337 research outputs found
Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle
Using the technique of the elliptic Frobenius determinant, we construct new
elliptic solutions of the -algorithm. These solutions can be interpreted as
elliptic solutions of the discrete-time Toda chain as well. As a by-product, we
obtain new explicit orthogonal and biorthogonal polynomials in terms of the
elliptic hypergeometric function . Their recurrence coefficients
are expressed in terms of the elliptic functions. In the degenerate case we
obtain the Krall-Jacobi polynomials and their biorthogonal analogs
Dual -1 Hahn polynomials: "classical" polynomials beyond the Leonard duality
We introduce the -1 dual Hahn polynomials through an appropriate
limit of the dual q-Hahn polynomials. These polynomials are orthogonal on a
finite set of discrete points on the real axis, but in contrast to the
classical orthogonal polynomials of the Askey scheme, the -1 dual Hahn
polynomials do not exhibit the Leonard duality property. Instead, these
polynomials satisfy a 4-th order difference eigenvalue equation and thus
possess a bispectrality property. The corresponding generalized Leonard pair
consists of two matrices each of size . In the eigenbasis
where the matrix is diagonal, the matrix is 3-diagonal; but in the
eigenbasis where the matrix is diagonal, the matrix is 5-diagonal.Comment: 12 pages, 14 reference
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