )$
...

**0**

votes

**0**answers

258 views

### Lambert W-function

I asked this question MSE, but didn't get any answers. Maybe here someone can help.
I need to solve
$$
\theta \rho^{\theta}+r \theta>v
$$
where $\theta \in \mathbb{R}^{+}, -1 < r,v<1, \ ...

**4**

votes

**0**answers

420 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) ...

**14**

votes

**2**answers

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 ...

**2**

votes

**2**answers

453 views

### Dilogarithm, tetrahedrons, and hyperbolic space

The Bloch-Wigner function $D(z)$, gives the volume of a ideal tetrahedron in hyperbolic space $\mathbb H3$, where z is the cross-ratio (z1,z2,z3,z4) parametrizing the tetrahedron in $\mathbb CP1$.
...

**10**

votes

**2**answers

765 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,
...

**23**

votes

**2**answers

950 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} ...

**3**

votes

**1**answer

524 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}
...

**29**

votes

**7**answers

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 ...

**0**

votes

**1**answer

137 views

### Convolution operators defined by compactly supported distribtion

Let $\mu_{n}$ be the unit measure over $S^{n-1}$,and consider the convolution operator$$Tf=\mu_{n}\ast f,\quad f\in \mathcal{S}$$
then,it's well-known that T can be extend to a bounded operator on ...

**0**

votes

**1**answer

209 views

### Inequality with even powers of trigonometric functions

For $m>0$,
$0 < n\leqslant m+1$ ($m,n\in \mathbb{Z} $) , and $0 < a < 1$ , prove that
$$2^{n}\cdot \left( a^{n}\cos ^{2m}\dfrac {\pi a} {2}+\left( 1-a\right) ^{n}\sin ^{2m}\dfrac {\pi ...

**2**

votes

**2**answers

2k views

### Numerical Computation of arcsin and arctan for real numbers [closed]

I'm coding some numerical methods and I do not know what the correct analysis would be for choosing the implementation for $arcsin$ and $arctan$ for real numbers. Here's what I know:
Both functions ...

**0**

votes

**1**answer

468 views

### The zeros of the digamma function

I wonder what work have been done on the zeros of the digamma function and on the values of the gamma function at such points (on the negative real axis). Any help please :)

**1**

vote

**1**answer

347 views

### Product of densities of a wrapped normal distribution

The density of a wrapped normal distribution is given by
$$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right]$$
Considering two ...

**5**

votes

**3**answers

589 views

### Proof of a combinatorial identity (possibly using trigonometric identities)

For integers $n \geq k \geq 0$, can anyone provide a proof for the following identity?
$$\sum_{j=0}^k\left(\begin{array}{c}2n+1\\\ 2j\end{array}\right)\left(\begin{array}{c}n-j\\\ ...

**3**

votes

**2**answers

255 views

### cyclic polygons & trigonometry

I posted this question to stackexchange, where it's generated some comments but no progress toward answering it. I'm going to say somewhat more here than I did there.
At one vertex of a pentagon ...

**5**

votes

**1**answer

771 views

### Request for the proof of a result from Ramanujan's letter to Hardy.

Srinivasa Ramanujan in his first letter to G.H. Hardy stated many results for which he didn't give proofs. Among them the result taken from this link seems interesting :
If ...

**9**

votes

**7**answers

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 ...

**3**

votes

**0**answers

170 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 $x$?