An introduction to second degree forms (Q1899279)

The author studies linear functionals \(u\) on the space of polynomials with complex coefficients in a real variable \(x\), for which the formal Stieltjes function defined from the moments \[ S(u)(z):= - \sum^\infty_{n= 0} {\langle u, x^n\rangle\over z^{n+ 1}} \] satisfies a quadratic equation of the form \[ B(z) S^2(u)(z)+ C(z) S(u)(z)+ D(z)= 0,\tag{1} \] for certain polynomials \((B, C)\), where \(D\) depends on \(B\), \(C\) and \[ B\neq 0;\;C^2- 4BD\neq 0;\;D\neq 0.\tag{2} \] The form \(u\) is then called a regular (2) second degree form (1). Through a chain of lemmas, the author derives a classification of these forms in case the pair \((B, C)\) is `primitive', i.e., it satisfies either \(B\) and \(C\) have no common root, or for any common root \(c\) we have \(\langle u, \theta_c C\rangle- \langle u^2, \theta_0\theta_c B\rangle\neq 0\). (Here \((\theta_c h)(x)= {h(x)- h(c)\over x- c}\) for complex \(c\) and \(h\) a polynomial; operations on forms are defined in the usual way reminiscent of `partial integration'.) The classification uses the degrees of the polynomials in the unique (!) primitive pair associated with the form. For \(\deg B= r\), \(\deg C= m\) there appear to be 5 classes: \(m\geq 1\), \(r= 0\); \(m\geq r+ 1\), \(r\geq 1\); \(m= r\), \(r\geq 1\); \(m= r- 1\), \(r\geq 2\); \(0\leq m< r- 1\), \(r\geq 2\). An example is given for certain instances of a Tchebycheff form (not all cases treated in full because of the increasing difficulties for growing degree) and the paper concludes with an equivalent classification in terms of the so-called `associated forms' \(u^{(n+ 1)}\).