**5**

votes

**0**answers

205 views

### Recurrence Formula for Zernike polynomials

I'm not sure if this is research level, so if this result is known, please excuse the intrusion. I am trying to find a relation between solutions of the Laplacian equation in $4$ dimensions and those ...

**5**

votes

**1**answer

83 views

### A special function solution of a fourth-order ODE

I want to consider the solutions of the following fourth-order ODE:
$$
f^{(4)}(t)+a tf^{(1)}(t)+b f(t)=0,
\tag{$\ast$}$$
where $a,b$ are complex parameters. It turns out that with a Fourier ...

**2**

votes

**0**answers

21 views

### Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let $n\ge 1$ be an integer, let
$$F(x,y)=\int_0^\infty u^{n(x+y)} (K_{x-y}(u))^n du$$
for $x,y\ge 0$.
When $n=1$, this is just Mellin transform of the Bessel K function. When $n=2$, $F(x,y)$ has ...

**5**

votes

**1**answer

109 views

### On a Sum of Gamma Functions

I am working on a problem where the following sum appears:
$$F(s, t)=\frac{1}{\Gamma(1+2\alpha)}\sum_{n=0}^{\infty}{\frac{s^{n} ...

**1**

vote

**0**answers

104 views

### What is the connection between the Riemann Xi-function and n-sphere? [closed]

Riemann's Xi-function is defined as
$$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$
At the same time we have the following formulas for n-sphere's area and volume:
...

**14**

votes

**1**answer

310 views

### Why does $\sum_{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)/\prod_{k=1,k\neq p}^n\sin\left(\frac{\pi (k-p)}{2m}\right)$ vanish?

While trying to prove some identities for generating functions, I ended up needing to show that
$$\sum_{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)\prod_{\substack{k=1\\k\neq ...

**2**

votes

**0**answers

48 views

### Challenging problems concerning Jacobian elliptic functions with complex modulus

I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since ...

**1**

vote

**1**answer

91 views

### An extreme of Jacobi elliptic function on an interval

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...

**10**

votes

**0**answers

230 views

### Inverse Mellin of the exponential of the digamma function

I'm looking for a function $f_p(x)$ with real parameter $p>0$ satisfying
$$ \int_0^\infty f_p(x)x^{s-1}dx=e^{-p\psi(s)} $$
where $\psi(s)$ is the usual digamma function. The inverse Mellin formula ...

**6**

votes

**1**answer

116 views

### Asymptotic behaviour of an integral

For $k\in\mathbb{N}_{0}$ and $x\in\mathbb{R}$, define
$$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$
where
...

**3**

votes

**0**answers

39 views

### Root Polylogarithm Dominance Questions

Motivation: I am trying to work on a problem related to computing the roots of a certain family of polynomials related to integer partition theory. In particular, I have been trying to ``Bridge the ...

**4**

votes

**0**answers

45 views

### Jacobi elliptic functions with modulus on the unit circle

I am gathering some available informations on Jacobi elliptic functions $sn(z,k)$, $cn(z,k)$, $dn(z,k)$ with $k\in\mathbb{C}$, $|k|=1$. I can not find much on them in standard references ...

**4**

votes

**0**answers

61 views

### Is there a nice way to invert this expression?

Let us first define the Euler polynomials to be the polynomials $P_n(q)$ that satisfy
$$
\frac{qP_n(q)}{(1 - q)^{n+1}} = \Big(q\frac{d}{dq}\Big)^n\frac{q}{1 - q}.
$$
For example, $P_0(q) = P_1(q) = ...

**6**

votes

**1**answer

73 views

### Modern comprehensive account of the Barnes G-Function

I previously put this forward on the math.stackexchange community with little luck:
I am looking for a comprehensive account of the properties and applications of the Barnes G-Function. Everything ...

**3**

votes

**0**answers

60 views

### Resource needed on Lerch's transcendent

I am looking for resources in english which discuss basic properties of the Lerch's transcendent function.
The Lerch Transcendant is defined by:
...

**10**

votes

**1**answer

459 views

### A question on Ramanujan's $1/\pi$ formulas

It is known that Ramanujan discovered a number of formulas for $1/\pi$. All of these formulas are of the form $$\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(1/2)_n(s)_n(1-s)_n}{(1)_n^3}(a+bn)z^n,$$where ...

**1**

vote

**0**answers

70 views

### Proving injectivity of a multivariable function

Let $I$ denote the interval $(0,\infty)$, we define the function $f:I^2\to I^2$ by,
$$f(x,y)=\left({\Gamma(4x+y)\Gamma(y)\over {\Gamma(2x+y)}^2},{\Gamma(4x+y)\Gamma(2x+y)\over ...

**3**

votes

**1**answer

107 views

### Closed Form Expression for Nested Series Summation?

Just wandering if there are any criteria that can decide whether a finite series summation has closed form or not. for example, In the following nested summation, $n$ is some even integer that will be ...

**2**

votes

**0**answers

33 views

### Rearrangement of summation expression

Referring to the spherical harmonics expansion in this article:
Méléard, P., Pott, T., Bouvrais, H., & Ipsen, J. H. (2011). Advantages of statistical analysis of giant vesicle flickering for ...

**1**

vote

**0**answers

57 views

### Rearrangement of a spherical harmonics expansion

Referring to this article:
http://i.stack.imgur.com/sfQ1C.png
and
http://i.stack.imgur.com/LelKb.png
How is it that they get from equation 2 to equation 3?
Whenever I do it, I get only cosine ...

**1**

vote

**0**answers

175 views

### system with solutions $\{x-a:0\leqslant a\leqslant z-1\}$ [closed]

What must be $F$ there where $0=F(1,x,0)=F(x-0,x,z)=F(x-1,x,z)=F(x-2,x,z)=F(x-3,x,z)=$ $\dots$ $=f(x-z-1,x,z)=0$?
Define $F$ in the domain where a continuous function exists that behaves so for ...

**5**

votes

**1**answer

420 views

### Roots of characteristic function of “reciprocal gamma measure”

Let us call a measure $\mu$ on the Borel $\sigma$-algebra $\mathfrak{B}_{(0,\infty)}$ of subsets of $(0,\infty)$ a reciprocal gamma measure if it is absolutely continuous with respect to the Lebesgue ...

**5**

votes

**4**answers

181 views

### Integrals involving the Tricomi hypergeometric function

I am looking for a reference for the two following equalities involving the Tricomi function $U$ and the Meijer function $G$. I have found these formulas on the website http://functions.wolfram.com/, ...

**-1**

votes

**2**answers

329 views

### What conditions imply that a function over $\mathbb{Z}$ is a polynomial? [closed]

How would one prove that a function is a polynomial? I can't seem to find anything about this on the internet. I would like to know if there are any unique properties that only polynomials can ...

**2**

votes

**0**answers

155 views

### Conway's box function iterated to produce a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...

**0**

votes

**0**answers

72 views

### Energy Oscillations in a One Dimensional Crystal

Good day!
Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)?
article, that I have
Especially ...

**1**

vote

**0**answers

110 views

### Four kinds of generalized hypergeometric formulas for $\pi$

Given,
$$\begin{array}{|c|c|c|c|}
\hline
n&a_n&b_n&c_n\\
\hline
1 & 6541681608 & 163096908 & -640320^3\\
\hline
2 & 85840 & 4492 & -14112^2\\
\hline
3 & 28302 ...

**5**

votes

**2**answers

277 views

### What is a “generalized zeta function”?

Out of procrastination I computed $$\sum_{k=1}^\infty k^{-k^2}\sim 1.06255080549625593786944593879.$$
The inverse symbolic calculator identified this number as "From generalized Zeta function". I do ...

**6**

votes

**2**answers

559 views

### A (likely) positivity property of the Lerch zeta-function

The problem is to show that $\Re L(b/2,1/2,p+1)>0$ for all real $b\ne0$ and all real $p>-1$, where
$$L(\lambda,c,s):=\sum_{k=0}^\infty\frac{\exp(2\pi i\lambda k)}{(k+c)^s}$$
is the Lerch ...

**3**

votes

**0**answers

62 views

### how understand periodicity in a combination of power, gamma and zeta functions?

Riemann's functional equation may be written:
$$
\frac{\zeta(s)}{\zeta(1-s)} = 2^s \pi^{s-1} \sin(\frac{\pi s}2) \Gamma(1-s) \tag{1}
$$
and so by symmetry:
$$
\frac{\zeta(1-s)}{\zeta(s)} = 2^{1-s} ...

**-2**

votes

**2**answers

154 views

### On Bohr-MollerupTheorem [closed]

In http://mathworld.wolfram.com/Bohr-MollerupTheorem.html, Bohr-Mollerup Theorem is given where it is stated that $\Gamma$ function is the unique log convex function that satisfies ...

**6**

votes

**1**answer

117 views

### An identity of complicated combinations of gamma functions (related to hypergeometric functions)

Can somebody help me in proving the following equation?
\begin{align*}&\textstyle \sum _{d=0} ^{n} \frac{1}{d!(n-d)!} \frac{\Gamma (b+d) \Gamma (b+n-d) \Gamma (c-n+d) \Gamma (c-b+1-n + 2d) ...

**6**

votes

**0**answers

189 views

### Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define
$$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$
...

**0**

votes

**1**answer

88 views

### Upper bound for a ratio of modified Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...

**5**

votes

**0**answers

125 views

### Elementary function relative to erf

The modified Bessel function of the 1st kind $I_0$ is defined by
$$
I_0(z)=\frac1\pi\int_0^{2\pi}e^{z\cos\theta}\,d\theta
$$
and arises, among other places, in the probability density function of a ...

**1**

vote

**0**answers

223 views

### What is the status on questions related to Bhargava's factorial function?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like:
For $k, l \in \mathbb{Z}$, we have $k! \times ...

**1**

vote

**0**answers

97 views

### What is $\int (1-e^{-x})^n dx$? [closed]

For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is
$(-1)^n B(e^x; -n, n+1)$
I ...

**1**

vote

**1**answer

153 views

### How to prove that $(1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$) for $x\ll 1$ in this specific case

After multiple plots I noticed that function $h(x)= (1-x)^b$ $_2F_1(a,b;c;x)$ can be approximated to $1-\alpha x$ (with $\alpha \approx 1$), for $x\ll 1$ (specifically $0<x<0.1$) and ...

**1**

vote

**2**answers

135 views

### Is there any simpler form of this function

Assume that $n$ is a positive integer. Is there any simple form of this hypergeometric value $$_2\mathrm{F}_1\left[\frac{1}{2},1,\frac{3+n}{2},-1\right]?$$

**1**

vote

**0**answers

105 views

### A hypergeometric identity [closed]

Is there any simple proof of this identity
$${_4F_3}[\{\frac{1}{2}+\frac{n}{4},1+\frac{n}{4},-p,p\},\{\frac{1}{2},\frac{3}{2},\frac{1}{2}+\frac{n}{2}\},1]=\frac{\Gamma[\frac{1+n}{2}] ...

**1**

vote

**1**answer

113 views

### Approximation of $ _2F_1((b-1)a,b;ba;x) $

Is there any (simple) approximation of this Hypergeometric function: $ _2F_1((b-1)a,b;ba;x) $, where $0<x<1$ and $b>a>1$.
Thanks!

**1**

vote

**0**answers

78 views

### q-Hermite polynomials

It is well known that the q-Hermite polynomials defined by $$H_n(\theta; q)= \sum\limits_{k=0}^n \frac{(q;q)_n}{(q;q)_k(q;q)_{n-k}}e^{i(n-2k)\theta}$$
are orthogonal in $\theta \in [0, \pi]$ with ...

**2**

votes

**2**answers

198 views

### How to calculate one Cauchy type determinant

As we know, a Cauchy determinant of size n admits the following explicit formula:
$$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y ...

**4**

votes

**1**answer

631 views

### How to prove this identity on double summation series?

I suspect the following identity is valid, but I can not prove it. I just calculate it numerically.
...

**0**

votes

**1**answer

129 views

### Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared:
$$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$
where $\rho,s>1$, ...

**5**

votes

**0**answers

206 views

### Legendre polynomials and formal groups

Let $P_n(x)$ be Legendre polynomials:
$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$
Usual arguments from the theory of formal groups allow to
prove that for any $n$
...

**3**

votes

**0**answers

155 views

### Analysing functions on zero-length intervals and super-small values

Suppose a function that has a pole in $x=0$:
Here we see the graphic of the real part of the Gamma function.
As we can see on it, there is a vertical line at $x=0$ that comes from $-\infty$ to ...

**5**

votes

**0**answers

127 views

### Are these identities Newton series?

Newton series is the following expansion of a function:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$
Now ...

**2**

votes

**0**answers

174 views

### Integral involving a Meijer-G function

I am having trouble with calculating the following integral:
$$
\int_{0}^{\infty} \ln{(1 + \alpha x)\, G^{k,0}_{k,k}\left[e^{-x}\left|^{(a_k)}_{(b_k)} \right. \right]} \, dx,
$$
where $\alpha > ...

**2**

votes

**1**answer

152 views

### Eigenfunctions of an infinite summation operator

I would like to find ALL eigenfunctions to the operator, for $f$ a real function on R+*:
$f \rightarrow \sum_{1}^{\infty} f(nx)$
So to find $f$ such that: $\sum_{1}^{\infty} f(nx) = \lambda f(x)$
...