Explicit formulas for the associated Jacobi polynomials and some applications.

*(English)*Zbl 0643.33009The associated Jacobi polynomials are the polynomials satisfying the same recurrence relation as the Jacobi polynomials, but with n replaced by $n+c$, for some fixed $c\ge 0$. In this well written paper, the author provides an explicit expression for the associated Jacobi polynomials, involving ${}_{4}{F}_{3}$ hypergeometric functions. He also determines explicitly the weight function with respect to which these polynomials are orthogonal, under mild restrictions on c and the Jacobi parameters $\alpha $, $\beta $.

Finally, he provides a generating function, and derives a representation for the $n-1$, $n$ Padé approximant to

$$f\left(z\right):=F\left(\begin{array}{c}a+1,b+1\\ c+1\end{array};z\right)/F\left(\begin{array}{c}a,b\\ c\end{array};z\right)\xb7$$

Reviewer: D.S.Lubinsky