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.

Read more about this topic:  Hurwitz Zeta Function

Famous quotes containing the words taylor and/or series:

    “Every schoolboy knows it.”
    —Jeremy Taylor (1613–1667)

    “Rosalynn said, “Jimmy, if we could only get Prime Minister Begin and President Sadat up here on this mountain for a few days, I believe they might consider how they could prevent another war between their countries.” That gave me the idea, and a few weeks later, I invited both men to join me for a series of private talks. In September 1978, they both came to Camp David.”
    —Jimmy Carter (James Earl Carter, Jr.)