# 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.

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### Linear eta product identities - how many are there?

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 $e_k:=\eta(q^k)$. With this notation, a blog entry of Michael ...
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### 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 ...
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### Combination of Generating Functions

Suppose I have the following generating functions: $$\frac{x^ke^{\left(z-\frac{1}{N}\right)x}}{N^{k-1}k!\sum_{j=0}^{N-1}w_N^{-jk}e^{\frac{w_N^jx}{N}}}=\sum_{j=0}^\infty H_{N,k,j}(z)\frac{x^j}{j!}$$ ...
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### 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 p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~, \end{...
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### 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 ...
In "A seventeenth-order polylogarithm ladder", on page 6 (eq 25), Bailey et al give a dilogarithmic ladder that begins with, $$0 = \operatorname{Li_2(\alpha_1^{-630})}-2\operatorname{Li_2(\alpha_1^{-... 0answers 146 views ### Are numbers h_{r,s} = \sum_{k} P(r;s;k) \frac{1}{n^{2k}} \bigg(1-\frac{1}{n}\bigg)^{n-2k} irrational? I asked this question on MSE and Mike Spivey gave an insightful answer. I decided to put it here nevertheless in case someone else gets interested. If this violates rules on MO, please let me know, I'... 0answers 125 views ### Error Function limes How can i calculate \prod_{n=1}^{\infty}{erf(n)}  with erf(z) = \frac{2}{\sqrt{\pi}}\int_0^z e^{-z^{2}} \mathrm{d}z? I know it's something like 0,84. And i see that only the first terms are ... 0answers 200 views ### Can Bernoulli polynomials be extended to fractional orders without losing elementarity? Can Bernoulli polynomials B_s(x) be extended to fractional s in such a way so that for any given s the function B_s(x) still could be expressed in elementary functions of x? 0answers 793 views ### Can one represent a generalized hypergeometric function 1F2 as a product of two confluent hypergeometric functions? I am trying to somewhat simplify a series, whose coefficients feature generalised hypergeometric functions {}_1F_2(1;a,a+1;z). I was unable to find useful functional relations for this specific ... 0answers 827 views ### Proof that derivative of Hurwitz Zeta by the first argument is not expressable in terms of Hurwitz Zeta The set of elementary functions is defined so that it to be closed against operation of differentiation. It is also evidently close against discrete differentiation. In the discrete calculus there is ... 0answers 783 views ### Cubic polynomials with “nice” roots, which can be expressed by trig functions of rational angles Consider the cubic polynomial x^3-ax+b for a,b\in\mathbb N. It has three real roots which, by Cardano's formula, can of course be written in closed form using thirds of angles or cube roots of ... 0answers 541 views ### Mathematica package for obtaining hypergeometric function In my current research in electromagnetics I am encountering integrals of the form$$ \int_0^\infty dt J_0( r t) \frac{\exp(-h \sqrt{t^2 - a^2})}{\sqrt{t^2 - b^2}} t .  $a$ and $b$ are complex ...
If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...