Mixed Type Multiple Orthogonal Polynomials for Two Nikishin Systems

Abstract

We study the logarithmic and ratio asymptotics of linear forms constructed from a Nikishin system which satisfy orthogonality conditions with respect to a system of measures generated by a second Nikishin system. This construction combines type I and type II multiple orthogonal polynomials. The logarithmic asymptotics of the linear forms is expressed in terms of the extremal solution of an associated vector valued equilibrium problem for the logarithmic potential. The ratio asymptotics is described by means of a conformal representation of an appropriate Riemann surface of genus zero onto the extended complex plane.

A canonical family of multiple orthogonal polynomials for Nikishin systems

“…In [5], ratio asymptotic was proved (see [16] for the case of generating measures with mass points, and also [10,12]) under assumptions on the generating measures similar to the hypothesis in Rakhmanov's theorem. Let us state [5,Theorem 1.2] reduced to the case of Nikishin systems of two measures and the sequence I of multi-indices.…”

“…Substituting A 0 in terms of A 1 in the first equality of (15) and A 1 in terms of A 0 in the second equality, we see that A 1 and A 0 satisfy the same algebraic equations as F 1 and F 2 , respectively. Taking into account the degrees of the polynomials Q 2n+i , Q 2n+i,2 , from (11) it is easy to deduce that A 0 (∞) ∈ C \ {0} and A 1 (∞) = 0. Therefore, A 1 = F 1,1 and A 0 = F 2,1 .…”

“…Moreover, the zeros of Q n are simple and lie in (a 1 , b 1 ) (we say that n, |n| = n, is strongly normal). Regarding normality of general Nikishin systems, recently in [12] it has been established that they are strongly perfect, which means that all multi-indices are strongly normal (see also [7,9,11] …”

Multiple orthogonal polynomials associated with the exponential integral

“…Proof. The Mellin transform of the normalized type I function 𝐹 By working out the convolution formula for the type I function in (6), we may obtain explicit expressions for the underlying type I polynomials 𝐴 𝑛,𝑚 and 𝐵 𝑛,𝑚 . However, since this is rather tedious and not very insightful, we would not do it here.…”

“…Later, they were also studied in Refs. [5][6][7]. Other applications of multiple orthogonal polynomials can be found in analytic number theory (Diophantine approximation), for example, in Apéry's proof 8 of the irrationality of 𝜁(3) as explained by Beukers in Ref.…”