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 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 and the series Θκ(α) (κ = 1, 2, …, t), where 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.
[1]
S. Schelkunoff.
Proposed symbols for the modified cosine and exponential integrals
,
1944
.
[2]
L. Goddard,et al.
A problem in the summation of series
,
1943,
Mathematical Proceedings of the Cambridge Philosophical Society.
[3]
D. B. Scott.
Invariant groups associated with an algebraic surface
,
1940,
Mathematical Proceedings of the Cambridge Philosophical Society.
[4]
J. H. PEARCE,et al.
Functions of a Complex Variable
,
1947,
Nature.
[5]
Eleanor Pairman,et al.
Tables of the digamma and trigamma functions
,
2022
.
[6]
G. M..
An Introduction to the Theory of Infinite Series
,
1908,
Nature.