In this chapter we introduce some classes of special functions that play important roles in applied mathematics, physics, etc. Most of these functions are polynomials, and they appear as solutions to various differential equations. The analysis of these special functions is intimately connected with the main theme of the book: in fact, for each of the considered classes of differential equations, the associated polynomial solutions form an orthonormal basis for a related L 2-space. The study of the differential equations and their solutions can easily cover an entire book. We do not aim at a complete description with full proofs, but will focus on the relationship with the theory we have developed for series expansions in Hilbert spaces.
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