**3**

votes

**0**answers

146 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, I'...

**1**

vote

**1**answer

360 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

360 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 ...

**3**

votes

**1**answer

652 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}} K_{v-\mu-...

**2**

votes

**1**answer

611 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 ...

**11**

votes

**0**answers

522 views

### Connection between Infinite continued fractions, elliptic integrals 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 + \frac{3^{...

**3**

votes

**3**answers

463 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, let'...

**6**

votes

**2**answers

475 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

440 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

482 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

107 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 $m\in\mathbb{N}$ , $b\...

**3**

votes

**0**answers

125 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

180 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

259 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

286 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, \ 0&...

**5**

votes

**0**answers

712 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

2k 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

648 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 $\mathbb CP1$.
$D(...

**10**

votes

**2**answers

838 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,
$$A(q)...

**25**

votes

**2**answers

1k views

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

Given the Dedekind eta function $\eta(\tau)$. Define $m = (p-1)/2$ and a $24$th root of unity $\zeta = e^{2\pi i/24}$.
Let p be a prime of form $p = 12v+5$. Then for $n = 2,4,8,14$:
$$\sum_{k=0}^{...

**4**

votes

**1**answer

723 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}
\ff(b+k;b;z)\...

**31**

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

142 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 $L^{...

**0**

votes

**1**answer

215 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

3k 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

609 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

447 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 ...

**6**

votes

**3**answers

638 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\\\ k-j\end{array}\...

**3**

votes

**2**answers

296 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

802 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 $$\int\limits_{0}^{\infty}...

**11**

votes

**8**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

199 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 $x$?

**2**

votes

**0**answers

278 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}; ...

**3**

votes

**1**answer

421 views

### p-adic poly-Bernoulli numbers

We can define p-adic Bernoulli polynomials by using q-integral on $\mathbb{Z}_p$ and Taekyun Kim's method.
But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $\...

**2**

votes

**2**answers

637 views

### Resources for special functions, integral identities

In the past weeks, I have struggled with finding suitable tables for integral indentities for Beta functions, Chebyshev polynomials and their like.
I would like to ask for online/offline resources ...

**4**

votes

**0**answers

270 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}} = \frac{2^k}{2^...

**11**

votes

**1**answer

466 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 ...

**3**

votes

**0**answers

783 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 ...

**1**

vote

**0**answers

2k views

### An inverse Laplace transform involving Error function

Dear MOs,
I need to calculate the inverse Laplace transform of the following function
$$
g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0.
$$
I have checked, among many ...

**0**

votes

**2**answers

203 views

### A certain sum with q by the power of binomial (n 2)

Is there a closed form to the following sum: $\sum_{n=0}^{\infty}a^nq^{n(n-1)/2}$
for all $a>0$ and $0\lt q\lt 1$ ?

**3**

votes

**1**answer

647 views

### Fourier and Bessel

Oliver Heaviside, on page 387 of Electrical Papers, Vol. I, Macmillan and Co., 1892, available here, writes
$$v = 1 - \frac{n^2r^2}{2^2} + \frac{n^4r^4}{2^2 4^2} - \frac{n^6r^6}{2^24^26^2} + \ldots =...

**35**

votes

**1**answer

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 ...

**7**

votes

**4**answers

546 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 ...

**4**

votes

**1**answer

839 views

### q-Pochhammer Symbol Identity

Is this identity or an equivalent one already referenced in the litterature? In particular, is it even true?
${\frac{\left ( -1 ; e^{-4\pi} \right) ^2_{\infty}}{\left ( e^{-2\pi} ; e^{-2\pi} \right) ^...

**4**

votes

**3**answers

544 views

### Monotonicity of a combination of Bessel functions

Prove that the following function is decreasing (as a function of a) for a > 0 when 0 < r < 1:
$${K_2(ar)I_2(a)-I_2(ar)K_2(a)\over I_2(a)}I_2(ar).$$
The problem arose in the analysis of a model ...

**1**

vote

**1**answer

232 views

### A uniqueness proposition involving Erf, the error function

This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function".
Consider the system of equations:
$$1/2 + {\rm Erf}(x) - \alpha {\rm Erf}(\frac{x+y}{...

**0**

votes

**1**answer

610 views

### Sum over Hypergeometric function 2F1 (generating function)

Dear mathematicians,
in my current research project I came accross this very bothersome sum over a rather simple hypergeometric function, or formulated differently: a sum over squared binomial ...

**4**

votes

**2**answers

751 views

### Reducing system of equations involving Erf, Error Function

I have a system of equations:
$$1/2 + {\rm Erf}(x) - {\rm Erf}(\frac{x+y}{2})=0$$
$$-1/2 + {\rm Erf}(y) - {\rm Erf}(\frac{x+y}{2})=0,$$
Where $x \le y$ and ${\rm Erf}$ is the Error Function.
By ...

**7**

votes

**2**answers

998 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 q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\...

**2**

votes

**1**answer

260 views

### Infinite Series of 2F1

I will be grateful for any ideas to solve the series
$$\sum^\infty_{k=0}\frac{x^k z^k}{k!} \frac{\Gamma(1+a+2k)}{\Gamma(2+k)}{}_2F_1(1,1+a+2k;2+k;z)$$
$a$ is a nonegative integer, $z$ and $x$ are ...