and Applied Analysis 3 real and there is a need to locate their position. Moreover, the usual methods M1 – M3 mentioned before for the study of the zeros of Pn x may not apply at all, when Pn x are complex, or they may need serious modifications. Instead, the M4 method can be used directly. Such a functional analytic methodwas introduced in 10 andwas successfully used in a series of papers by t...

We give expansions of reproducing kernels the Christoffel-Darboux type in terms Schur polynomials. For this, we use evaluations averages characteristic polynomials and random matrix ensembles. explicitly compute new averages, such as average a $q$-Laguerre ensemble, ensuing kernels. In addition to classical $q$-deformed cases on real line, extensions Dotsenko-Fateev integrals obtain expressions...

The discrete orthogonality relations hold for all the orthogonal polynomials obeying three term recurrence relations. We show that they also multi-indexed Laguerre and Jacobi polynomials, which are new obtained by deforming these classical polynomials. could be considered as a more encompassing characterization of than As start at positive degree ℓD≥1, broken. extra ℓD “lower polynomials,” nece...

The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, ...