**9**

votes

**0**answers

334 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 ...

**0**

votes

**1**answer

237 views

### derivative of a special function in integral form

What is the derivative of $Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right)$ with respect to $x$, i.e,
$$\frac{\partial}{\partial x}Q_m\left(\frac{\alpha}{x^a},\frac{\beta}{x^b}\right),
\quad
...

**2**

votes

**0**answers

318 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 ...

**3**

votes

**1**answer

263 views

### The relationship between Stirling number of the second kind and the polylogarithm

It is shown here on Mathworld's page on Stirling number of the second kind that
$$
\sum_{k=1}^n S(n,k) (k-1)! z^k = (-1)^n \text{Li}_{1-n}(1+\frac{1}{z})
$$
where $S(n,k)$ is Stirling number of the ...

**0**

votes

**0**answers

191 views

### Convert 2F1 to polynomial

Is there any transformation to convert each of the following versions of ${}_2F_1$ to a polynomial?
The first one is
$${}_2F_1\left(\frac{1-a}{2}, -\frac{a}{2}; b;\frac{4z}{(1+z)^2} \right), \quad ...

**5**

votes

**1**answer

509 views

### Contour Integral with Gamma functions and 2F1

Given the following contour integral
$$\frac{1}{2\pi j}\int^{c+j\infty}_{c-j\infty} \frac{\Gamma(-1+a+s)\Gamma(b+s)}{\Gamma(3+a-s)}\cos(-1+a+s)\,
{}_2F_1\Big(-1-a+s,-1+a+s;\frac{1}{2};z\Big) y^s\: ...

**2**

votes

**1**answer

544 views

### Quadratic Transformation of the Hypergeometric Function 2F1

The function ${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)$ can be transformed (as reported by A. Erdélyi) by the following formula
${}_2F_1\Big(\frac{a-b}{2},\frac{a+b-1}{2};c;y\Big)=
...

**1**

vote

**1**answer

188 views

### About one series. Are there some related special functions?

Hello,
I have the following series:
$$
\sum_{n=2}^\infty \frac{t^n}{\Gamma(a n)} = ?,\qquad t\ge 0,
$$
where the parameter $a\in (0,1]$, $\Gamma$ is the Gamma function. When $a=1$, the above sum ...

**4**

votes

**1**answer

1k views

### minimal polynomials of trig functions of ($k \pi/p$) and divisibility of coefficients by p

Take an odd prime $p$ and put $x_0:=\sum\limits_{j=0}^{p-1}\left(a_{j}\sqrt{p}\cos\dfrac{j\pi}p+b_{j}\sin\dfrac{j\pi}p +c_{j}\tan\dfrac{j\pi}p\right)$, where the $a_{ij}$ are integers. If $f$ denotes ...

**2**

votes

**0**answers

721 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 ...

**9**

votes

**2**answers

869 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 ...

**7**

votes

**3**answers

1k views

### a dilogarithm identity: known or new?

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

**1**

vote

**0**answers

287 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, ...

**3**

votes

**2**answers

752 views

### Inverse Hankel Transform

I was reading through Akhiezer's book Lectures on Integral Transforms and in chapter nine, he states that the Hankel transform is unitary for $\nu > -1$, so that for a suitable function, $f$,
...

**2**

votes

**1**answer

245 views

### is this bound on a kummer function known?

Is it already known that ${}_1F_1(a;b;x) \leq \Gamma(b)(1+|x|)^{-a}$ when $a$ is an integer, $a <0,$ and $b>0?$ If it is, what is a reference?
my proof:
Since the Kummer function can be ...

**0**

votes

**1**answer

351 views

### Upper bounds on generalized Laguerre polynomials

I evaluated an integral and obtained an expression with a Laguerre polynomial. I'd like something more explicit and useable.
Are there any known simple (e.g. exponential) upper bounds on the ...

**3**

votes

**1**answer

542 views

### Generalized trigonometric functions $Cos(n) v$ and $Sin(n) v$.

I just discovered a paper from 1948, Eine Verallgemeinerung der Kreis-und Hyperbelfunktionen by R. Grammel which introduces functions he calls Cos(n) and Sin(n), representing a parameterization of the ...

**1**

vote

**1**answer

587 views

### infinite series with Hypergeometric functions

Can we get a closed form for the series
$\sum^\infty_{k=0} \frac{ t^k}{k!} \Gamma(k+a)\Gamma(k+\frac{1}{2}){}_2F_1(k+a,k+\frac{1}{2};n+1,x)$
any hints or clues are welcomed.

**5**

votes

**1**answer

1k views

### Two-variable generating functions for Laguerre polynomials

Where can I find generating functions for orthogonal polynomials in two variables?
Lebedev's book (Special Functions and their Applications, Dover, 1972) gives a closed form for
$$
\sum_{n=0}^\infty ...

**5**

votes

**2**answers

1k views

### How to do integrals involving two Bessel functions and another function?

I often encounter the integrals in the following form:
$\int_0^\infty{\rm Bessel}(ax)\cdot{\rm Bessel}(bx)\cdot f(cx)dx$,
where Bessel can be $J$, $N$, $H^{(1)}$, $H^{(2)}$, $I$, or $K$; and $f(x)$ ...

**2**

votes

**0**answers

862 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 ...

**4**

votes

**3**answers

596 views

### Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$

I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression
$
\sum_{k=0}^{\infty} \frac{z^k}{k!k!} = F_{1}[;1;z]
$
This is not very helpful though, and I 'd ...

**0**

votes

**3**answers

710 views

### How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times ...

**15**

votes

**2**answers

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 $n=2m$, ...

**1**

vote

**0**answers

422 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 ...

**0**

votes

**0**answers

474 views

### modified bessel fucntion of the third kind

Hi I'm doing a computation where the modified bessel function of the third kind is the main source of computational strain, we are using a 10,000 dimension's for our distribution, is there any easier ...

**1**

vote

**1**answer

504 views

### Legendre Polynomial Identity

I have encountered the following sum involving Legendre polynomials, which I hope to reduce to something involving a $\delta$-function:
$$
\frac{d^2}{dx^2} \sum_{\ell=0}^{\infty} \frac{2 \ell + 1}{2 ...

**6**

votes

**2**answers

765 views

### On a polynomial related to the Legendre function of the second kind

The Legendre function of the second kind, $Q_n(z)$, along with the usual Legendre polynomial $P_n(z)$, are the two linearly independent solutions of the Legendre differential equation.
$Q_n(z)$ can ...

**2**

votes

**3**answers

474 views

### Differential equation with some constraints

I posted this to stackexchange, and after some hours got a comment that was so pessimistic about finding some neat orderly solution, that I'm posting it here too. (In case anyone cares, this is ...

**0**

votes

**0**answers

250 views

### High dimensional beta integral (question following the previous post)

Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} ...

**1**

vote

**2**answers

459 views

### High dimensional beta integral (a typo in Stein's book “singular integrals”)

Hello,
When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake:
$$
\int_{R^n} |x-y|^{-n+\alpha} ...

**11**

votes

**1**answer

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 ...

**1**

vote

**3**answers

482 views

### Pairing function monotonic respect to product of arguments [closed]

Has anyone ever created a "pairing function" (possibly non-injective)
with the property to be nondecreasing wrt to product of arguments, integers n>=2, m>=2. (We can also assume that n and m are ...

**1**

vote

**1**answer

2k views

### looking for monotonically increasing functions with range in [0,1] [closed]

Ideally, the function f(x) would tend toward zero as x tends towards negative infinity, and f(x) would tend towards 1 as x tends towards infinity, all the while being monotonically increasing.
for ...

**7**

votes

**2**answers

367 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
$$
...

**0**

votes

**0**answers

227 views

### Non-standard addition theorem for Legendre function of the first kind

There is known link text addition theorem for the Legendre functions of the first kind $P_{\nu}^m(x)$, which establishes the connection between $P_{\nu}^m(x)P_{\nu}^{m}(y)$ and $P_{\nu}(F(x,y))$ for ...

**3**

votes

**0**answers

499 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 ...

**2**

votes

**1**answer

287 views

### antiderivative involving modified bessel function

This little integral has been holding me up for weeks. Has anyone come across a similar integral in their work.
$\int {\frac{d x}{c-I_0(x)}} $

**1**

vote

**1**answer

187 views

### Can you control the amplitudes of a finite collection of sine curves just by controlling the amplitude of their superposition?

Dear all,
I would like to know whether the following claim is true. In particular, if it is true, then I would like to know if there is some textbook that contains the statement and maybe even the ...

**3**

votes

**0**answers

326 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 ...

**1**

vote

**1**answer

296 views

### Maximum of a series of integrals of Hermite functions

Given the function $$f(A) := \sum_{n=1}^{\infty}\left( \int_A \varphi_0\varphi_n \right)^2,$$ where $A$ is any measurable subset of $\mathbb{R}$, and $\varphi_n$ is the $n$th Hermite function, I want ...

**0**

votes

**2**answers

779 views

### Functions defined as infinite products

Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too!
The original motivation for this is the ...

**0**

votes

**2**answers

679 views

### Cosine of a Partial Sum

Does anyone know of a closed formula for $cos(\displaystyle\sum_{n=0}^m a_{n})$? I've seen formulas for $cos(\displaystyle\sum_{n=0}^\infty a_{n})$ and $tan(\displaystyle\sum_{n=0}^m a_{n})$, but the ...

**4**

votes

**1**answer

894 views

### What's the difference between a Riemann theta and a Siegel theta function?

One of the things I'm working on has required me to look into the literature of multidimensional theta functions, and I've gotten a bit confused on a few naming details.
A look at the DLMF says that ...

**5**

votes

**1**answer

909 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 ...

**5**

votes

**1**answer

701 views

### Integral identity for Legendre polynomials

How does one prove the following integral identity, where $P_n(x)$ is the $n$th Legendre polynomial?
$$
\int_0^1 P_n(2t^2-1) dt = \frac{(-1)^n}{2n+1}
$$
Notes & Background
A variant of this ...

**2**

votes

**1**answer

308 views

### Legendre Function “Types”

Legendre functions can be of first and second kinds; $P$, $Q$.
They can have order $\mu$ and degree $\nu$; $P^\mu_\nu$ $Q^\mu_\nu$
But do they also have Types? Some of the numerical software I am ...

**7**

votes

**4**answers

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

**4**answers

2k 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 ...

**3**

votes

**3**answers

2k views

### Integral over error function and normal distribution

Help me understand why
$\int_{-\infty}^{\infty}\frac{1}{2}[1+\operatorname{erf}(\frac{\theta-x}{\sqrt{2q^2}})]\frac{1}{\sqrt{2\pi\sigma^2}}{\exp(-\frac{(x-\mu)^2}{2\sigma^2})}dx \approx ...