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.

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