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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "with(MS):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }{XPPEDIT 18 0 "s
(x) = (x^2+1)*sum(b[i]*x^i/i!,i = 0 .. infinity);" "6#/-%\"sG6#%\"xG*&
,&*$F'\"\"#\"\"\"\"\"\"F,F,-%$sumG6$*(&%\"bG6#%\"iGF,)F'F5F,-%*factori
alG6#F5!\"\"/F5;\"\"!%)infinityGF," }{TEXT -1 12 ", where the " }
{XPPEDIT 18 0 "b[i];" "6#&%\"bG6#%\"iG" }{TEXT -1 38 "s are the Fibona
cci numbers satisfing " }{XPPEDIT 18 0 "b[i] = b[i-1]+b[i-2];" "6#/&%
\"bG6#%\"iG,&&F%6#,&F'\"\"\"\"\"\"!\"\"F,&F%6#,&F'F,\"\"#F.F," }{TEXT 
-1 24 " with initial values of " }{XPPEDIT 18 0 "b[0] = 0;" "6#/&%\"bG
6#\"\"!F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[1] = 1;" "6#/&%\"bG6#
\"\"\"\"\"\"" }{TEXT -1 74 ".  This example shows how to determine the
 linear recurrence relation for " }{XPPEDIT 18 0 "s(x) = sum(d[i]*x^i/
i!,i = 0 .. infinity);" "6#/-%\"sG6#%\"xG-%$sumG6$*(&%\"dG6#%\"iG\"\"
\")F'F/F0-%*factorialG6#F/!\"\"/F/;\"\"!%)infinityG" }{TEXT -1 12 ", w
here the " }{XPPEDIT 18 0 "d[i];" "6#&%\"dG6#%\"iG" }{TEXT -1 39 " are
 to be written as functions of the " }{XPPEDIT 18 0 "b[i];" "6#&%\"bG6
#%\"iG" }{TEXT -1 37 ".  But this can just be rewritten as " }
{XPPEDIT 18 0 "sum(b[i]*x^(i+2)/i!,i = 0 .. infinity)+sum(b[i]*x^i/i!,
i = 0 .. infinity);" "6#,&-%$sumG6$*(&%\"bG6#%\"iG\"\"\")%\"xG,&F+F,\"
\"#F,F,-%*factorialG6#F+!\"\"/F+;\"\"!%)infinityGF,-F%6$*(&F)6#F+F,)F.
F+F,-F26#F+F4/F+;F7F8F," }{TEXT -1 16 ", which is just " }{XPPEDIT 18 
0 "sum(b[i]*(i+2)*(i+1)*x^(i+2)/(i+2)!,i = 0 .. infinity)+sum(b[i]*x^i
/i!,i = 0 .. infinity);" "6#,&-%$sumG6$*,&%\"bG6#%\"iG\"\"\",&F+F,\"\"
#F,F,,&F+F,\"\"\"F,F,)%\"xG,&F+F,\"\"#F,F,-%*factorialG6#,&F+F,\"\"#F,
!\"\"/F+;\"\"!%)infinityGF,-F%6$*(&F)6#F+F,)F2F+F,-F66#F+F:/F+;F=F>F,
" }{TEXT -1 20 ", or in other words " }{XPPEDIT 18 0 "sum((b[i-2]*i*(i
-1)+b[i])*x^i/i!,i = 0 .. infinity);" "6#-%$sumG6$*(,&*(&%\"bG6#,&%\"i
G\"\"\"\"\"#!\"\"F.F-F.,&F-F.\"\"\"F0F.F.&F*6#F-F.F.)%\"xGF-F.-%*facto
rialG6#F-F0/F-;\"\"!%)infinityG" }{TEXT -1 150 ".  There is a facility
 in Maple to make procedures with this additional information of the $
p(x)$ in Theorem \\ref\{thm:factor\}, (in this case $x^2+1$)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "t := b(i) = b(i-1) + b(i-2),
 b, i, [b(0)=0,b(1)=1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG6&/-%
\"bG6#%\"iG,&-F(6#,&F*\"\"\"!\"\"F/F/-F(6#,&F*F/!\"#F/F/F(F*7$/-F(6#\"
\"!F9/-F(6#F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "T := `eg
f/makeproc`(t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S := `eg
f/makeproc`(t,i^2+1):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Check th
e first few cases to see if it is correct." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "seq(i*(i-1)*T(i-2)+T(i),i=0..10);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6-\"\"!\"\"\"F$\"\")\"#:\"#X\"#)*\"$B#\"$p%\"$q*\"%X>
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(S(i),i=0..10);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6-\"\"!\"\"\"F$\"\")\"#:\"#X\"#)*\"$B#\"
$p%\"$q*\"%X>" }}}}{MARK "8" 0 }{VIEWOPTS 1 1 0 1 1 1803 }