APPLICATIONES MATHEMATICAE 29,1 (2002), pp. 97–116 Stanislaw Lewanowicz (Wroclaw) RECURRENCES FOR THE COEFFICIENTS OF SERIES EXPANSIONS WITH RESPECT TO CLASSICAL ORTHOGONAL POLYNOMIALS Abstract. Let {P k } be any sequence of classical orthogonal polynomials. Further, let f be a function satisfying a linear differential equation with polynomial coefficients. We give an algorithm to construct, in a compact form, a recurrence relation satisfied by the coefficients a k in f = ∑ k a k P k . A systematic use of the basic properties (including some nonstandard ones) of the polynomials {P k } results in obtaining a low order of the recurrence. 1. Introduction. Let {P k (x)} be any system of classical orthogonal polynomials, i.e. associated with the names of Jacobi, Laguerre, Hermite or Bessel. Given a function f , a series expansion (1.1) f =  k a k [f ]P k is a matter of interest in numerical analysis, applied mathematics and math- ematical physics. Important special cases are connection and linearization problems, where f = P n and f = P m P n (m,n nonnegative integers), re- spectively, and {P k } is a family of polynomials (orthogonal or not). In par- ticular, positivity of the connection coefficients c k = a k [P n ], or the lineariza- tion coefficients l k = a k [P m P n ] is of great importance. See [1, 2, 4, 5, 9, 12, 13, 19–28, 33]. Usually, determination of the expansion coefficients a k [f ] requires a deep knowledge of special (hypergeometric) functions. See, e.g., [1, 2, 4, 9, 15, 18, 19, 25, 26]. It is important to note that, even in the case when it is possible 2000 Mathematics Subject Classification : 33C45, 42A16, 42C10, 42C15, 65Q05. Key words and phrases : classical orthogonal polynomials, Fourier coefficients, recur- rences. Supported by Komitet Bada´ n Naukowych (Poland) under the grant 2P03A02717. [97]