General linearization formulae for products of continuous hypergeometric-type polynomials

The linearization of products of wavefunctions of exactly solvable potentia ls often reduces to the generalized linearization problem for hypergeometri c polynomials (HPs) of a continuous variable, which consists of the expansi on of the product of two arbitrary HPs in series of an orthogonal HP set. H ere, this problem is algebraically solved directly in terms of the coeffici ents of the second-order differential equations satisfied by the involved p olynomials. General expressions for the expansion coefficients are given in integral form, and they are applied to derive the connection formulae rela ting the three classical families of hypergeometric polynomials orthogonal on the real axis (Hermite, Laguerre and Jacobi), as well as several general ized linearization formulae involving these families. The connection and li nearization coefficients are generally expressed as finite sums of terminat ing hypergeometric functions, which often reduce to a single function of th e same type; when possible, these functions are evaluated in closed form. I n some cases, sign properties of the coefficients such as positivity or non -negativity conditions are derived as a by-product from their resulting exp licit representations.