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Connection between the polylogarithm and the Bernoulli polynomials.
I have been studying the polylogarithm function and came across its relation with Bernoulli polynomials, as Wikipedia site asserts:
For positive integer polylogarithm orders $s$, the Hurwitz zeta ...
- 812
4
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How to derive this polylogarithm identity (involving Bernoulli polynomials)?
How can one derive the following identity, found here, relating the polylogarithm functions to Bernoulli polynomials?
$$\operatorname{Li}_n(z)+(-1)^n\operatorname{Li}_n(1/z)=-\frac{(2\pi i)^n}{n!}B_n\!...
- 6,672
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Why is $B_{2n}(\frac12+ix)\in\mathbb R$ whenever $x\in\mathbb R$?
I just noticed that $B_{2n}(\frac12+ix)\in\mathbb R$, where:
$x\in\mathbb R$, $n\in\mathbb N$, and $B_n(x)$ is the $n$th Bernoulli Polynomial.
Why? Is there a simple, slick proof? Does it follow from ...
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