Solvability of the Hankel determinant problem for real sequences

TY - CHAP

T1 - Solvability of the Hankel determinant problem for real sequences

AU - Bakan, Andrew

AU - Berg, Christian

PY - 2018

Y1 - 2018

N2 - To each nonzero sequence s := {sn}n≥0 of real numbers, we associate the Hankel determinants Dn = detHn of the Hankel matrices Hn := (si+j)n i,j=0, n ≥ 0, and the nonempty set Ns := {n ≥ 1 Dn-1 ≠ 0}. We also define the Hankel determinant polynomials P0 := 1, and Pn, n ≥ 1 as the determinant of the Hankel matrix Hn modified by replacing the last row by the monomials 1, x, xn. Clearly Pn is a polynomial of degree at most n and of degree n if and only if n ε Ns. Kronecker established in 1881 that if Ns is finite then rank Hn = r for each n ≥ r - 1, where r := max Ns. By using an approach suggested by Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence {tn}n≥0 to be of the form tn = Dn, n ≥ 0 for a real sequence {sn}n≥0. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial Pn satisfying deg Pn = n ≥ 1 is preceded by a nonzero polynomial Pn-1 whose degree can be strictly less than n - 1 and which has no common zeros with Pn. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that D0 > 0, Dr-1 > 0 and Dn = 0 for all n ≥ r.

AB - To each nonzero sequence s := {sn}n≥0 of real numbers, we associate the Hankel determinants Dn = detHn of the Hankel matrices Hn := (si+j)n i,j=0, n ≥ 0, and the nonempty set Ns := {n ≥ 1 Dn-1 ≠ 0}. We also define the Hankel determinant polynomials P0 := 1, and Pn, n ≥ 1 as the determinant of the Hankel matrix Hn modified by replacing the last row by the monomials 1, x, xn. Clearly Pn is a polynomial of degree at most n and of degree n if and only if n ε Ns. Kronecker established in 1881 that if Ns is finite then rank Hn = r for each n ≥ r - 1, where r := max Ns. By using an approach suggested by Iohvidov in 1969 we give a short proof of this result and a transparent proof of the conditions on a real sequence {tn}n≥0 to be of the form tn = Dn, n ≥ 0 for a real sequence {sn}n≥0. This is the Hankel determinant problem. We derive from the Kronecker identities that each Hankel determinant polynomial Pn satisfying deg Pn = n ≥ 1 is preceded by a nonzero polynomial Pn-1 whose degree can be strictly less than n - 1 and which has no common zeros with Pn. As an application of our results we obtain a new proof of a recent theorem by Berg and Szwarc about positive semidefiniteness of all Hankel matrices provided that D0 > 0, Dr-1 > 0 and Dn = 0 for all n ≥ r.

KW - Frobenius rule

KW - Hankel matrices

KW - Kronecker theorem

KW - Orthogonal polynomials

UR - http://www.scopus.com/inward/record.url?scp=85045712122&partnerID=8YFLogxK

U2 - 10.1142/9789813228887_0005

DO - 10.1142/9789813228887_0005

M3 - Book chapter

AN - SCOPUS:85045712122

SN - 9789813228870

T3 - Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes

SP - 85

EP - 117

BT - Frontiers In Orthogonal Polynomials and Q-series

A2 - Nashed, M Zuhair

A2 - Li, Xin

PB - World Scientific

ER -