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Abstract:

The goal of this thesis is to study the nearest neighbor recurrence rela tions for multiple orthogonal polynomials, which are a generalization of the concept of orthogonal polynomials. They satisfy higher order recurr ence relations. Most of the people work with the so-called step-line rec urrence relation, however it is not necessarily the most natural one cou ld consider. It is also one of the objectives of the thesis to convince the reader that the nearest neighbor recurrence relations are the more i nteresting object in this context. We believe that many important properties of orthogonal polynomials can be generalized using the nearest neighbor recurrence relation as the pro per analog of the three term recurrence relation. As an example one coul d point out the Christoffel-Darboux type formula for multiple orthogonal polynomials, which can be obtained in a similar way as in the classical case. Another example is the interlacing property, which can be derived from the recurrence relation, just as in the case of orthogonal polynom ials. Since this system of recurrences was less studied in the past compared t o the step-line recurrence relation, we investigate the relation between these two main recurrences for multiple orthogonal polynomials and also the relation with the classical orthogonal polynomials. We should not talk about these objects without giving a few concrete exa mples. We introduce some multiple orthogonal polynomials and compare the recurrence coefficients of the step-line, nearest-neighbor recurrence r elations as well as the coefficients of the classical orthogonal polynom ials.