Partial Fraction Decompositions and Trigonometric Sum Identities

The partial fraction decomposition method is explored to establish several interesting trigonometric function identities, which may have applications to the evaluation of classical multiple hypergeometric series, trigonometric approximation and interpolation. 1. Outline and introduction Recently, in an attempt to prove, through the Cauchy residue method, Dougall’s theorem (Dougall [6, 1907], see [5] also) on the well-poised bilateral hypergeometric 5H5-series, the author found the following trigonometric sum identity: sin a sin b sin c sin d sin e = cot b sin(a− 2b) sin(c− b) sin(d− b) sin(e− b) (1a) + cot c sin(a− 2c) sin(b− c) sin(d− c) sin(e− c) (1b) + cot d sin(a− 2d) sin(b− d) sin(c− d) sin(e− d) (1c) + cot e sin(a− 2e) sin(b− e) sin(c− e) sin(d− e) . (1d) Consider the trigonometric fraction defined by R(z) := e cot z sin(a+ 2z) sin(b+ z) sin(c+ z) sin(d+ z) sin(e+ z) . According to the Euler formulae cos z = e + e−iz 2 and sin z = e − e−iz 2i , this rational function is essentially a fraction in e with the degree of the numerator polynomial less than that of the denominator polynomial by one. Therefore we can Received by the editors October 25, 2006. 2000 Mathematics Subject Classification. Primary 42A15; Secondary 65T40.