{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 
0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }
{CSTYLE "" -1 256 "terminal" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "
" -1 257 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 
{CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Co
urier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 
0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE 
"" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 
0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 
0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 
0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 
256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 
-1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 257 1 {CSTYLE "" -1 -1 "He
lvetica" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "R3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 
0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 32 "/home/m262f99/A=B/Maple/i
nt3.mws" }{MPLTEXT 1 0 0 "" }}{PARA 256 "" 0 "" {TEXT 257 86 "Math 262
a, Fall 1999, Glenn Tesler\nCreative Telescoping for a triple integral
\n10/27/99" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "read TRIPLE_
INTEGRAL;\nP:=1:\nF := 1/(1-z*(w+1/w+x+1/x))/(x*w)/z^(n+1);\n \nORDER:
=3:\n#resh:=[n,ca1,ca2,k]:\nS1:=1/x^2/w^2/z^2:\nS2:=S1:\nS3:=S1:\nresh
 := [n]:\ngu:=ikar(P,F,n,z,w,x,ORDER,resh,N,S1,S2,S3);\n" }}{PARA 7 "
" 1 "" {TEXT -1 44 "Warning, `mekh` is implicitly declared local" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG**,&\"\"\"F'*&%\"zGF',*%\"wGF'*$
F+!\"\"F'%\"xGF'*$F.F-F'F'F-F-F.F-F+F-)F),&%\"nGF'F'F'F-" }}{PARA 0 "
" 0 "" {TEXT -1 401 "We seek a polynomial P(N,n) and rational function
s Rx,Rw,Rz of (n,x,w,z), such that\n P(N,n) F(n,x,w,z) = (d/dx)(Rx F) \+
+ (d/dw) (Rw F) + (d/dz) (Rz F).\nExpanding derivatives and dividing b
y F gives\n(P F)/F = (d/dx Rx  +  d/dw Rw  +  d/dz Rz)  +  (Rx * dF/dx
  +  Rw * dF/dw  +  Rz * dF/dz)\nHere the variables are numbered: z, w
, x are vars. 1, 2, 3;  Rz, Rw, Rx are R1, R2, R3;  d/dz Rz  is  R1Y1;
 etc.\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%&eq~isG,0*&,*\"\"\"F'*&%#a
1GF'%\"zG!\"\"F'*&%#a2GF'F*!\"#F'*&%#a3GF'F*!\"$F'F'%%SmolGF'F'%%Y1R1G
F+**%#R1GF',6*(F*F'%\"xGF'%\"wG\"\"#F:*&F*F'F8F'F:*(F*F'F8F:F9F'F:*&F*
F'F9F'F:*(%\"nGF'F8F'F9F'F+**F?F'F*F'F8F'F9F:F'*(F?F'F*F'F8F'F'**F?F'F
*F'F8F:F9F'F'*(F?F'F*F'F9F'F'*&F8F'F9F'F+F',,FDF+F7F'F;F'F<F'F=F'F+F*F
+F'%%Y2R2GF+*(%#R2GF'FEF+,*F8F+*(F*F'F8F'F9F'F:*&F*F'F8F:F'F*F'F'F'%%Y
3R3GF+*(%#R3GF'FEF+,*F9F+*&F*F'F9F:F'F*F'FJF:F'F'" }}{PARA 0 "" 0 "" 
{TEXT -1 1061 "No good theorems for the exact form of Rx, Rw, Rz are p
resently known.  You \"guess\" the denominators by hand, and form rati
onal functions Sx, Sw, Sz containing the denominators (required) and a
ny numerator factors that you also manage to guess (optional, just to \+
speed things up if you have the knack for it).\nThis makes the rationa
l functions Rx, Rw, Rz have the form Sx, Sw, Sz times mystery polynomi
als in x,w,z, whose coefficients are  in Q(n). The degrees of these my
stery polynomials are all unknown.  Also, P is a polynomial in N (whos
e order is specified by the input variable ORDER, here 3) with unknown
 coefficients in Q(n).  The program gets a lower bound on the degree D
 to try for the mystery polynomials, and then using the method of unde
termined coefficients, lets these polynomials be generic polynomials w
ith all terms of total degree <=D.  If this fails, it tries again with
 D+1, then D+2, etc., until it succeeds or the program is stopped or c
rashes.\n\nNow it tries various degrees, and cryptically tells us it f
ailed (\"0\") or succeeded (\"1\")." }}{PARA 11 "" 1 "" {XPPMATH 20 "6
$%&gu~isG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%&gu~isG\"\"\"" }}
{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT -1 0 "" }}{PARA 6 "" 1 "" 
{TEXT -1 51 "Theorem:\n\nLet G(   z   ,   w   ,   x   ,   n   ) be" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$**,&\"\"\"F%*&%\"zGF%,*%\"wGF%*$F)!\"
\"F%%\"xGF%*$F,F+F%F%F+F+F,F+F)F+)F',&%\"nGF%F%F%F+%!G" }}{PARA 6 "" 
1 "" {TEXT -1 171 "\nand a(   n   ) be its triple integral w.r.t to   \+
z   w   and   x   .\n\nLet   N   be the forward shift operator in   n \+
  .\n\nThe sequence a(   n   ) satisfies the recurrence" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&,&*$,&%\"nG\"\"\"F)F)\"\"#\"#;*&,&F(F)F*F)F*%
\"NGF*!\"\"F)-%\"aG6#F(F)%#0.G" }}{PARA 6 "" 1 "" {TEXT -1 39 "\nProof
: It is routinely verifiable that" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*
&,&*$,&%\"nG\"\"\"F(F(\"\"#\"#;*&,&F'F(F)F(F)%\"NGF)!\"\"F(-%\"GG6&%\"
zG%\"wG%\"xGF'F(" }}{PARA 6 "" 1 "" {TEXT -1 13 "\n= D_   z   (" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*,,0**%\"zG\"\"#%\"wGF(%\"xGF(%\"nG
\"\"\"\"#;*(F)F(F*F(F+F,!\"\"*&F'F,F*F,\"\"%*(F'F(F)F(F*F(\"#C*$F'F(!#
7*&F'F(F*F(F1*&F)F(F*F(!\"#F,F)F8F*F8F'F/-%\"GG6&F'F)F*F+F,F/" }}
{PARA 6 "" 1 "" {TEXT -1 24 "          )\n\n+D_   w   (" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#*,,D*(%\"wG\"\"#%\"xGF'%\"nG\"\"\"!\"\"**%\"zGF'
F&F'F(F'F)F*\"\")*(F(F*F&F*F)F*!\"#*(F-F*F&F*F)F*F'*(F-F'F(F'F)F*\"\"%
**F-F'F&F*F(F*F)F*F.**F-F*F(F'F&F*F)F*F3*&F-F'F)F*!\"%*(F-F*F(F*F)F*F'
*&F(F*F&F*F7*$F-F'F7*(F-F*F(F'F&F*F.*&F-F*F&F*F3*(F-F'F&F*F(F*F.*&F&F'
F(F'F0*&F-F'F(F'F3*(F-F'F&F'F(F'F.F*F-F0F(F0F&F+-%\"GG6&F-F&F(F)F*" }}
{PARA 6 "" 1 "" {TEXT -1 12 "\n+D_   x   (" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#*,,B*(%\"zG\"\"\"%\"wGF'%\"nGF'!\"#**F&\"\"#F(F,%\"xGF,
F)F'\"\")*(F&F,F-F,F)F'!\"%*(F&F'F-F'F)F'F,*&F&F,F)F'F0*(F(F,F-F,F)F'!
\"\"**F&F'F-F,F(F'F)F'F0*(F-F'F(F'F)F'F,*&F&F'F-F'F.*&F&F'F(F'F0*(F&F,
F(F,F-F,F.*&F&F,F-F,F0*&F-F'F(F'\"\"%*$F&F,F0*&F(F,F-F,F**(F&F'F-F,F(F
'!\")F'F&F*F(F*F-F4-%\"GG6&F&F(F-F)F'" }}{PARA 6 "" 1 "" {TEXT -1 91 "
          ),\n\nand the result follows by triple integrating w.r.t to
\nz   ,   w   and   x   ." }}{PARA 0 "" 0 "" {TEXT -1 3 "\n\n\n" }
{TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#guG6',&*$,&%\"nG\"
\"\"F*F*\"\"#\"#;*&,&F)F*F+F*F+%\"NGF+!\"\",$**,0**%\"zGF+%\"wGF+%\"xG
F+F)F*F,*(F6F+F7F+F)F*F0*&F5F*F7F*\"\"%*(F5F+F6F+F7F+\"#C*$F5F+!#7*&F5
F+F7F+F:*&F6F+F7F+!\"#F*F6FAF7FAF5F0F0**,DF8F0F4\"\")*(F7F*F6F*F)F*FA*
(F5F*F6F*F)F*F+*(F5F+F7F+F)F*F:**F5F+F6F*F7F*F)F*FD**F5F*F7F+F6F*F)F*F
:*&F5F+F)F*!\"%*(F5F*F7F*F)F*F+*&F7F*F6F*FKF=FK*(F5F*F7F+F6F*FD*&F5F*F
6F*F:*(F5F+F6F*F7F*FDF@FAF?F:F;FDF*F5FAF7FAF6F0**,BFFFAF4FDFGFKFLF+FJF
KF8F0FIFKFEF+F9FDFOFKF;FDF?FKFMF:F=FKF@FAFN!\")F*F5FAF6FAF7F0$\"&]/%!
\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 
13 }{VIEWOPTS 1 1 0 1 1 1803 }