On a generalization of Schur theorem concerning resultants
Let \(K\) be a field, \(\mathbb{N}\) the set of non-negative integers, and \(\mathcal{A} := \{(i, j, k, m) \in \mathbb{N}^4: i \le j\) and \(m \le k\}\). For each \(A \in \mathcal{A}\), the paper under review considers the sequence \(r_{A,n}(x))_{n \in \mathbb{N}}\) of polynomials defined by the recurrence \[ r_{A,n}(x) = f_n(x)r_{A,n-1}(x) - v_nx^mr_{A,n-2}(x), \quad n \ge 2, \] where the initial polynomials \(r_{A,0}\), \(r_{A,1} \in K[x]\) are of degree \(i, j\), respectively, and \(f_n(x) \in K[x]\) is of degree \(n\) with variable coefficients, for any \(n \ge 2\). The paper under review finds a formula for the resultant \(\operatorname{Res}(r_{A,n}(x), r_{A,n-1}(x))\), which yields in the special case of \(A = (0, 1, 1, 0)\)) the classical Schur formula. The significance of the obtained formula is due to the fact (established by Favard) that each family of orthogonal polynomials corresponds to the sequence \((r_{A,n}(x))_{n \in \mathbb{N}}\), for suitably chosen sequences \((a_n)_{n \in \mathbb{N}}\), \((b_n)_{n \in \mathbb{N}}\) and \((c_n)_{n \in \mathbb{N}}\) (for a proof of Favard's theorem, see Theorem 4.4 of: \textit{T. S. Chihara} [An introduction to orthogonal polynomials. New York etc.: Gordon and Breach Science Publishers (1978; Zbl 0389.33008)]). As an application of the main result of the paper under review, the author gives the formula for the resultant \(\operatorname{Res}(r_{A,n}, r_{A,n-2})\), where \((r_{A,n}(x))_{n \in \mathbb{N}}\) is the sequence of orthogonal polynomials corresponding to a moment functional which is symmetric, i.e. its moments of odd order are zero. This condition is satisfied by the Legendre, Hermite, Chebyshev, Bessel, Lommel and many other sequences of orthogonal polynomials (see Chapter V of the above cited book). The author presents three illustrative examples which highlight the latter formula for the sequence of Legendre polynomials and the one of Hermite polynomials as well as for a certain sequence of polynomials with combinatorial coefficients, which is not a sequence of orthogonal polynomials.
scientific article; zbMATH DE number 7377566