Orthogonal polynomials with exponential weight in a finite interval and application to the optical model

A quadrature procedure is developed which makes the construction of momentum-space meson-nucleus optical potentials more accurate. We deal with numerical evaluation of integrals with finite t-integration range which contain exp(Dt) explicitly, where D is a parameter. The Gaussian rule is used with abscissas determined as roots of orthogonal polynomials with exponential weight function in the interval [−1,1]. Recurrence relations and inequalities for these polynomials are obtained. A nonlinear recursion is derived, which permits the evaluation of abscissas and weights without accumulation of roundoff error. The nonlinear recursion is solved by means of an iterative procedure, the convergence properties of which are established. The quadrature procedure is summarized as an easily implementable algorithm. The rate of convergence is demonstrated for several test integrals.

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