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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 53 "Maple 6 worksheet for testing the
 Maple package zeilb" }}{PARA 0 "" 0 "" {TEXT -1 75 "(summation of ter
minating hypergeometric series by Zeilberger's algorithm)." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Copyright 1992, 1
996, 1998, 2000 by Tom H. Koornwinder (thk@science.uva.nl)." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "The 1992 versi
on was a competely rewritten version of a Maple procedure ident_prover
.maple by D. Zeilberger.\nThe 1996 version was adapted to Maple V, Rel
ease 3 and 4." }}{PARA 0 "" 0 "" {TEXT -1 57 "The 1998 version was ada
pted to Maple V, Release 4 and 5." }}{PARA 0 "" 0 "" {TEXT -1 95 "The \+
present version, 9 December 2000,  works in Maple 6 as well as in Mapl
e V, Release 4 and 5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 58 "The present worksheet was last modified on 5 January 2
001." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 
"if SearchText(APPLE,kernelopts(version))>0 then read `:mylib:zeilb:ze
ilb` else read zeilb fi:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "
t:=4:" }}}{PARA 0 "" 0 "" {TEXT -1 60 "Gosper's algorithm already appl
ies in the following example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "zeilb([b],[ ],1,f(n),0,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>
zeilb:~computing~P,~R1~and~R2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*ti
me=.100G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Ezeilb:~computing~maximal
~degree~of~PG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*time=.600G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%9zeilb:~solving~equationsG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%*time=.750G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#*&*&,&%\"bG\"\"\"%\"nGF'F'-%$facG6$F&F(F'F'*&-F*6$F'F(F'F&F'!\"
\"" }}}{PARA 0 "" 0 "" {TEXT -1 148 "Above we might specialize the out
put to b=-n (n noonegative integer). In the present implementation thi
s specialization cannot be made in the input:" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "zeilb([-n],[ ],1,f(n),0,t);" }}{PARA 8 "" 1 "" 
{TEXT -1 56 "Error, (in zeilb) order=0 while some index depends on n\n
" }}}{PARA 0 "" 0 "" {TEXT -1 113 "No error message is obtained if we \+
change the order of recurrence into 1, but then an arbitrary constant \+
appears:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "zeilb([-n],[ ],1
,f(n),1,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>zeilb:~computing~P,~R
1~and~R2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,time=.17e-1G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%Ezeilb:~computing~maximal~degree~of~PG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%,time=.50e-1G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9zeilb:~solving~equationsG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%,time=.67e-1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Wri
te~the~input~series~asG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%
\"nG-%$SumG6$-F%6$F'%\"kG/F-;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%&Then~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%$SumG6$,&-%\"fG6$%\"n
G%\"kG\"\"\"*&,&*&F+F-&%\"aG6#F-F-F-F1!\"\"F--F)6$,&F+F-F-F4F,F-F-/F,;
\"\"!%\"mG*&*(,&F;F-F-F-F-,*F0F4F1F-F-F4*&F1F-F;F-F-F--F)6$F+F>F-F-F+F
42F:F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+time~=.834G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%2Hence~f(n)~equalsG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$,$*(&%\"aG6#\"\"\"F(,&%\"nGF(F(!\"\"F(-%\"fG6#F)F(F+2\"
\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "zeilb([-n,b,c+1],[
b+1,c],1,f(n),1,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>zeilb:~comput
ing~P,~R1~and~R2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(time=0.G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%Ezeilb:~computing~maximal~degree~of~P
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*time=.400G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9zeilb:~solving~equationsG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%*time=.400G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Write
~the~input~series~asG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"n
G-%$SumG6$-F%6$F'%\"kG/F-;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
&Then~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%$SumG6$,&-%\"fG6$%\"nG%
\"kG\"\"\"*&*&F+F--F)6$,&F+F-F-!\"\"F,F-F-,&%\"bGF-F+F-F3F3/F,;\"\"!%
\"mG*&**,(F5F-F9F-F-F-F-,&F9F-F-F-F-,&*&%\"cGF-F4F3F3*&*&F+F-F9F-F-,**
&F5F-F+F-F-F5F3*$)F+\"\"#F-F-F+F3F3F3F--F)6$F+F=F-F-*&,(F@F-F9F-F-F-F-
F+F-F32F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+time~=.634G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%2Hence~f(n)~equalsG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$*&*&%\"nG\"\"\"-%\"fG6#,&F%F&F&!\"\"F&F&,&%\"bGF&F%F&F+
2F&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "zeilb([-n,b],[c],z
,f(n),2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>zeilb:~computing~P,~R
1~and~R2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(time=0.G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%Ezeilb:~computing~maximal~degree~of~PG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%,time=.84e-1G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%9zeilb:~solving~equationsG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%,time=.84e-1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Wri
te~the~input~series~asG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%
\"nG-%$SumG6$-F%6$F'%\"kG/F-;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%&Then~G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/-%$SumG6$,(-%\"fG6$%\"n
G%\"kG\"\"\"*&*&,.*&%\"zGF-F+F-F-*&\"\"#F-F+F-!\"\"*&F2F-%\"bGF-F-F2F5
F4F-%\"cGF5F--F)6$,&F+F-F-F5F,F-F-,(F8F-F-F5F+F-F5F-*&*(,&F2F-F-F5F-F;
F--F)6$,&F+F-F4F5F,F-F-F<F5F5/F,;\"\"!%\"mG,$*&*(,&F8F-FFF-F-,&FFF-F-F
-F--F)6$F+FKF-F-*&F<F-F+F-F5F52F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%+time~=.467G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Hence~f(n)~equalsG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&*&,.*&%\"zG\"\"\"%\"nGF)F)*&\"
\"#F)F*F)!\"\"*&F(F)%\"bGF)F)F(F-F,F)%\"cGF-F)-%\"fG6#,&F*F)F)F-F)F),(
F0F)F)F-F*F)F-F-*&*(,&F(F)F)F-F)F4F)-F26#,&F*F)F,F-F)F)F5F-F)2F)F*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "zeilb([-n,n+3,-y],[2,z],1,g
(n),2,t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>zeilb:~computing~P,~R1~
and~R2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(time=0.G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%Ezeilb:~computing~maximal~degree~of~PG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%*time=.134G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%9zeilb:~solving~equationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
*time=.134G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Write~the~input~serie
s~asG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"nG-%$SumG6$-F%6$F
'%\"kG/F-;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&Then~G" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6$/-%$SumG6$,(-%\"gG6$%\"nG%\"kG\"\"\"*&
*&,**&%\"zGF-F+F-\"\"#*(\"\"%F-F+F-%\"yGF-F-F2F-*&F3F-F6F-F-F--F)6$,&F
+F-F-!\"\"F,F-F-,,*$)F+F3F-F-F1F-F+F-*&F3F-F2F-F-F3F;F;F;*&*&,,F=F-F1F
;F+F-F2F-F3F;F--F)6$,&F+F-F3F;F,F-F-F<F;F;/F,;\"\"!%\"mG,$*&*,,&F3F-FI
F-F-,&F2F-FIF-F-,&FIF-F-F-F-,(F=F3F+F;F-F;F--F)6$F+FOF-F-*,,(F+F-F2F-F
-F;F-F+F-,(F+F-\"\"$F-FIF-F-F:F-,(F+F-F3F-FIF-F-F;!\"#2F-F+" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%,time~=1.350G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%2Hence~g(n)~equalsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&*(,&
%\"zG\"\"\"*&\"\"#F(%\"yGF(F(F(,&%\"nGF*F(F(F(-%\"gG6#,&F-F(F(!\"\"F(F
(*&,&F-F(F*F(F(,(F-F(F'F(F(F2F(F2F(*&*(F1F(,(F-F(F'F2F*F(F(-F/6#,&F-F(
F*F2F(F(*&F4F(F5F(F2F(2F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
38 "zeilb([-n,n+3,y+z],[2,z],1,h(n),2,t); " }{TEXT -1 0 "" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%>zeilb:~computing~P,~R1~and~R2G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%,time=.17e-1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%Ezeilb:~computing~maximal~degree~of~PG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%*time=.150G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9zeilb
:~solving~equationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*time=.150G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Write~the~input~series~asG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"nG-%$SumG6$-F%6$F'%\"kG/F-
;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&Then~G" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6$/-%$SumG6$,(-%\"hG6$%\"nG%\"kG\"\"\"*&*&,**&%\"zGF-F
+F-\"\"#*(\"\"%F-F+F-%\"yGF-F-F2F-*&F3F-F6F-F-F--F)6$,&F+F-F-!\"\"F,F-
F-,,*$)F+F3F-F-F1F-F+F-*&F3F-F2F-F-F3F;F;F-*&*&,,F=F-F1F;F+F-F2F-F3F;F
--F)6$,&F+F-F3F;F,F-F-F<F;F;/F,;\"\"!%\"mG,$*&*,,&F3F-FIF-F-,&F2F-FIF-
F-,&FIF-F-F-F-,(F=F3F+F;F-F;F--F)6$F+FOF-F-*,,(F+F-F2F-F-F;F-F+F-,(F+F
-\"\"$F-FIF-F-F:F-,(F+F-F3F-FIF-F-F;!\"#2F-F+" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%+time~=.784G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Henc
e~h(n)~equalsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&*(,&%\"zG\"\"\"*
&\"\"#F(%\"yGF(F(F(,&%\"nGF*F(F(F(-%\"hG6#,&F-F(F(!\"\"F(F(*&,&F-F(F*F
(F(,(F-F(F'F(F(F2F(F2F2*&*(F1F(,(F-F(F'F2F*F(F(-F/6#,&F-F(F*F2F(F(*&F4
F(F5F(F2F(2F(F-" }}}{PARA 0 "" 0 "" {TEXT -1 74 "Thus for the last two
 inputs g(n)=(-1)^n * h(n). (Check case n=1 by hand.)" }}{PARA 0 "" 0 
"" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 83 "The next test cases f
ollow Table 6.2 in W. Koepf's book \"Hypergeometric summation\"." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "binomial:
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "zeilb([-n],[],x,f(n),1,t
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>zeilb:~computing~P,~R1~and~R2G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(time=0.G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%Ezeilb:~computing~maximal~degree~of~PG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%,time=.34e-1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%9zeilb:~solving~equationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,time=
.50e-1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Write~the~input~series~as
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"nG-%$SumG6$-F%6$F'%\"
kG/F-;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&Then~G" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$/-%$SumG6$,&-%\"fG6$%\"nG%\"kG\"\"\"*&,&%\"xGF-
F-!\"\"F--F)6$,&F+F-F-F1F,F-F-/F,;\"\"!%\"mG,$*&*&,&F8F-F-F-F--F)6$F+F
<F-F-F+F1F12F7F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+time~=.167G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Hence~f(n)~equalsG" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6$,$*&,&%\"xG\"\"\"F'!\"\"F'-%\"fG6#,&%\"nGF'F'F(F'F(2
\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Hence~f(n)~equalsG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#),&%\"xG!\"\"\"\"\"F'%\"nG" }}}{PARA 
0 "" 0 "" {TEXT -1 16 "Chu-Vandermonde:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "zeilb([-n,b],[c],1,f(n),1,t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%>zeilb:~computing~P,~R1~and~R2G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%(time=0.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Ezeilb:~
computing~maximal~degree~of~PG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,ti
me=.50e-1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%9zeilb:~solving~equatio
nsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,time=.66e-1G" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%:Write~the~input~series~asG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%\"fG6#%\"nG-%$SumG6$-F%6$F'%\"kG/F-;\"\"!F'" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&Then~G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$/-%$SumG6$,&-%\"fG6$%\"nG%\"kG\"\"\"*&*&,*F+F-%\"bG!\"\"%\"cGF-F
-F2F--F)6$,&F+F-F-F2F,F-F-,(F3F-F-F2F+F-F2F2/F,;\"\"!%\"mG,$*&*(,&F3F-
F;F-F-,&F;F-F-F-F--F)6$F+F@F-F-*&F7F-F+F-F2F22F:F+" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%+time~=.216G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2He
nce~f(n)~equalsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$*&*&,*%\"nG\"\"\"%
\"bG!\"\"%\"cGF'F'F)F'-%\"fG6#,&F&F'F'F)F'F',(F*F'F'F)F&F'F)2\"\"!F&" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Hence~f(n)~equalsG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&-%$facG6$,&%\"bG!\"\"%\"cG\"\"\"%\"nGF+-F%6$F*F,
F)" }}}{PARA 0 "" 0 "" {TEXT -1 166 "The algorithm also works correctl
y if we might have taken the order of recursion higher than necessary.
 But then one or more arbitrary constants appear in the output:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "zeilb([-n,b],[c],1,f(n),2,t)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>zeilb:~computing~P,~R1~and~R2G
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%(time=0.G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%Ezeilb:~computing~maximal~degree~of~PG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%,time=.66e-1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%9zeilb:~solving~equationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,time=
.83e-1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:Write~the~input~series~as
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"nG-%$SumG6$-F%6$F'%\"
kG/F-;\"\"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&Then~G" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6$/-%$SumG6$,(-%\"fG6$%\"nG%\"kG\"\"\"*&*&,6*&F+F
-&%\"aG6#F-F-!\"$*&)F+\"\"#F-F2F-F-*&)%\"cGF8F-F2F-F-*(\"\"$F-F;F-F2F-
!\"\"*&F8F-F2F-F-**F8F-F+F-F2F-F;F-F-F-F-F+F>%\"bGF-F;F>F--F)6$,&F+F-F
-F>F,F-F-,(F;F-F-F>F+F-F>F-*(,*!\"#F-FAF>F+F-F;F-F-F2F--F)6$,&F+F-F8F>
F,F-F>/F,;\"\"!%\"mG*&**,&F;F-FOF-F-,&FOF-F-F-F-,&*&,0F>F-*&F;F-F2F-F>
F6F-*(F8F-F+F-F2F-F>*(F+F-F2F-F;F-F-F2F-F+F-F-FEF>F>*&F2F-FOF-F-F--F)6
$F+FSF-F-*&F+F-FDF-F>2F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%+time~=
.416G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%2Hence~f(n)~equalsG" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6$,&*&*&,6*&%\"nG\"\"\"&%\"aG6#F)F)!\"$*
&)F(\"\"#F)F*F)F)*&)%\"cGF0F)F*F)F)*(\"\"$F)F3F)F*F)!\"\"*&F0F)F*F)F)*
*F0F)F(F)F*F)F3F)F)F)F)F(F6%\"bGF)F3F6F)-%\"fG6#,&F(F)F)F6F)F),(F3F)F)
F6F(F)F6F6*(,*!\"#F)F9F6F(F)F3F)F)F*F)-F;6#,&F(F)F0F6F)F)2F)F(" }}}
{PARA 0 "" 0 "" {TEXT -1 7 "Kummer:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "zeilb([-n,a],[1+a+n],-1,f(n),1,t);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%>zeilb:~computing~P,~R1~and~R2G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%(time=0.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Ezeilb:~
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