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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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Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$...

None of these functions have real zeroes, because they can be written as moment generating functions of certain random variables $Y_\alpha$. More precisely, for $0<\alpha <1$ $$E_{\alpha,1}(z)=\mathb …
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5 votes

(Sharp) inequality for Beta function

One can also use Jensen's inequality. Let (for $\sigma>0$) $G_\sigma$ denote a random variable with $\Gamma(1,\sigma)$-distribution, i.e. having Lebesgue density $$f_\sigma(t)=\frac{t^{\sigma-1}}{\G …
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Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a C...

The asymptotic distribution of the number of nodes at maximal height in a random tree is known. The following was known as "Wilf's conjecture" (H. S. Wilf stated it in 1991, evidently unaware of the …
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