Let be analytic within an ellipse with foci , and
18.18.1 | |||
Then
18.18.2 | |||
when lies in the interior of . Moreover, the series (18.18.2) converges uniformly on any compact domain within .
Alternatively, assume is real and continuous and is piecewise continuous on . Assume also the integrals and converge. Then (18.18.2), with replaced by , applies when ; moreover, the convergence is uniform on any compact interval within .
This is the case of Jacobi. Equation (18.18.1) becomes
18.18.3 | |||
Assume is real and continuous and is piecewise continuous on . Assume also converges. Then
18.18.4 | |||
, | |||
where
18.18.5 | |||
The convergence of the series (18.18.4) is uniform on any compact interval in .
Assume is real and continuous and is piecewise continuous on . Assume also converges. Then
18.18.6 | |||
, | |||
where
18.18.7 | |||
The convergence of the series (18.18.6) is uniform on any compact interval in .
In all three cases of Jacobi, Laguerre and Hermite, if is on the corresponding interval with respect to the corresponding weight function and if are given by (18.18.1), (18.18.5), (18.18.7), respectively, then the respective series expansions (18.18.2), (18.18.4), (18.18.6) are valid with the sums converging in sense. See Szegő (1975, Theorems 3.1.5 and 5.7.1). See also (18.2.24), (18.2.25).
18.18.10 | |||
18.18.11 | |||
18.18.12 | |||
18.18.13 | |||
18.18.14 | ||||
18.18.15 | ||||
18.18.18 | ||||
18.18.19 | ||||
18.18.20 | |||
18.18.21 | |||
18.18.22 | |||
18.18.23 | |||
The coefficients in the expansions (18.18.22) and (18.18.23) are positive, provided that in the former case . See (18.17.45) and (18.17.49) for integrated forms of (18.18.22) and (18.18.23), respectively. See Rahman (1981) for the linearization formula for Jacobi polynomials and Zeng (1992) for the linearization coefficients for Laguerre polynomials.
With
18.18.24 | |||
18.18.25 | |||
18.18.26 | |||
See (18.2.41) for the Poisson kernel in case of general OP’s.
In this subsection the variables and are not confined to the closures of the intervals of orthogonality; compare §18.2(i).
18.18.29 | |||
18.18.30 | |||
18.18.31 | ||||
18.18.32 | ||||
18.18.33 | ||||
18.18.34 | ||||
18.18.35 | ||||
18.18.36 | |||
18.18.37 | |||
18.18.38 | |||
18.18.39 | |||
18.18.40 | |||
See also (18.38.3) for a finite sum of Jacobi polynomials.