Sponsor:
The research of both authors was supported by research grant MTM 2006-13000-C03-02 of Ministerio de Ciencia e Innovación, Spain. I.A. Rocha also received support from Universidad Politécnica de Madrid through Grupo de Investigación TACA.

Review:
NonPeerReviewed

Keywords:
Hermite-Padé orthogonal polynomials
,
Multiple orthogonal polynomials
,
Nikishin system
,
Varying measures
,
Ratio asymptotics

For any pair of compact intervals Δ1 and Δ2 of the real line such that Δ1∩Δ2 = ø we obtain two pairs of absolutely continuous probability measures (μ1,μ2) and (τ1,τ2) supported on Δ1 and Δ2, respectively, such that:For any pair of compact intervals Δ1 and Δ2 of the real line such that Δ1∩Δ2 = ø we obtain two pairs of absolutely continuous probability measures (μ1,μ2) and (τ1,τ2) supported on Δ1 and Δ2, respectively, such that:[+][-]

- for appropriate constants C1 and C2, (μ1,μ2) is the Nikishin system generated by (μ1,C1τ1) and (τ1,τ2) the Nikishin system generated by (τ1,C2μ1),- for appropriate constants C1 and C2, (μ1,μ2) is the Nikishin system generated by (μ1,C1τ1) and (τ1,τ2) the Nikishin system generated by (τ1,C2μ1),[+][-]

- the polynomials of multiple orthogonality with respect to the Nikishin system
(μ1,μ2) and indices {..., (n,n), (n+1,n), ...} satisfy a recurrence relations with constant coe±cients of period 2,- the polynomials of multiple orthogonality with respect to the Nikishin system
(μ1,μ2) and indices {..., (n,n), (n+1,n), ...} satisfy a recurrence relations with constant coe±cients of period 2,[+][-]

- 1/hat-μ1(z) and 1/hat-μ2(z) are the functions which describe the ratio asymptotics of multiple orthogonal polynomials with respect to an arbitrary Nikishin system N(σ1,σ2) verifying supp(σi) = Δi, and σi' > 0, i = 1,2, almost everywhere on Δi. Analogously,
- 1/hat-μ1(z) and 1/hat-μ2(z) are the functions which describe the ratio asymptotics of multiple orthogonal polynomials with respect to an arbitrary Nikishin system N(σ1,σ2) verifying supp(σi) = Δi, and σi' > 0, i = 1,2, almost everywhere on Δi. Analogously,
1/hat-τ1(z) and 1/hat-τ2(z) give the ratio asymptotics for N(σ1,σ2).[+][-]