The Generating Function for the Legendre Polynomials

We derive the formula for the generating function of the Legendre polynomials. Starting with the Bonnet Recursion Formula ( • Bonnet's Recursion For... ), we write down a Maclaurin series whose coefficients are the Legendre polynomials. Shifting the series by one term and using the Bonnet recursion formula, we arrive at an integral equation for the series. Differentiating that equation yields the equality of two logarithmic derivatives, which in turn yields the formula for the generating function for the Legendre polynomials. This generating function is extremely important in the study of electrostatic and gravitational potentials as well as the study of spherical harmonics.
#mikethemathematician, #mikedabkowski, #profdabkowski

[@OnlyGodalone]

Great job

[@mikethemathematician]

@J_H2961 Thanks so much! I like to present alternative ways (classical) of constructing the generating function of orthogonal polynomials!