For the first right-hand sides of (22.12.2)–(22.12.4)
see Lawden (1989, (8.8.3), (8.8.4), and (8.8.7)). The first right-hand sides
of (22.12.5)–(22.12.13) can be obtained from the first right-hand
sides of (22.12.2)–(22.12.4) by translations of the variable in
accordance with the intersection of the first four rows and first four columns of
Table 22.4.3. For example, in the case of (22.12.5) we apply
the translation to (22.12.2). The
second right-hand sides of (22.12.2)–(22.12.13) can be obtained from
the corresponding first right-hand sides by substituting, as appropriate, the expansions
or
; compare
(4.22.5) and (4.22.3).
See also Walker (1996, §5.1) and Weil (1999, pp. 14–47).
The double sums in (22.12.2)–(22.12.4) are convergent but
not absolutely convergent, hence the order of the summations is important.
Compare §20.5(iii).