§31.9 Orthogonality

§31.9(i) Single Orthogonality

With

31.9.1

we have

Here is an arbitrary point in the interval . The integration path begins at , encircles once in the positive sense, followed by once in the positive sense, and so on, returning finally to . The integration path is called a Pochhammer double-loop contour (compare Figure 5.12.3). The branches of the many-valued functions are continuous on the path, and assume their principal values at the beginning.

The normalization constant is given by

where

and denotes the Wronskian (§1.13(i)). The right-hand side may be evaluated at any convenient value, or limiting value, of in since it is independent of .

For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64).

§31.9(ii) Double Orthogonality

Heun polynomials , , satisfy

where

and the integration paths , are Pochhammer double-loop contours encircling distinct pairs of singularities , , .

For further information, including normalization constants, see Sleeman (1966a). For bi-orthogonal relations for path-multiplicative solutions see Schmidt (1979, §2.2). For other generalizations see Arscott (1964b, pp. 206–207 and 241).