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

For the polynomial families {P-n(x)}(n) belonging to the Askey scheme or to its q-analogue, the hypergeometric representation provides a natural expansion of the form P-n(x) = Sigma D-n(m=0)m(n)theta (m)(x), where the expanding basis theta (m)(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 I-m(n) in the expansion theta (n)(x) = Sigma I-n(m=0)m(n)P-m(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. (C) 2001 Elsevier Science B.V. All rights reserved.