Some extensions of the Lanczos-Ortiz theory of canonical polynomials in the Tau Method

Lanczos and Ortiz placed the canonical polynomials (c.p.’s) in a central position in the Tau Method. In addition, Ortiz devised a recursive process for determining c.p.’s consisting of a generating formula and a complementary algorithm coupled to the formula. In this paper a) We extend the theory so as to include in the formalism also the ordinary linear differential operators with polynomial coefficients D with negative height h = max n∈N {mn − n} < 0, where mn denotes the degree of Dxn. b) We establish a basic classification of the c.p.’s Qm(x) and their orders m ∈ M , as primary or derived , depending, respectively, on whether ∃n ∈ N : mn = m or such n does not exist; and we state a classification of the indices n ∈N, as generic (mn = n+h), singular (mn < n+h), and indefinite (Dxn ≡ 0). Then a formula which gives the set of primary orders is proved. c) In the rather frequent case in which all c.p.’s are primary, we establish, for differential operators D with any height h, a recurrency formula which generates bases of the polynomial space and their multiple c.p.’s arising from distinct xn, n ∈ N , so that no complementary algorithmic construction is needed; the (primary) c.p.’s so produced are classified as generic or singular , depending on the index n. d) We establish the general properties of the multiplicity relations of the primary c.p.’s and of their associated indices. It becomes clear that Ortiz’s formula generates, for h ≥ 0, the generic c.p.’s in terms of the singular and derived c.p.’s, while singular and derived c.p.’s and the multiples of distinct indices are constructed by the algorithm.