Due to their complexity and lack of space, we will not present a number of more advanced
theorems related to calculation of Meijer G Function Representations. We just
say that these theorems go by names such as
Paired Index Theorems
(similar to theorems in Adamchik
[3]), a
PFD Duplication Formula
(related to a similar formula in Roach
[15]), and an
Expansion Theorem
(a generalization of Slater's Theorem).
One of our long term goals is to enlarge our
Lookup
routine to the point that our algorithm should basically reproduce nearly all 879
(roughly) of the formula representations for the Meijer G function listed in chapter
8 of
Integrals and Series Volume 3: More Special Functions
[13]. We are not at that point yet, but progress is good. Every formula in
this book through our algorithm turns into infinitely many formulas. We also
envision that our algorithm will appear as an important subroutine inside general
routines which solve integration problems.
In the course of this work, we discovered mistakes in formulas 2(19), 12(7), 12(8),
15(7), 15(8), 18(15), 18(16), 20(8), 20(45), 22(15), 22(16), 22(21), 22(22), 23(34),
25(5), 29(15), 29(16), 40(6), 40(22), 40(23), 43(1), 43(2), 46(9), 46(10), 49(2),
49(8), 49(41), 49(42), and 49(44) of section 8.4 of
Integrals and Series Volume 3: More Special Functions
[13]. We have not inspected sections 41 and 42 discussing the Legendre functions
and
closely enough yet to comment about their correctness, but otherwise this list of
errors may be nearly comprehensive.
