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.
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).
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).