hgfred and generalized hypergeometric functions
- Subject: hgfred and generalized hypergeometric functions
- From: Edmond Orignac
- Date: Sat, 15 Apr 2006 22:41:45 +0200
I am using Maxima 5.9.1 on Debian Linux 3.1. I have recently discovered
possible bugs in the hgfred function when applied to generalized
hypergeometric functions such as 3F2 function.
More specifically, I have compared the output of maxima with the
identities listed on functions.wolfram.com for the generalized
hypergeometric functions.
Generally, when the expressions reduce to a rational fraction,
maxima gives the correct denominator, but the numerator differs from
the formulas quoted in functions.wolfram.com except for the constant
term.
I am unsure whether these potential bugs are fixed in the more recent
versions of maxima (5.9.2/5.9.3) since I haven't seen bug reports
on the sourceforge site dealing with the generalized hypergeometric
functions.
Below is a copy of the maxima session.
Maxima restarted.
(%i1) batch(hypergeometric2);
batching #p/home/edmond/hypergeometric2
- 1 1
(%i2) HGFRED([---, 1], [-, 2], x)
2 2
1 1
(%o2) %F ([- -, 1], [-, 2], x)
2, 2 2 2
5 5 1 1
(%i3) HGFRED([-, -, 3], [-, -], x)
2 2 2 2
2
20 x 8 x 1
(%o3) 30 x (-------- + -------- + --------)
6 5 4
(1 - x) (1 - x) (1 - x)
2 2
2 30 x 10 x 1 12 x 6 x 1
+ 140 x (-------- + -------- + --------) + -------- + -------- +
--------
7 6 5 5 4 3
(1 - x) (1 - x) (1 - x) (1 - x) (1 - x) (1 - x)
(%i4) RADCAN(%)
4 3 2
2557 x + 1320 x + 290 x + 32 x + 1
(%o4) - ---------------------------------------------------
7 6 5 4 3 2
x - 7 x + 21 x - 35 x + 35 x - 21 x + 7 x - 1
(%i5) FACTOR(%)
4 3 2
2557 x + 1320 x + 290 x + 32 x + 1
(%o5) - -------------------------------------
7
(x - 1)
%
(%i6) -
3
4 3 2
2557 x + 1320 x + 290 x + 32 x + 1
(%o6) - -------------------------------------
7
3 (x - 1)
5 5 1 1
(%i7) HGFRED([-, -, 3], [-, -], x)
2 2 2 2
2
20 x 8 x 1
(%o7) 30 x (-------- + -------- + --------)
6 5 4
(1 - x) (1 - x) (1 - x)
2 2
2 30 x 10 x 1 12 x 6 x 1
+ 140 x (-------- + -------- + --------) + -------- + -------- +
--------
7 6 5 5 4 3
(1 - x) (1 - x) (1 - x) (1 - x) (1 - x) (1 - x)
(%i8) RATSIMP(%)
4 3 2
2557 x + 1320 x + 290 x + 32 x + 1
(%o8) - ---------------------------------------------------
7 6 5 4 3 2
x - 7 x + 21 x - 35 x + 35 x - 21 x + 7 x - 1
(%i9) FACTOR(%)
4 3 2
2557 x + 1320 x + 290 x + 32 x + 1
(%o9) - -------------------------------------
7
(x - 1)
5 5 1
(%i10) HGFRED([-, -, 3], [-, 1], x)
2 2 2
2
63 x 7 x 1
(%o10) 15 x (------------- + ---------- + ----------)
11/2 9/2 7/2
4 (1 - x) (1 - x) (1 - x)
2
2 99 x 9 x 1
105 x (------------- + ----------- + ----------)
13/2 11/2 9/2 2
4 (1 - x) (1 - x) (1 - x) 35 x
+ ------------------------------------------------- + ------------
2 9/2
4 (1 - x)
5 x 1
+ ---------- + ----------
7/2 5/2
(1 - x) (1 - x)
(%i11) RATSIMP(%)
4 3 2
5903 x + 3458 x + 898 x + 128 x + 8
(%o11) ----------------------------------------------------------------
6 5 4 3 2
SQRT(1 - x) (8 x - 48 x + 120 x - 160 x + 120 x - 48 x + 8)
(%i12) FACTOR(%)
4 3 2
5903 x + 3458 x + 898 x + 128 x + 8
(%o12) --------------------------------------
6
8 SQRT(1 - x) (x - 1)
5 5 1 3
(%i13) HGFRED([-, -, 3], [-, -], x)
2 2 2 2
4 x 1 2 5 x 1 3 x
(%o13) 10 x (-------- + --------) + 28 x (-------- + --------) +
--------
5 4 6 5 4
(1 - x) (1 - x) (1 - x) (1 - x) (1 - x)
1
+ --------
3
(1 - x)
(%i14) RADCAN(%)
3 2
84 x + 45 x + 10 x + 1
(%o14) -------------------------------------------
6 5 4 3 2
x - 6 x + 15 x - 20 x + 15 x - 6 x + 1
(%i15) FACTOR(%)
2
(4 x + 1) (21 x + 6 x + 1)
(%o15) ---------------------------
6
(x - 1)
6
(%i16) % (x - 1)
2
(%o16) (4 x + 1) (21 x + 6 x + 1)
(%i17) RADCAN(%)
3 2
(%o17) 84 x + 45 x + 10 x + 1
5 5 3 3
(%i18) HGFRED([-, -, 3], [-, -], x)
2 2 2 2
4 x 1 3 x 1
(%o18) 5 x (-------- + --------) + -------- + --------
5 4 4 3
(1 - x) (1 - x) (1 - x) (1 - x)
(%i19) RADCAN(%)
2
13 x + 6 x + 1
(%o19) - -----------------------------------
5 4 3 2
x - 5 x + 10 x - 10 x + 5 x - 1
(%i20) FACTOR(%)
2
13 x + 6 x + 1
(%o20) - ---------------
5
(x - 1)
1
(%i21) HGFRED([3, 3, 3], [-, 1], x)
2
[About a 1000 lines of output deleted]
- 1 5
(%i22) HGFRED([---, 1, -], [2, 2], x)
2 2
1 5
(%o22) %F ([- -, 1, -], [2, 2], x)
3, 2 2 2
- 1 5 3
(%i23) HGFRED([---, 1, -], [-, 2], x)
2 2 2
2 - x 2
3 SQRT(%PI) %I %Q (-----) %P (SQRT(1 - x)) x (x - 1)
0, 3/2 x - 2, - 2
(%o23) ------------------------------ -
----------------------------------
3/4 1/4 32
4 (x - 1) (x + 1)
(%i24) RADCAN(%)
3/4 1/4 3
(%o24) - ((x - 1) (x + 1) (%P (SQRT(1 - x)) x
- 2, - 2
x - 2
- %P (SQRT(1 - x)) x) - 24 SQRT(%PI) %I %Q (- -----))
- 2, - 2 0, 3/2 x
3/4 1/4
/(32 (x - 1) (x + 1) )
- 1 5 3
(%i25) HGFRED([---, 1, -], [-, 3], x)
2 2 2
SIMP2F1-WILL-CONTINUE-IN
2 - x
5 SQRT(%PI) %I %Q (-----) x
1 1, 3/2 x
(%o25) %F ([- -, 1], [3], x) - --------------------------------
2, 1 2 3/4 5/4
48 (x - 1) (x + 1)
- 1 5
(%i26) HGFRED([---, 1, -], [3, 2], x)
2 2
1 5
(%o26) %F ([- -, 1, -], [3, 2], x)
3, 2 2 2
(%i27) HGFRED([a, b, c], [d, e], x)
(%o27) %F ([a, b, c], [d, e], x)
3, 2
(%i28) HGFRED([a, b, c], [d, e], 1)
(%o28) %F ([a, b, c], [d, e], 1)
3, 2
1
(%i29) HGFRED([-, b, 1 - b], [d, 2 - d], x)
2
1
(%o29) %F ([-, b, 1 - b], [d, 2 - d], x)
3, 2 2
1 1 3 5 3 7
(%i30) HGFRED([-, -, -, 1], [-, -, -], x)
4 2 4 4 2 4
1 1 3 5 3 7
(%o30) %F ([-, -, -, 1], [-, -, -], x)
4, 3 4 2 4 4 2 4
3 5 7 9
(%i31) HGFRED([-, 1, 1, -], [-, 2, -], x)
4 4 4 4
3 5 7 9
(%o31) %F ([-, 1, 1, -], [-, 2, -], x)
4, 3 4 4 4 4
(%i32) HGFRED([2, 2, 2, 2], [1, 1, 1], x)
4 x 1
9 x (-------- + --------)
5 4
(1 - x) (1 - x) 3 x 1
(%o32) 8 x (------------------------- + -------- + --------)
2 4 3
(1 - x) (1 - x)
3 x 1 2 x 1
+ 4 x (-------- + --------) + -------- + --------
4 3 3 2
(1 - x) (1 - x) (1 - x) (1 - x)
(%i33) RATSIMP(%)
3 2
85 x + 47 x + 11 x + 1
(%o33) - -----------------------------------
5 4 3 2
x - 5 x + 10 x - 10 x + 5 x - 1
(%i34) FACTOR(%)
2
(5 x + 1) (17 x + 6 x + 1)
(%o34) - ---------------------------
5
(x - 1)
5
(%i35) % (x - 1)
2
(%o35) - (5 x + 1) (17 x + 6 x + 1)
(%i36) RATSIMP(%)
3 2
(%o36) - 85 x - 47 x - 11 x - 1
(%i37)
Process maxima processus arrét?