Rational Zeta Series - Polygamma-related Series

Polygamma-related Series

A number of additional relationships can be derived from the Taylor series for the polygamma function at z = 1, which is

\psi^{(m)}(z+1)= \sum_{k=0}^\infty (-1)^{m+k+1} (m+k)!\; \zeta (m+k+1)\; \frac {z^k}{k!}.

The above converges for |z| < 1. A special case is

\sum_{n=2}^\infty t^n \left = -t\left

which holds for |t| < 2. Here, ψ is the digamma function and ψ(m) is the polygamma function. Many series involving the binomial coefficient may be derived:

\sum_{k=0}^\infty {k+\nu+1 \choose k} \left = \zeta(\nu+2)

where ν is a complex number. The above follows from the series expansion for the Hurwitz zeta

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

taken at y = −1. Similar series may be obtained by simple algebra:

\sum_{k=0}^\infty {k+\nu+1 \choose k+1} \left = 1

and

\sum_{k=0}^\infty (-1)^k {k+\nu+1 \choose k+1} \left = 2^{-(\nu+1)}

and

\sum_{k=0}^\infty (-1)^k {k+\nu+1 \choose k+2} \left = \nu \left - 2^{-\nu}

and

\sum_{k=0}^\infty (-1)^k {k+\nu+1 \choose k} \left = \zeta(\nu+2)-1 - 2^{-(\nu+2)}

For integer n ≥ 0, the series

can be written as the finite sum

The above follows from the simple recursion relation Sn + Sn + 1 = ζ(n + 2). Next, the series

may be written as

for integer n ≥ 1. The above follows from the identity Tn + Tn + 1 = Sn. This process may be applied recursively to obtain finite series for general expressions of the form

for positive integers m.

Read more about this topic:  Rational Zeta Series

Famous quotes containing the word series:

    “Every Age has its own peculiar faith.... Any attempt to translate into facts the mission of one Age with the machinery of another, can only end in an indefinite series of abortive efforts. Defeated by the utter want of proportion between the means and the end, such attempts might produce martyrs, but never lead to victory.”
    —Giuseppe Mazzini (1805–1872)