Mehler-Heine asymptotics for multiple orthogonal polynomials

Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard edge involves Bessel functions $J_\alpha$. We show that the asymptotic behavior near the endpoint of the interval of (one of) the measures for multiple orthogonal polynomials involves a generalization of the Bessel function. The multiple orthogonal polynomials considered are Jacobi-Angelesco polynomials, Jacobi-Pi\~neiro polynomials, multiple Laguerre polynomials, multiple orthogonal polynomials associated with modified Bessel functions (of the first and second kind), and multiple orthogonal polynomials associated with Meijer $G$-functions.Comment: 15 pages. Typos corrected, references updated, section "concluding remarks" adde

Majorization results for zeros of orthogonal polynomials

We show that the zeros of consecutive orthogonal polynomials $p_n$ and $p_{n-1}$ are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for the zeros of $p_n$ and the associated polynomial $p_{n-1}^{(1)}$ and for the zeros of the polynomial obtained by deleting the $k$th row and column $(1 \leq k \leq n)$ in the corresponding Jacobi matrix.Comment: 15 page

Compact Jacobi matrices: from Stieltjes to Krein and M(a,b)

In a note at the end of his paper {\it Recherches sur les fractions continues}, Stieltjes gave a necessary and sufficient condition when a continued fraction is represented by a meromorphic function. This result is related to the study of compact Jacobi matrices. We indicate how this notion was developped and used since Stieltjes, with special attention to the results by M. G. Krein. We also pay attention to the perturbation of a constant Jacobi matrix by a compact Jacobi matrix, work which basically started with Blumenthal in 1889 and which now is known as the theory for the class $M(a,b)$

Modularity and the Organization of International Production

In many globalized industries, vertical outsourcing seems to co-evolve with horizontal integration in the component sector. In order to account for this phenomenon, I incorporate modularity into an industry-equilibrium model with monopolistic competition and perfect contracts that allows the organization of the firm to be endogenous in both the vertical and horizontal dimensions of production. The model illustrates that the co-evolution is most likely to occur in industries with modular product architectures and high increasing returns to scale in the intermediate good sector. This paper also provides a theoretical legitimation of Stigler's contentious conjecture that firm production structures become vertically disintegrated as an industry expands.

Zero distribution of polynomials satisfying a differential-difference equation

In this paper we investigate the asymptotic distribution of the zeros of polynomials $P_{n}(x)$ satisfying a first order differential-difference equation. We give several examples of orthogonal and non-orthogonal families.Comment: 26 pages, 2 figure

Modularity and the Organization of International Production

In recent decades, complex manufacturing sectors such as electronics have transformed from an industry dominated by vertically integrated firms that source locally to an industry dominated by horizontally specialized firms that source globally. To account for this, we build an two-country industry-equilibrium model in which firms concurrently choose (i) a product architecture, (ii) an ownership structure and (iii) a location for production. In industries with partial input specificity and economies of scale in input production, we find that the industry transformation can be explained by a reduction in synergistic specificity, a reduction in the cost of internationalizing and an increase in industry demand. Au cours des dernières décennies, les secteurs manufacturiers complexes, tels que celui de l’électronique, se sont transformés, passant d’une industrie dominée par des firmes intégrées verticalement et s’approvisionnant localement, à une industrie dominée par des firmes spécialisées horizontalement et s’approvisionnant sur les marchés mondiaux. Pour expliquer ce phénomène, nous construisons un modèle d’équilibre industriel entre deux pays, dans lequel les entreprises choisissent simultanément (i) l’architecture du produit, (ii) la structure de propriété et (iii) le lieu de production. Dans les industries caractérisées par une spécificité partielle d’intrants et des économies d’échelle liées à la production de ceux-ci, nous constatons que la transformation qu’a connue l’industrie peut s’expliquer par une réduction de la spécificité synergique, une réduction du coût d’internationalisation et une augmentation de la demande au sein de l’industrie.input specificity, modularity, outsourcing, product architecture, architecture du produit, impartition, modularité, spécificité des intrants

Orthogonal polynomials and Laurent polynomials related to the Hahn-Exton q-Bessel function

Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are $q$-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent $q$-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orthogonal polynomials the orthogonality measure is determined using the three-term recurrence relation as a starting point. The relation between Chebyshev polynomials of the second kind and the Laurent $q$-Lommel polynomials and related functions is used to obtain estimates for the latter