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

Abstract:

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 \begin{equation*}h=\underset {{n\in N}}{\max } \{m_{n}-n\}<0, \end{equation*} where $m_{n}$ denotes the degree of $Dx^{n}$. b) We establish a basic classification of the c.p.’s $Q_{m}(x)$ and their orders $m\in M$, as primary or derived, depending, respectively, on whether $\exists n\in \mathbf {N}\colon m_{n}=m$ or such $n$ does not exist; and we state a classification of the indices $n\in \mathbf {N}$, as generic $(m_{n}=n+h)$, singular $(m_{n}<n+h)$, and indefinite $(Dx^{n}\equiv 0)$. 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 $x^{n}$, $n\in N$, 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 $h\ge 0$, 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.