In the present note, which is introductory to the following paper, closed expressions, suitable for computational purposes, are found for the sums of the series
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100022507/resource/name/S0305004100022507_eqnU001.gif?pub-status=live)
where α > 1, t = 1, 2, 3, …, and n is a positive integer. In each case a recurrent relation is found giving the values of
and
for t > 2 in terms of ![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20151103034928968-0531:S0305004100022507_inline3.gif?pub-status=live)
and the series Θκ(α) (κ = 1, 2, …, t), where
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100022507/resource/name/S0305004100022507_eqnU002.gif?pub-status=live)
When κ is even the last series is expressed in closed form in terms of the Bernoullian polynomial φκ(l/α) and, when κ is odd and α is rational, a closed form is found involving the polygamma function Ψ(κ)(z), where
The general expressions for
and
involve Ψ(z) and Ψ′(z) when α is rational, but for special values of α they reduce to a form independent of the Ψ-function.
and
are independent of n and are expressible as simple rational functions of α.