**3**

votes

**1**answer

174 views

### An inequality involving Bessel functions of imaginary order

The following inequality: $$\frac{\pi k}{\sinh{(\pi k)}}\;|J_{ik}(\tau)|^2\le 1,\;\;\;k,\tau\ge 0,$$ for Bessel function $J_{ik}(\tau)$, I found in http://link.springer.com/article/10.1134%2F1.558677 ...

**1**

vote

**3**answers

373 views

### State of the Art in Approximating Fresnel Integrals

Background of my question is, that I need to calculate Clothoids and I found an AMS article "Chebyhev Approximations for Fresnel Integrals" by W.J. Cody from 1968 ...

**3**

votes

**1**answer

178 views

### Numerical Evaluation of Some Triple Integral involving Negative Powers

Let $\beta_i\in (-1/2,0)$, $i=1,2,3,4$. I'm interested in obtaining numerical value of the following integrals:
$$
\int_{0<u_1<u_2<u_3<1} (1-u_1)^{\beta_1}(1-u_2)^{\beta_2} ...

**3**

votes

**2**answers

256 views

### Recognize this sum

I have strong feeling that the above function $$f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)}$$ is a known special function but I can't seem to recognize it. Here $\Gamma(x)$ ...

**11**

votes

**2**answers

371 views

### 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
...

**4**

votes

**1**answer

163 views

### Homeomorphisms that admit a decomposition

Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$.
If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ ...

**1**

vote

**2**answers

175 views

### Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero?

I want to know under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where
${E_{\alpha ,1}}(z) = \sum\limits_{k = 0}^\infty ...

**1**

vote

**2**answers

220 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 ...

**1**

vote

**1**answer

105 views

### Inversion of incomplete elliptic integral of third kind

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)?
That means that given $\Pi(n,u,m) = f(x)$, I would like to obtain ...

**2**

votes

**0**answers

78 views

### Jacobi triple product for multidimensional lattices

The Jacobi triple product identity gives as a special case a product formula for the theta function of a 1-dimensional lattice. Is there a more general product formula for the theta function of an ...

**9**

votes

**1**answer

880 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 $a=1/4$.
...

**7**

votes

**1**answer

178 views

### Inequality for Laguerre polynomials

Let $L_n$ be the $n$-th Laguerre polynomial defined by
$\quad
L_n
(x)=\frac{e^x}{n!}\frac{d^n}{dx^n}(x^n e^{-x}).\quad
$
I want to prove that
$$
\forall n\in \mathbb N,\forall x\ge 0,\quad \sum_{0\le ...

**1**

vote

**2**answers

421 views

### Is there a class of functions closed against differentiation besides elementary? [closed]

Is there a finite set $P$ of non-elementary functions $f_n$ such that the derivative of any function $f$ from that set is not elementary, but expressible with functions from the same set $P$ plus ...

**1**

vote

**1**answer

168 views

### The asymptotic behavior of hypergeometric function around -1

Recently, in studing some specific orthogonal polynomials on unit circle, I was lead to study the asymptotic behavior of the following hypergeometric function at the neighberhood of $-1$:
$$ ...

**3**

votes

**1**answer

353 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 ...

**7**

votes

**4**answers

234 views

### Is there an oscillating analog of the Gaussian distribution?

It frequently happens that, in some famillies of polynomials with positive coefficients, the coefficients of large polynomials look like a bell curve and tend to the distribution function of the ...

**4**

votes

**2**answers

481 views

### What is known about this power series?

In the course of some calculations, I came across the following powers series.
For fixed $C>1$ let
$$
f_C(u)=\sum_{k=0}^\infty\frac{u^k}{C^{k^2}}.
$$
This series converges for all $u\in\mathbb C$, ...

**6**

votes

**1**answer

248 views

### Under which constraints are there only finite numbers of irreducible eta product identities?

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 $e_k:=\eta(q^k)$.
An eta product identity (or eta identity for ...

**15**

votes

**2**answers

698 views

### Is there a known solution to $f(x) = (1-x)f(x^2)$?

The functional equation $f(x) = (1-x)f(x^2)$ (with $f(0)=1$) has a simple solution that can be expressed as a rapidly converging infinite product
$$f(x) = \prod_{n=0}^\infty (1 - x^{2^n}) = ...

**1**

vote

**1**answer

87 views

### What function is “$U_{\nu}(\cdot, \cdot)$”?

I was searching in the Prudnikov (vol. 2) how to solve an integral and I finally found it. However, I didn't recognized a function that appears in the answer.
Integral 1.8.2.4:
$$
\int_0^x x^{\nu+1} ...

**1**

vote

**2**answers

137 views

### Quantifying simplicity, in the case of trigonometric and exponential functions

The pair of identities the sine and cosine of a sum of two terms as functions of the sines and cosines of the terms separately is not as simple as the identity that expresses the exponential of a sum ...

**3**

votes

**0**answers

182 views

### Solvable parametric $7$th and $13$th degree equations using $\eta(q)/\eta(q^p)$?

Q: Why is that some polynomial relations between eta quotients have a solvable Galois group, even if the deg is $n>4$?
For example, we have the well-known modular equation,
$$u^6 - v^6 + ...

**36**

votes

**1**answer

2k views

### There's something strange about $\sqrt{d\big(j(\tau)-1728\big)}$

Given the j-function $j(\tau)$, I was looking at,
$$F(\tau) = \sqrt{d\big(j(\tau)-1728\big)}$$
which appears in Ramanujan-type pi formulas. Let $C_d$ be the prime factors of the constant term of the ...

**3**

votes

**1**answer

144 views

### On Continuous Replicative Functions

I asked this question on math.stackexchange here, but it did not receive much attention. Thus, I was suggested to post it here.
Knuth, in The Art of Computer Programming Vol. 1, defines a replicative ...

**2**

votes

**0**answers

66 views

### Associated Legendre/Gegenbauer functions with complex degree at larger order

I am interested in approximating the associated Legendre function (also known as conical function)
\begin{equation}
P_{-1/2 + i p}^{\frac{2-N}{2}}(x)
\end{equation} when $N \to \infty$. The real ...

**1**

vote

**3**answers

205 views

### A definite integral of hypergeometric function 2F1

I am wondering whether there exists a closed form for the definite integral
$$F(x)=\int_0^1t^{-a}(1-t)^{N}(1-xt)^{-a}{}_2F_1(-a,k-a-1/2,k-a;4xt(1-xt))dt,$$
where $a\in(0,1)$ and $N,k$ are positive ...

**1**

vote

**1**answer

166 views

### Accurate bounds for derivatives of Legendre polynomials

Let $P_n(x)$ denote the $n$th Legendre polynomial. What bounds can one give for $d_{n,m}(x) = |\frac{d^m}{dt^m}P_n(t)|_{t=x}$ assuming that $|x| \le 1$? Clearly
$$d_{n,m}(x) \le d_{n,m}(1) = ...

**1**

vote

**0**answers

87 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

89 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

145 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

446 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

277 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}} ...

**1**

vote

**0**answers

168 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

60 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

166 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

134 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

313 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

64 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

323 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

161 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 ...

**6**

votes

**1**answer

217 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

280 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

113 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

254 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 ...

**2**

votes

**0**answers

115 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 = ...

**5**

votes

**0**answers

216 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

219 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

155 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

179 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

240 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 ...