Duality of q-polynomials, orthogonal on countable sets of points

Details Duality of q-polynomials, orthogonal on countable sets of points Duality of q-polynomials, orthogonal on countable sets of points

2004-11-11

arxiv.org/abs/math/0411249v2

Mathematics

Classical Analysis and ODEs

71 pages, corrected typos, added references

Scientific paper

We review properties of q-orthogonal polynomials, related to their orthogonality, duality and connection with the theory of symmetric (self-adjoint) operators, represented by a Jacobi matrix. In particular, we show how one can naturally interpret the duality of families of q-polynomials, orthogonal on countable sets of points. In order to obtain orthogonality relations for dual sets of polynomials, it is proposed to use two symmetric (self-adjoint) operators, representable (in some distinct bases) by Jacobi matrices. This approach is applied to several pairs of dual families of q-polynomials, orthogonal on countable sets, from the q-Askey scheme. For each such pair, the corresponding operators, representable by Jacobi matrices, are explicitly given. These operators are employed in order to find explicitly sets, on which the polynomials are orthogonal, and orthogonality relations for them.

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