On a generalization of Schur theorem concerning resultants
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On a generalization of Schur theorem concerning resultants Maciej Ulas1 Accepted: 20 June 2020 © The Author(s) 2020
Abstract Let K be a field and put A := {(i, j, k, m) ∈ N4 : i ≤ j and m ≤ k}. For any given A ∈ A we consider the sequence of polynomials (r A,n (x))n∈N defined by the recurrence r A,n (x) = f n (x)r A,n−1 (x) − vn x m r A,n−2 (x), n ≥ 2, where the initial polynomials r A,0 , r A,1 ∈ K [x] are of degree i, j respectively and f n ∈ K [x], n ≥ 2, is of degree k with variable coefficients. The aim of the paper is to prove the formula for the resultant Res(r A,n (x), r A,n−1 (x)). Our result is an extension of the classical Schur formula which is obtained for A = (0, 1, 1, 0). As an application we get the formula for the resultant Res(r A,n , r A,n−2 ), where the sequence (r A,n )n∈N is the sequence of orthogonal polynomials corresponding to a moment functional which is symmetric. Keywords Resultant · Recurrence relations · Orthogonal polynomials Mathematics Subject Classification Primary: 12E10 · 12E05; Secondary: 13P05
1 Introduction Let N denotes the set of non-negative integers, N+ the set of positive integers and for given k ∈ N+ we write N≥k for the set of positive integers ≥ k. Let K be a field and consider the polynomials F, G ∈ K [x]. The resultant Res(F, G) of the polynomials F, G is an element of K which gives the information of possible common roots. More precisely, Res(F, G) = 0 if and only if the polynomials F, G has a common factor of positive degree. The computation of resultants is, in general, a difficult task. Of special interest is the computation of resultants of pairs of polynomials which are interesting from either a number theoretic or analytic point of view. The classical result is the computation of resultant of two cyclotomic polynomials m , n . More precisely, Apostol proved the formula p ϕ(n) if mn is a power of a prime p, Res(m , n ) = 1 otherwise,
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Maciej Ulas [email protected] Faculty of Mathematics and Computer Science, Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
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M. Ulas
where ϕ is the Euler phi function [1]. On the other side, we have a result of Schur which allow computation of resultants of consecutive terms in the sequence (rn (x))n∈N of the polynomials defined by a linear recurrence of degree two. More precisely, if r0 (x) = 1, r1 (x) = a1 x + b1 and we define rn (x) = (an x + bn )rn−1 (x) − cn rn−2 (x), n ≥ 2, with an , bn , cn ∈ C satisfying an cn = 0. Under these assumptions, we have the following compact formula proved by Schur [9] (see also [10, p. 143]): Res(rn , rn−1 ) = (−1)
n(n−1) 2
n−1
2(n−i) i ci+1 .
ai
i=1
In nfactm Schur obtained a slightly different result, i.e., he obtained the expression for i=1 rn−1 (x i,n ), where x i,n is the ith root of the polynomial rn . The importance of the Schur method lies in its applications in the computation of discriminants of orthogonal polynomials. Indeed, Favard proved that each family of orthogonal polynomials corresponds with the sequenc
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