For finding the generating function
of the sequence of the Legendre polynomials
we have to present as the general coefficient of Taylor series in , i.e. as the th derivative of some in the origin, divided by the factorial . The Cauchy integral formula offers the chance to implement that.
Starting from the Rodrigues formula of Legendre polynomials, we may write
where the contour runs anticlockwise once around the point . The change of variable
where must go round the origin clockwise, but in
anticlockwise. This is, by Cauchy integral formula again,
This means that
is the searched generating function of the Legendre polynomials:
Cf. the generating function of the Bessel functions.