Orthogonal polynomials with discontinuous weights

Details Orthogonal polynomials with discontinuous weights Orthogonal polynomials with discontinuous weights

2005-01-22

arxiv.org/abs/math-ph/0501057v1

J. Phys. A: Math. Gen. 38 (12), L191--L198 (2005)

Physics

Mathematical Physics

9 pages, 2 figures, JPA style

Scientific paper

10.1088/0305-4470/38/12/L01

In this paper we present a brief description of a ladder operator formalism applied to orthogonal polynomials with discontinuous weights. The two coefficient functions, A_n(z) and B_n(z), appearing in the ladder operators satisfy the two fundamental compatibility conditions previously derived for smooth weights. If the weight is a product of an absolutely continuous reference weight w_0 and a standard jump function, then A_n(z) and B_n(z) have apparent simple poles at these jumps. We exemplify the approach by taking w_0 to be the Hermite weight. For this simpler case we derive, without using the compatibility conditions, a pair of difference equations satisfied by the diagonal and off-diagonal recurrence coefficients for a fixed location of the jump. We also derive a pair of Toda evolution equations for the recurrence coefficients which, when combined with the difference equations, yields a particular Painleve IV.

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