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14 votes
3 answers
664 views

(Sharp) inequality for Beta function

I am trying to prove the following inequality concerning the Beta Function: $$ \alpha x^\alpha B(\alpha, x\alpha) \geq 1 \quad \forall 0 < \alpha \leq 1, \ x > 0, $$ where as usual $B(a,b) = \...
Ester Mariucci's user avatar
11 votes
1 answer
566 views

Integral representation of product of two Whittaker functions

Does anyone know anything about the following formula involving special functions: $$\begin{multline*} W_{\kappa,\mu}(z)W_{\lambda,\mu}(w)=\frac{e^{-(z+w)/2}(zw)^{\mu+1/2}}{\Gamma(1-\kappa-\lambda)}\...
Y.Okuyama's user avatar
  • 373
10 votes
1 answer
580 views

About certain infinite products with the property $f(a)=f(1/a)$

In the paper Transformations of infinite series, Bryden Cais gives the following transformations of infinite products Theorem 4. If $$ f(t) = \frac{\cosh(\pi t)-1}{\sinh(\pi t)}\frac{\cosh(2\pi t)+1}{...
Nemo's user avatar
  • 5,624
7 votes
4 answers
4k 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 ...
Tom LaGatta's user avatar
  • 8,512
5 votes
1 answer
1k 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} ...
C.S.'s user avatar
  • 4,795
4 votes
1 answer
387 views

$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$

The Fejer-Jackson inequality as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ I conjecture that the inequality as follows holds: $$\sum_{...
Đào Thanh Oai's user avatar
3 votes
2 answers
780 views

An integral identity evaluating the gamma function

While reading a number theory paper I encountered the identity $$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$ ...
Frank Thorne's user avatar
  • 7,347
3 votes
1 answer
183 views

On integral representation of Whittaker $W$ functions

According to NIST, the integral representation of Whittaker $W$ functions $$ W_{\kappa,\mu}\left(z\right)=\frac{z^{\mu+\frac{1}{2}}2^{-2\mu}}{\Gamma\left(% \frac{1}{2}+\mu-\kappa\right)}\int_{1}^{\...
Y.Okuyama's user avatar
  • 373
3 votes
0 answers
203 views

Identity for the product of two different associated Legendre polynomials

In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated: $$ \small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}...
Zurab Silagadze's user avatar
2 votes
1 answer
662 views

Clausen’s identity for associated Legendre polynomials

Clausen’s identity for Legendre polynomials has the form (see, for example, A generating function of the squares of Legendre polynomials, by Wadim Zudilin: https://arxiv.org/abs/1210.2493) $$P_n(\cos{\...
Zurab Silagadze's user avatar
2 votes
2 answers
467 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 ...
Steven's user avatar
  • 75
1 vote
1 answer
285 views

Why are the two ODE solutions linearly independent?

I notice that some second-order ODEs can be related to the triconfluent Heun's equation $$u''(z)-(3z^2+\gamma)u'(z)+(\alpha-(3-\beta) z)u(z)=0.$$ And people usually say the general solution of the ...
xiaohuamao's user avatar
1 vote
1 answer
196 views

Taylor expansion of Modified Mathieu functions

Do we know the Taylor expansion at $0$ of the radial Mathieu functions $(\mathsf{Mc}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 0}$ and $(\mathsf{Ms}_n^{(j)}(\,\cdot\,, \sqrt{q}))_{n \ge 1}$, for $q \in \...
Zoïs Moitier's user avatar
0 votes
1 answer
125 views

A slight generalization of triconfluent Heun equation: what is known?

I have recently come across an ODE of the form $$y''+(a+b x^2)y'+(c+dx+h/x^2)y=0 \hspace{30mm} (*)$$ where $y=y(x)$ and $a,b,c,d,h$ are arbitrary constants. As far as I understand (please correct ...
just-someone's user avatar