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On the summation of certain trigonometric series

Published online by Cambridge University Press:  24 October 2008

L. S. Goddard
L. S. Goddard
Affiliation:
St John's CollegeCambridge
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 41 , Issue 2 , June 1945 , pp. 145 - 160
Copyright
Copyright © Cambridge Philosophical Society 1945

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References

REFERENCES

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