  In the first part, by the first author's work of 1972, an integral
representation for an ultraspherical polynomial of higher index in terms of one
of lower index and an infinite series was obtained. While this representation
works well from a theoretical point of view, it is not numerically satisfactory
as it involves polynomials of high degree, which are numerically unstable. Here
we sum this series to obtain an integral, which is numerically tractable.
