Some connection and linearization problems for polynomials in and beyond the Askey scheme (Q5946628)

The main aim of the present work is the study of the so called ``linearization'' and ``connection'' problems, i.e., the expansion of the product of two polynomials (not necessary orthogonal) \(h_n\) and \(q_m\) in terms of an orthogonal hypergeometric polynomial sequence \(\{p_k\}_k\) NEWLINE\[NEWLINE r_n(x) q_m(x) =\sum_{j=0}^{n+m} c_{n,m,r} p_j(x). NEWLINE\]NEWLINE In the paper the authors are interested in finding closed analytical formulas for the linearization coefficients \(c_{n,m,r}\), as well as for the connection coefficients which correspond to the case \(n\) or \(m\) equal 0 (i.e., \(r_n(x)=1\), or \(q_m(x)=1\)). The main tool used is the \textit{J. L. Fields} and \textit{J. Wimp} result on expansions of hypergeometric functions in generalized hypergeometric functions [Math. Comput. 15, 390-395 (1961; Zbl 0107.05902)]. Using this formulas the authors solve several connection problems involving the Wilson and Racah polynomials (Askey tableau) as well as several families of polynomials beyond the Askey tableau (e.g. Celine polynomials, Sobolev-type, etc). Also some special case of linearization formulas (when \(r_n=q_m\)) are considered.