Abstract:
We discuss the relationship between the recurrence coefficients of orthogonal
polynomials with respect to a generalized Freud weight
w(x; t ) = |x|2λ+1 exp(−x4 + tx2), x ∈ R,
with parameters λ > −1 and t ∈ R, and classical solutions of the fourth
Painlev´e equation. We show that the coefficients in these recurrence relations
can be expressed in terms of Wronskians of parabolic cylinder functions that
arise in the description of special function solutions of the fourth Painlev´e
equation. Further we derive a second-order linear ordinary differential
equation and a differential-difference equation satisfied by the generalized
Freud polynomials.