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Legendre polynomials are the most familiar orthogonal polynomials, and are 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 PDEs: Laplace's equation, the wave equation, etc. However, the Legendre functions in common use are of integer degree and order. In this talk, we explain why many Legendre functions and spherical harmonics of fractional degree and order are not really mysterious: they too are elementary functions. They can be used in solving PDEs in wedge geometries, and elsewhere. The many identities, such as recurrences, that relate these new functions will be explained.