Closed form evaluation of some reciprocal trigonometric sums via partial fraction decomposition

Talha Arıkan, Helmut Prodinger

Abstract


Recently, Melham computed some nite sums in which the denominator of the
summand includes products of `sine' or `cosine'. In this paper, generalizations of the sums,
which he studied in 2016, are presented, by allowing arbitrary factors in the denominator of
the summand. Note more than the elementary technique of partial fraction decomposition
method is used. Furthermore, some of the sums, which he studied in 2017, are treated in
the same style.


Keywords


partial fraction decomposition; reciprocal sum identities; trigonometric functions.

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References


Chu, W. (2007) Partial fraction decompositions and trigonometric sum identities. Proc. Amer. Math. Soc. 136(1):229--237.

Kılıç, E. & Prodinger, H. (2017) Closed form evaluation of Melham's reciprocal sums. Miskolc Math. Notes 8:251--264.

Melham, R.S. (2016) Closed forms for nite sums in which the denominator of the summand is a product of trigonometric functions. Fibonacci Quart. 54(3):196--203.

Melham, R.S. (2017) Finite reciprocal sums of products involving squares of sines or cosines with arguments in arithmetic progression. Integers 17(A33):1--12.


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