We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painlevé equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions which arise in the description of special function solutions of the fourth Painlevé equation.
We discuss the relationship between the recurrence coefficients of orthogonal
polynomials with respect to a generalized Freud weight
\[w(x;t)=|x|^{2\lambda+1}\exp\left(-x^4+tx^2\right),\qquad x\in\mathbb{R},\]
with parameters $\lambda>-1$ and $t\in\mathbb{R}$, 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.Comment: 22 pages, Studies in Applied Mathematics, accepted for publicatio
We prove results for the interlacing of zeros of Jacobi polynomials of the same or adjacent degree as one or both of the parameters are shifted continuously within a certain range. Numerical examples are given to illustrate situations where interlacing fails to occur.
We derive upper bounds for the smallest zero and lower bounds for the largest zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed three term recurrence relations satisfied by polynomials corresponding to different parameter(s) within the same classical family. We prove that interlacing properties of the zeros impose restrictions on the possible location of common zeros of the polynomials involved and deduce strict bounds for the extreme zeros of polynomials belonging to each of these three classical families. We show numerically that the bounds generated by our method improve known lower (upper) bounds for the largest (smallest) zeros of polynomials in these families, notably in the case of Jacobi and Gegenbauer polynomials.
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