Orthogonal Polynomials: Connection and Linearization Coefficients A. Ronveaux

Let fP n (x)g 1 n=0 and fQ m (x)g 1 m=0 be two families of orthogonal polynomials. The linearization problem involves only one family via the relation: P i (x) P j (x) = i+j X k=ji?jj L ijk P k (x) and the connection problem mixes both families: P n (x) = n X m=0 C m (n) Q m (x): In many cases, it is possible to build a recurrence relation involving only m , satissed by the linearization coeecients L ijk and connection coeecients C m (n) , in particular if the families P n (x) and Q m (x) are classical: continuous or discrete. We intend to examine these coee-cients from the point of view of recurrence properties, emphasizing mainly the structure of these recurrence relations. This article contains some results already published, in print or in preparation, Let fP n (x)g 1 n=0 and fQ m (x)g 1 m=0 be two sequences of polynomials of degree exactly equal to n. The so-called Connection Problem asks to nd the coeecients C m (n) in the expansion: P n (x) = n X m=0 C m (n) Q m (x) ; (1) and goes back to Stirling 35] as pointed out by Askey in his 1975 survey 4]. Let us represent the polynomials P n (x) and Q m (x) in the monomial basis fx k g 1 k=0 , and analyze the structure of the connection coeecients C m (n) as depending on several assumptions on families P n (x) and Q m (x)

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