**1**

vote

**0**answers

95 views

### Quadratic transformation of hypergeometric function 2F1

I want to know whether there is some transformation between
$_2F_1(a,b;c;x)$ and $_2F_1(a',b';c';x(1-x))$.
Here is an example called the Kummer quadratic transformation, which may be known to most of ...

**2**

votes

**0**answers

102 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

**0**answers

156 views

### Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as
\begin{equation}
p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~,
...

**5**

votes

**1**answer

530 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 ...

**2**

votes

**1**answer

385 views

### About large z behavior of hypergeometric function $_2F_1(1/2,1/2,1;z)$

The hypergeometric function $_2F_1(\large \frac{1}{2},\frac{1}{2},1;\frac{1-\frac{u}{\Lambda^2}} {2} \large)$ at large $\mid u\mid$ can be approximated by
$$ -\frac{\Lambda}{\pi} \sqrt{\frac{2}{u}} ...

**0**

votes

**0**answers

205 views

### Simplify this expression with modified Bessel functions of the second kind

I'm interested in the function
$$\frac{K^{(0,2)}(0,x)}{K(0,x)}$$
where the numerator is the modified Bessel function of the second kind twice differentiated with respect to $\alpha$ and taken at ...

**1**

vote

**0**answers

69 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

171 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 ...

**2**

votes

**0**answers

150 views

### A generalization of Macdonald functions?

I am interested in finding a set of functions $f(z_1,\cdots ,z_k;q,\,t)$, conjecturally polynomials, which depend on two parameters $(q,t)$ and an integer $k$, and are orthogonal under the following ...

**6**

votes

**0**answers

334 views

### Why does the Rogers-Ramanujan continued fraction $R(q)$ appear in Emma Lehmer's quintic?

Define the Ramanujan theta function $f(a,b)$ as,
$$f(a,b) = \sum_{n=-\infty}^{\infty} a^{n(n+1)/2}\,b^{n(n-1)/2}$$
and the Dedekind eta function,
$$\eta(\tau) = q^{1/24}\prod_{n=1}^{\infty} ...

**2**

votes

**0**answers

69 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}})} = ...

**7**

votes

**0**answers

391 views

### Inequality between incomplete beta and gamma functions; or when is binomial distribution function above/below its limiting Poisson

Please note: this question was posted first (September 4) in math.stackeschange.com and then (September 16) in stats.stackeschange.com. It got no answers in neither of those sites.
Let the ...

**2**

votes

**2**answers

173 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 ...

**5**

votes

**1**answer

223 views

### Logarithm of the hypergeometric function

For $F(x)={}_2F_1 (a,b;c;x)$, with $c=a+b$, $a>0$, $b>0$, it has been proved in [1] that $\log F(x)$ is convex on $(0,1)$.
I numerically checked that with a variety of $a,\ b$ values, $\log ...

**1**

vote

**0**answers

306 views

### complex contour integral calculation after Möbius transformation

Good day to everyone.
In my scientific research I've got stuck with a contour integration problem.
I would like to evaluate the following integral:
$$I=\int_0^{\infty } \frac{e^{\frac{\alpha -\mathrm ...

**1**

vote

**1**answer

117 views

### $q$-differential equation for the Rodgers polynomials?

The Rodgers polynomials $C_{\alpha,q}$ are a particular family of well-known $q$-hypergeometric function. For example, a description can be found here on Wikipedia. For the special case of $q=1$, we ...

**4**

votes

**0**answers

263 views

### Why does the plastic constant appear in the snub icosidodecadodecahedron?

The golden ratio,
$$\phi =\frac{1+\sqrt{5}}{2}$$
appears (among other polyhedra) in the Platonic solids icosahedron and dodecahedron, and it's quite easy to see the significance of the discriminant ...

**3**

votes

**0**answers

126 views

### A second polylogarithm ladder for the tribonacci and n-nacci constants

In "A seventeenth-order polylogarithm ladder", on page 6 (eq 25), Bailey et al give a dilogarithmic ladder that begins with,
$$0 = ...

**6**

votes

**0**answers

235 views

### Identification of a curious function

The following question was asked on math.stackexchange, but there were no replies.
During computation of some Shapley values (details below), I encountered the following function:
$$
f\left(\sum_{k ...

**6**

votes

**1**answer

245 views

### Conjectured bound on Kummer's function (confluent hypergeometric function)

I believe the following bound to be correct:
${}_1F_1(-a;1+a;-z) a z^{-a} \gamma(a,z) \leq 1$
for real-valued a > 0 and z $\geq 0$. $\gamma(a,z)$ is the lower incomplete gamma function.
Apart from ...

**2**

votes

**1**answer

173 views

### A definite integral related to hypergeometric function

I obtained the following integral when looking for a probability density function:
$$\int_0^1 x^{\alpha-1} \,(1-x) ^{-A}\, {}_2F_1 (1-A, \alpha -1-A, \alpha -A, x) \,dx$$
Can anyone please give me ...

**2**

votes

**1**answer

189 views

### Limit involving regularlized gamma function and its inverse

Let
$$L(x)=Q\left(\frac{x}{2},\frac{a}{a+f(x)/\sqrt{x}}Q^{-1}\left(\frac{x}{2},1-b^{1/g(x)}\right)\right)$$
where $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}$ is the upper incomplete gamma function ...

**4**

votes

**1**answer

244 views

### How to get an expression for this integral(Numerically/Analytically)

I have the following problem:
I need to evaluate the integral $$\int_{\cos(\alpha)}^{1} P_l(t)P_{l'}(t) dt $$ for $\alpha \in [0,\pi]$ and each combination of $l$ and $l'$, where $P_l$ is the l-th ...

**2**

votes

**2**answers

212 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 ...

**1**

vote

**0**answers

194 views

### Can the series $\sum\limits_{n=0}^\infty q^{F_n}$ be expressed in terms of theta functions?

Let $F_0=0,F_1=1,...$ be the Fibonacci numbers. Is there a known closed form for the sum $\sum\limits_{n=0}^\infty q^{F_n}$? By closed form, I mean in terms of well-known functions, the first ones to ...

**5**

votes

**1**answer

197 views

### Function transformation of exponentials

I came across the following function transformation:
$$
\sum_{j=-\infty}^{\infty} e^{(-j^2\cdot t)} = \sqrt{\frac{\pi}{t}} \cdot \sum_{j=-\infty}^{\infty} e^{(-\frac{\pi^2}{t}\cdot j^2)}
$$
where $ j ...

**2**

votes

**2**answers

422 views

### Summation of certain series

Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then ...

**1**

vote

**1**answer

269 views

### Asymptotic expansion of an integral with singularity, Hydrodynamic free-surface green function

I am frustrated by the asymptotic expansion of the free surface green function when epsilon (or H for I2)tends to zero and with a singularity K(K is a constant). Can anyone help me derive the formula ...

**8**

votes

**1**answer

471 views

### How to prove Lambert's W function is not elementary?

Liouville's theorem gives such a proof for antiderivatives of functions like $e^x/x$ or $e^{x^2}$, and differential Galois thory extends that to Bessel functions, say. But what tools exist for ...

**11**

votes

**1**answer

412 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) - ...

**5**

votes

**0**answers

324 views

### The Riemann Zeta Function summing over the Gamma Function

Has anyone studied a function of the form
$$\eta(s) = \sum_{n=1}^{\infty} \frac{1}{\Gamma(n)^{s}} = \sum_{n=0}^{\infty}\frac{1}{k!^s}$$
This series is appearing in my research on the volumetric ...

**0**

votes

**1**answer

285 views

### Integrating the complete elliptic integral K

I've run into the following integral:
$\int \frac{K(k)}{k} dk$
where $K$ is the complete elliptic integral of the first kind
$K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin\theta}}$.
I've ...

**3**

votes

**1**answer

235 views

### Extending the Shimura Lift to Non-Cuspidal Classical Modular Forms of Higher Level

The definition of the Shimura lift of a classical cusp form is well documented. Zagier and Kohnen define a modified version of the lift for a cusp form $g(z)=\sum a(n)q^n \in S_{k+1/2}^{+}(4)$ in the ...

**1**

vote

**1**answer

168 views

### hyperbolic functions and Gauss hypergeometric series

If the Gauss hypergeometric function
$F(1, 3/2, 5/2; z^2) = 3 [\tanh^{-1} (z) –z]/z^3$
what is the corresponding result for $F(1,15/8,23/8;z^8) $?

**2**

votes

**1**answer

130 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 ...

**2**

votes

**0**answers

209 views

### An integral with Gamma functions (Part 2)

I was wondering if there is a generalization of the integral discussed here to a case like,
\begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ...

**19**

votes

**3**answers

656 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

139 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

298 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

228 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!}
...

**9**

votes

**2**answers

991 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 ...

**2**

votes

**1**answer

584 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

166 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

477 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

228 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

395 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

145 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

145 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

492 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

309 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 ...