# Tagged Questions

Many special functions appear as solutions of differential equations or integrals of elementary functions. Most special functions have relationships with representation theory of Lie groups.

51 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 (Abramowitz&...
73 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, $P_0(q) = P_1(q) = 1$...
85 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 ...
74 views

571 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 ...
213 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/, ...
340 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 satisfy....
172 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 ...
77 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 ...
127 views

160 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 $\phi(x+1)=x\phi(x)$...
139 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) \...
199 views

### Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$ ...
106 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 ...
132 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 ...