Multiple orthogonal polynomials associated with an exponential cubic weight. (English) Zbl 1309.42037
The authors introduce two families of multiple orthogonal polynomials associated with an exponential cubic weight over contours in the complex plane. As main results, the Rodrigues formula and nearest-neighbour recurrence relations are found. The coefficients of the latter are connected with two sequences \(a_n\) and \(b_n\), which are related to the coefficients of the non-multiple orthogonal polynomials with the same weight and contours (presented also in this article). The asymptotic behaviour of these sequences \(a_n\) and \(b_n\) is also studied.
In the diagonal case (both indices of the multiple orthogonal polynomial with equal values) the zeros of these functions are placed over the contour of orthogonality. The authors describe the asymptotic distribution of the zeros in this case, as well as the asymptotic behaviour of the ratio of two consecutive diagonal polynomials as the degree tends to infinity.
In the diagonal case (both indices of the multiple orthogonal polynomial with equal values) the zeros of these functions are placed over the contour of orthogonality. The authors describe the asymptotic distribution of the zeros in this case, as well as the asymptotic behaviour of the ratio of two consecutive diagonal polynomials as the degree tends to infinity.
Reviewer: Pablo Sánchez-Moreno (Granada)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
Keywords:
multiple orthogonal polynomials; exponential cubic weight; Rodrigues formula; nearest-neighbor recurrence relations; string equations; discrete Painleve equation; zeros; asymptoticsReferences:
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