Some extensions of the Lanczos-Ortiz theory of canonical polynomials in the Tau Method
M. Bunchaft · Mathematics of Computation · 1997
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 D with negative height h = max n ∈ N { m n − n } > 0 , \begin{equation*}h=\underset {{n\in N}}{\max } \{m_{n}-n\}>0, \end{equation*} where m n m_{n} denotes the degree of D x n Dx^{n} . b) We establish a basic classification of the c.p.’s Q m ( x ) Q_{m}(x) and their orders m ∈ M m\in M , as primary or derived , depending, respectively, on whether ∃ n ∈ N : m n = m \exists n\in \mathbf {N}\colon m_{n}=m or such n n does not exist; and we state a classification of the indices n ∈ N n\in \mathbf {N} , as generic