Geodesic
A Fourier-Neumann series and its application to the reduction of triple cosine series
Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 198-201

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The Jacobi expansionis well known and easily obtained from the generating function of the Besselcoefficients. The sum of the series on the right of equation (1) when sin (n+1⁄2)x is replaced by cos (n+1⁄2)x cannot be found in this way but it can be expressed in terms of a definite integral as shown below. The result so obtained is useful in reducing certain triple cosine series to dual series and so simplifying the solution given by one of us for such series in an earlier paper [1]. 
Tranter, C. J.; Cooke, J. C. A Fourier-Neumann series and its application to the reduction of triple cosine series. Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 198-201. doi: 10.1017/S0017089500001968
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[1] 1.Tranter, C. J., Some triple trigonometrical series, Glasgow Math. J. 10 (1969), 121–125. Google Scholar | DOI

[2] 2.Magnus, W. and Oberhettinger, F. (translated by Wermer, J.), Special functions of mathematical physics (New York, 1949). Google Scholar

[3] 3.Watson, G. N., Theory of Bessel functions (Cambridge, 1944). Google Scholar

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