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The expansion of the two sides of a sum equality in terms of Polynomials in 
and 
, followed by closed form
summation in terms of 
and 
. For an example of the technique, see Bloom (1995).
References
Bloom, D. M. ``A Semi-Unfriendly Identity.'' Problem 10206. Solution by R. J. Chapman. Amer. Math. Monthly 102, 657-658, 1995.
Wilf, H. S. Generatingfunctionology, 2nd ed. New York: Academic Press, 1993.