15.9.1 | |||
15.9.2 | ||||
15.9.3 | ||||
15.9.4 | |||
15.9.5 | ||||
15.9.6 | ||||
15.9.7 | |||
15.9.8 | |||
; | |||
compare also §15.2(ii).
15.9.9 | |||
15.9.10 | |||
This is a generalization of Jacobi polynomials (§18.3) and has the representation
15.9.11 | |||
The Jacobi transform is defined as
15.9.12 | |||
with inverse
15.9.13 | |||
where the contour of integration is located to the right of the poles of the gamma functions in the integrand, and
15.9.14 | |||
For this result, together with restrictions on the functions and , see Koornwinder (1984a).
This is a generalization of Gegenbauer (or ultraspherical) polynomials (§18.3). It is defined by:
15.9.15 | |||
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 14.21(iii).
The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers.
15.9.16 | ||||
, and . | ||||
15.9.17 | ||||
and . | ||||
15.9.18 | ||||
. | ||||
15.9.19 | ||||
and . | ||||
15.9.20 | ||||
. | ||||
15.9.21 | ||||
. | ||||
For the case see (14.3.1). |
15.9.22 | |||
, , | |||
where the sign in the exponential is according as .
15.9.23 | |||
, , | |||
where the sign in the exponential is according as .
15.9.24 | |||
15.9.25 | |||
15.9.26 | |||