Hurwitz Zeta Function - Taylor Series

Taylor Series

The derivative of the zeta in the second argument is a shift:

Thus, the Taylor series has the distinctly umbral form:

$\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!} \frac {\partial^k} {\partial x^k} \zeta (s,x) = \sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).$

Closely related is the Stark–Keiper formula:

$\zeta(s,N) = \sum_{k=0}^\infty \left {s+k-1 \choose s-1} (-1)^k \zeta (s+k,N)$

which holds for integer N and arbitrary s. See also Faulhaber's formula for a similar relation on finite sums of powers of integers.

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