For the notation see §§18.3 and 18.19.
compare also §15.2(ii).
This is a generalization of Jacobi polynomials (§18.3) and has
The Jacobi transform is defined as
where the contour of integration is located to the right of the poles of the
gamma functions in the integrand, and
For this result, together with restrictions on the functions f(t) and
f~(λ), see Koornwinder (1984a).
This is a generalization of Gegenbauer (or ultraspherical) polynomials
(§18.3). It is defined by:
Any hypergeometric function for which a quadratic transformation exists can be
expressed in terms of associated Legendre functions or Ferrers functions. For
examples see §§14.3(i)–14.3(iii) and
The following formulas apply with principal branches of the hypergeometric
functions, associated Legendre functions, and fractional powers.
where the sign in the exponential is ± according as