# Tagged Questions

**6**

votes

**1**answer

157 views

+100

### 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
...

**1**

vote

**2**answers

191 views

### Looking for a limit related to the series in a previous post

Can any one show that the following limit?
$$
\lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1.
$$
If one uses the ...

**0**

votes

**0**answers

65 views

### An example of a homeomorphism on $[0,1]^2$ with constant Jacobian determinant $\pm1$ [migrated]

Let $T(x,y):=(t_1(x,y),t_2(x,y))$ be a continuous bijection, namely a homeomorphism on $[0,1]^2$.
I am trying to find a $T$ such that $\det(J_T)=1$. (*)
The trivial cases are
$T(x,y)=(x,y)$, ...

**7**

votes

**1**answer

488 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**3**

votes

**1**answer

107 views

### Asymptotic expansion of modified Bessel function $K_\alpha$

An integral representation for the Bessel function $K_\alpha$ for real $x>0$ is given by $$K_\alpha(x)=\frac{1}{2}\int_{-\infty}^{\infty}e^{\alpha h(t)}dt$$ where ...

**0**

votes

**0**answers

30 views

### Inverse Barnes G(n) function?

It is known that Barnes $G$-function is an analytic continuation of the $G$-function defined in the construction of the Glaisher-Kinkelin constant.
http://mathworld.wolfram.com/BarnesG-Function.html
...

**2**

votes

**0**answers

59 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 ...

**3**

votes

**1**answer

188 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 ...

**1**

vote

**0**answers

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 ...

**3**

votes

**2**answers

145 views

### Identifying a special function from its power series

Here is a power series, which looks a bit like a Hypergeometric function series, but I don't think that it is. Has anyone any idea what it is? Here $n,p,r$ are integers with $n\ge 0$ and $p\ge r\ge ...

**3**

votes

**0**answers

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}})} = ...

**2**

votes

**1**answer

108 views

### Estimate of a ratio of two incomplete gamma functions

I would like to bound from above the expression
$$
\frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)}
$$
for $x>y>0$. By plotting the above expression I have found that ...

**2**

votes

**2**answers

147 views

### Ewald's generalized theta function

Could anyone provide me some materials on the derivation of Ewald's generalized theta function (in English)? The original paper was written in German :-(
Die Berechnung optischer und ...

**11**

votes

**1**answer

270 views

### Is there something wrong with Hörmander's theorem on stationary phase method

It is well-know that the Bessel function has the asymptotic expansion $J_n(\omega) \sim \left( \frac 2 {\pi \omega} \right)^{1/2} \left( \cos \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) - ...

**2**

votes

**1**answer

118 views

### determining sign of function containing logarithm.

I would like to know the sign of the following term in general. I tried approximation for $\log$ function and it had negative sign. Is there any $m_0$ such that for all $n>m>m_0$, the following ...

**13**

votes

**1**answer

295 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

**0**answers

223 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

**1**answer

313 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 ...

**3**

votes

**0**answers

107 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 ...

**27**

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

204 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 ...

**5**

votes

**1**answer

750 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 ...

**3**

votes

**0**answers

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 $x$?

**0**

votes

**0**answers

694 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

**1**answer

171 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 $0\lt q\lt 1$ ?

**5**

votes

**2**answers

748 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 ...

**1**

vote

**0**answers

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 ...

**3**

votes

**1**answer

465 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

**0**answers

364 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 ...

**2**

votes

**3**answers

462 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

247 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

418 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} ...

**7**

votes

**2**answers

341 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

216 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

420 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

255 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

184 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 ...

**0**

votes

**2**answers

732 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 ...

**7**

votes

**4**answers

1k 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 ...

**4**

votes

**1**answer

217 views

### Positivity of a rational function

A rational function is called positive if all its Taylor coefficients are positive.
Friedrichs-Lewy conjecture states the positivity of the rational function
\begin{eqnarray*}\frac{1}{
(1-x)(1- ...

**4**

votes

**2**answers

366 views

### Analytic continuation of $_4F_3(1)$

The Gauss theorem
$${_2F_1}(a,b;c;1)=\frac{\Gamma(c-a)\Gamma(c-b)}{\Gamma(c)\Gamma(c-a-b)}$$
allows to compute the analytic continuation of ${_2F_1}(a,b;c;1)$ for $a+b>c$ when the series ...

**2**

votes

**0**answers

446 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 ...

**4**

votes

**2**answers

620 views

### Bounds on remainder term of power series of elementary functions

This is mainly a question about the remainder term of power series for elementary functions.
I'm very interested in aspects of calculating or computing elementary operations and functions, by which I ...

**0**

votes

**1**answer

450 views

### modular arithmetic of Hermite polynomials

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder
$R_{k,m} \equiv H_{k} ~ \mod H_m$
for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial ...

**1**

vote

**1**answer

547 views

### Asymptotics of Hermite and hypergeometric function

I am looking for the asymptotics of the following integral
$\int_{\mathbb{R}} H_m^2(x) {\rm e}^{-2 \alpha^2 x^2} {\rm d} x = 2^{m-1/2} \alpha^{-2m -1} (1-2\alpha^2)^m \ \Gamma(m+1/2) ~ ...

**4**

votes

**3**answers

2k views

### Spherical Harmonics - a bunch of questions about them

Hi there,
Please tell me if I should divide these into individual questions next time.
Short intro:
Spherical Harmonics are a nice collection of functions. They are orthogonal and allow you to take ...

**26**

votes

**0**answers

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} ...

**18**

votes

**0**answers

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 ...

**7**

votes

**2**answers

476 views

### Relation between full elliptic integrals of the first and third kind

I am working on a calculation involving the Ronkin function of a hyperplane in 3-space.
I get a horrible matrix with full elliptic integrals as entries. A priori I know that the matrix is symmetrical ...

**15**

votes

**5**answers

9k views

### Inverse gamma function?

This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this.
We have the ...