Example of the transformation of singular series
Euler starts with a hypergeometric series F(a, b, c, x), and differentiates it to get a functional relation. Then he integrates to get another relation and ends up with something like Legendre polynomials.
Original Source Citation
Nova Acta Academiae Scientiarum Imperialis Petropolitanae, Volume 12, pp. 58-70.
Opera Omnia Citation
Series 1, Volume 16B, pp.41-55.