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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "with(MS):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 238 "This example will demonstrate that proce
ss of multisectioning can be used where the recurrence has symbolic va
lues rather than simply numeric ones.  Consider the ``\{\\em Chebyshev
 T polynomials\}'', as polynomials in t, with the recurrence " }
{XPPEDIT 18 0 "T[n] = 2*t*T[n-1]-T[n-2];" "6#/&%\"TG6#%\"nG,&*(\"\"#\"
\"\"%\"tGF+&F%6#,&F'F+\"\"\"!\"\"F+F+&F%6#,&F'F+\"\"#F1F1" }{TEXT -1 
26 " with initial polynomials " }{XPPEDIT 18 0 "T[0] = 1;" "6#/&%\"TG6
#\"\"!\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "T[1] = t;" "6#/&%\"T
G6#\"\"\"%\"tG" }{TEXT -1 86 " \\cite\{Abramowitz\}.  Consider multise
ctioning this by 5 at 1, to get a recurrence for " }{XPPEDIT 18 0 "T[1
];" "6#&%\"TG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "T[6];" "6#&%\"
TG6#\"\"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "T[11];" "6#&%\"TG6#\"#6" }
{TEXT -1 2 ", " }{XPPEDIT 18 0 "T[16];" "6#&%\"TG6#\"#;" }{TEXT -1 5 "
, ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "egf := f(x) = 2*t*
f(x-1)-f(x-2),f,x,[f(0)=1, f(1)=t];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$egfG6&/-%\"fG6#%\"xG,&*&%\"tG\"\"\"-F(6#,&F*F.!\"\"F.F.\"\"#-F(6#,
&F*F.!\"#F.F2F(F*7$/-F(6#\"\"!F./-F(6#F.F-" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 18 "`egf/ms`(egf,5,1);" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6&/-%\"fG6#%\"xG,&-F%6#,&F'\"\"\"!#5F,!\"\"*&,(*$)%\"tG\"\"&\"\"\"\"#
K*$)F3\"\"$F5!#SF3\"#5F,-F%6#,&F'F,!\"&F,F,F,F%F'7,/-F%6#\"\"!FD/-F%6#
F,F3/-F%6#\"\"#FD/-F%6#F9FD/-F%6#\"\"%FD/-F%6#F4FD/-F%6#\"\"',**&F3F5,
(*&F3F5,(*&F3F5,&*&F3F,,&*$)F3FKF5FKF.F,F,FKF3F.F,FKF]o!\"#F,F,F,FKF[o
F_oF3F,F,FKFinF_oF]oFKF.F,/-F%6#\"\"(FD/-F%6#\"\")FD/-F%6#\"\"*FD" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "expand([%]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#7&/-%\"fG6#%\"xG,*-F&6#,&F(\"\"\"!#5F-!\"\"*&-F&6#
,&F(F-!\"&F-F-)%\"tG\"\"&\"\"\"\"#K*&F1F8)F6\"\"$F8!#S*&F1F8F6F-\"#5F&
F(7,/-F&6#\"\"!FD/-F&6#F-F6/-F&6#\"\"#FD/-F&6#F<FD/-F&6#\"\"%FD/-F&6#F
7FD/-F&6#\"\"',**$)F6FYF8F9*$)F6FRF8!#[*$)F6FKF8\"#=F/F-/-F&6#\"\"(FD/
-F&6#\"\")FD/-F&6#\"\"*FD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "This
 example is interesting in that it shows that the " }{XPPEDIT 18 0 "la
mbda[i];" "6#&%'lambdaG6#%\"iG" }{TEXT -1 167 " used in the definition
 of poly-exponential functions, (Definition \\ref\{def:\\Pe\}) can be \+
symbolic values in the complex numbers, as opposed to just the numeric
 values." }}}}{MARK "5 0 2" 166 }{VIEWOPTS 1 1 0 1 1 1803 }