Legendre Functions and Spherical Harmonics with Fractional Parameters
Legendre polynomials are orthogonal polynomials used in approximation theory. Their generalizations, called Legendre functions, are parametrized by an order as well as a degree, and are typically not polynomials. But they too can be used in series approximations. They appear most often in spherical harmonics, which are used in modal decomposition of functions on the sphere, and in series and closed-form solutions of ODEs and PDEs. But the Legendre functions most often used are of integer degree and order. In this talk, we explain why Legendre functions and spherical harmonics of fractional degree and order are not mysterious: they too can be used in constructing solutions, and can be elementary functions. Recurrences and other identities that relate these new functions will be revealed.