Solving connection and linearization problems within the Askey scheme and its q-analogue via inversion formulas

For the polynomial families {Pn(x)}n belonging to the Askey scheme or to its q-analogue, the hypergeometric representation provides a natural expansion of the form Pn(x)=∑m=0nDm(n)θm(x), where the expanding basis θm(x) is, in general, a product of Pochhammer symbols or q-shifted factorials. In this paper we solve the corresponding inversion problem, i.e. we compute the coefficients Im(n) in the expansion θn(x)=∑m=0nIm(n)Pm(x), which are then used as a tool for solving any connection and linearization problem within the Askey scheme and its q-analogue. Extensions of this approach for polynomials outside these two schemes are also given.