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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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On the $p$-adic valuation of the sum of the first $n$ factorials

This is more a curiosity than anything useful. Consider $n>3$ and define $A(n)= 1!+2!+\cdots+n!$ It seems that if $p$ is the largest prime divisor of $A(n)$, then the $p$-adic valuation of $A(n)$ is 1 …
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