> It looks pretty close to the right-hand side of
> http://en.wikipedia.org/wiki/Meijer_G-function#Polynomial_cases. I'm
> afraid I'm not good enough with the notation to tell if the indices
> match up or not, though. There are some other forumlas on that page
> with sums of G-functions on the right-hand side as well.
I tried to match against this formula. More on that approach later.
First let's see in more detail what Sympy does. Start with:
In [25]: pprint(_)
╭─╮1, 1 ⎛1 3 │ ⎞
│╶┐ ⎜ │ z⎟
╰─╯2, 2 ⎝2 0 │ ⎠
In [26]: hyperexpand(_, allow_hyper=True)
Out[26]: Piecewise((z**2/2, Abs(z) < 1), (1/2, Abs(1/z) < 1), (meijerg(((1,),
(3,)), ((2,), (0,)), z), True))
In [27]: pprint(_)
⎧ 2
⎪ z
⎪ ── for │z│ < 1
⎪ 2
⎪
⎪ │1│
⎨ 1/2 for │─│ < 1
⎪ │z│
⎪
⎪╭─╮1, 1 ⎛1 3 │ ⎞
⎪│╶┐ ⎜ │ z⎟ otherwise
⎪╰─╯2, 2 ⎝2 0 │ ⎠
⎩
where we obtain the following conditional expression which is fine so far (I
hope).
Now we try to substitute the actual values +1 and -1 for z:
In [28]: meijerg([1],[3],[2],[0],1)
Out[28]: meijerg(((1,), (3,)), ((2,), (0,)), 1)
In [29]: pprint(_)
╭─╮1, 1 ⎛1 3 │ ⎞
│╶┐ ⎜ │ 1⎟
╰─╯2, 2 ⎝2 0 │ ⎠
In [30]: hyperexpand(_, allow_hyper=True)
Out[30]: 1/2
Hmm, ok, but the condition on this branch is NOT fulfilled: abs(1/1) < 1
is not true for a strict inequality.
In [31]: meijerg([1],[3],[2],[0],-1)
Out[31]: meijerg(((1,), (3,)), ((2,), (0,)), -1)
In [32]: hyperexpand(_, allow_hyper=True)
Out[32]: meijerg(((1,), (3,)), ((2,), (0,)), -1)
In [33]: pprint(_)
╭─╮1, 1 ⎛1 3 │ ⎞
│╶┐ ⎜ │ -1⎟
╰─╯2, 2 ⎝2 0 │ ⎠
In case of z=-1 we get the correct but less helpful answer.
Now, why can we simplify the case z=1 although the condition
is not matched? Is this a bug?
If we go ahead never the less we find that:
In [35]: G1 = meijerg([1],[3],[2],[0],-1)
In [36]: pprint(g1)
╭─╮1, 1 ⎛1 3 │ ⎞
│╶┐ ⎜ │ -1⎟
╰─╯2, 2 ⎝2 0 │ ⎠
In [37]: G2 = meijerg([1],[3],[2],[0],1)
In [40]: pprint(G2)
╭─╮1, 1 ⎛1 3 │ ⎞
│╶┐ ⎜ │ 1⎟
╰─╯2, 2 ⎝2 0 │ ⎠
In [41]: G3 = meijerg([3,1],[],[],[2,0],-1)
In [42]: pprint(G3)
╭─╮0, 2 ⎛3, 1 │ ⎞
│╶┐ ⎜ │ -1⎟
╰─╯2, 2 ⎝ 2, 0 │ ⎠
In [43]: G4 = meijerg([3,1],[],[],[2,0],1)
In [44]: pprint(G4)
╭─╮0, 2 ⎛3, 1 │ ⎞
│╶┐ ⎜ │ 1⎟
╰─╯2, 2 ⎝ 2, 0 │ ⎠
In [45]: -G1/2 + G2/2-G3/2+G4/2
Out[45]: -meijerg(((1,), (3,)), ((2,), (0,)), -1)/2 + meijerg(((1,), (3,)),
((2,), (0,)), 1)/2 - meijerg(((3, 1), ()), ((), (2, 0)), -1)/2 + meijerg(((3,
1), ()), ((), (2, 0)), 1)/2
In [46]: pprint(_)
╭─╮1, 1 ⎛1 3 │ ⎞ ╭─╮1, 1 ⎛1 3 │ ⎞ ╭─╮0, 2 ⎛3, 1 │ ⎞ ╭─╮0,
2 ⎛3, 1 │ ⎞
│╶┐ ⎜ │ -1⎟ │╶┐ ⎜ │ 1⎟ │╶┐ ⎜ │ -1⎟ │╶┐
⎜ │ 1⎟
╰─╯2, 2 ⎝2 0 │ ⎠ ╰─╯2, 2 ⎝2 0 │ ⎠ ╰─╯2, 2 ⎝ 2, 0 │ ⎠ ╰─╯2,
2 ⎝ 2, 0 │ ⎠
- ─────────────────── + ────────────────── - ───────────────────────── +
────────────────────────
2 2 2
2
In [47]: hyperexpand(_, allow_hyper=True)
Out[47]: -meijerg(((1,), (3,)), ((2,), (0,)), -1)/2 - meijerg(((3, 1), ()),
((), (2, 0)), -1)/2 + 1/4
In [48]: pprint(_)
╭─╮1, 1 ⎛1 3 │ ⎞ ╭─╮0, 2 ⎛3, 1 │ ⎞
│╶┐ ⎜ │ -1⎟ │╶┐ ⎜ │ -1⎟
╰─╯2, 2 ⎝2 0 │ ⎠ ╰─╯2, 2 ⎝ 2, 0 │ ⎠ 1
- ─────────────────── - ───────────────────────── + ─
2 2 4
There is reason to believe that the two G in front should result in -1/4.
At this point the algorithm seems to be incomplete ...
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