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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) = sum(b[i]*x^i/i!,i = 0 .. infinity);" "6#/-%\"sG6#%\"xG-%$sumG6$*
(&%\"bG6#%\"iG\"\"\")F'F/F0-%*factorialG6#F/!\"\"/F/;\"\"!%)infinityG
" }{TEXT -1 8 ", where " }{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 5 " and " }{XPPEDIT 18 0 "b[0] = 2,b[1] = 1;" "6$/&%\"bG6#\"\"!\"\"
#/&F%6#\"\"\"\"\"\"" }{TEXT -1 76 ".  These are the Lucas numbers  as \+
defined by  Graham, Knuth and Patashnik, " }{TEXT -1 20 "\\cite\{Riord
an Knuth\}" }{TEXT -1 163 ".  To avoid confusion with the Lucas number
s as defined by Lehmer, call these the ``Lucas numbers, type I''  \\La
bel\{defn:Lucas I\}.\nNow multisection s(x) by 4 at 1." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "s := b(x) = b(x-1) + b(x-2), b, x, \+
[b(0) = 2, b(1) = 1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG6&/-%\"
bG6#%\"xG,&-F(6#,&F*\"\"\"!\"\"F/F/-F(6#,&F*F/!\"#F/F/F(F*7$/-F(6#\"\"
!\"\"#/-F(6#F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "First convert
 this to poly-exponential form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "pe := convert_pe(s)[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#
peG,&-%$expG6#*&%\"xG\"\"\",&#F+\"\"#F+*$-%%sqrtG6#\"\"&\"\"\"#!\"\"F.
F+F+-F'6#*&F*F4,&F-F+F/F-F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 
"Now multisection the poly-exponential function using the formula as g
iven in definition \\ref\{defn:\\Ms\}." }{MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "ms  := 1/4*sum(subs(x=x*exp(2*Pi*I/
4*i), pe)*exp(-2*Pi*I/4*i), i=0..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#msG,.-%$expG6#*&%\"xG\"\"\",&#F+\"\"#F+*$-%%sqrtG6#\"\"&\"\"\"#!
\"\"F.F+#F+\"\"%-F'6#*&F*F4,&F-F+F/F-F+F7*&%\"IGF4,&-F'6#*(F>F+F*F4F,F
4F+-F'6#*(F>F4F*F4F<F4F+F+#F6F8-F'6#,$F)F6FF-F'6#,$F;F6FF*&F>F4,&-F'6#
,$FBF6F+-F'6#,$FEF6F+F+F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Now \+
convert this back into an exponential generating function." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "convert_egf(ms, b, x);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6&/-%\"bG6#%\"xG,&-F%6#,&F'\"\"\"!\")F,!\"\"-F%6#
,&F'F,!\"%F,\"\"(F%F'7*/-F%6#\"\"!F8/-F%6#F,F,/-F%6#\"\"#F8/-F%6#\"\"$
F8/-F%6#\"\"%F8/-F%6#\"\"&\"#6/-F%6#\"\"'F8/-F%6#F3F8" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "From this it follows that s^1_4(x) = " }
{XPPEDIT 18 0 "sum(b[i]*x^i/i!,i = 0 .. infinity);" "6#-%$sumG6$*(&%\"
bG6#%\"iG\"\"\")%\"xGF*F+-%*factorialG6#F*!\"\"/F*;\"\"!%)infinityG" }
{TEXT -1 8 ", where " }{XPPEDIT 18 0 "b[i] = 7*b[i-4]-b[i-8];" "6#/&%
\"bG6#%\"iG,&*&\"\"(\"\"\"&F%6#,&F'F+\"\"%!\"\"F+F+&F%6#,&F'F+\"\")F0F
0" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[1] = 1,b[5] = 11;" "6$/&%\"bG
6#\"\"\"\"\"\"/&F%6#\"\"&\"#6" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[i
] = 0;" "6#/&%\"bG6#%\"iG\"\"!" }{TEXT -1 4 " if " }{XPPEDIT 18 0 "i <
> `mod`(1,4);" "6#0%\"iG-%$modG6$\"\"\"\"\"%" }{TEXT -1 18 ".  So  s^1
_4(x) = " }{XPPEDIT 18 0 "sum(b[4*i+1]*x^(4*i+1)/(4*i+1)!,i = 0 .. inf
inity);" "6#-%$sumG6$*(&%\"bG6#,&*&\"\"%\"\"\"%\"iGF-F-\"\"\"F-F-)%\"x
G,&*&\"\"%F-F.F-F-\"\"\"F-F--%*factorialG6#,&*&\"\"%F-F.F-F-\"\"\"F-!
\"\"/F.;\"\"!%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
90 "Alternatively there is automated code to achieve the same result, \+
using this naive method." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 
"`egf/ms/naive`(s,4,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%\"bG6#%
\"xG,&-F%6#,&F'\"\"\"!\")F,!\"\"-F%6#,&F'F,!\"%F,\"\"(F%F'7*/-F%6#\"\"
!F8/-F%6#F,F,/-F%6#\"\"#F8/-F%6#\"\"$F8/-F%6#\"\"%F8/-F%6#\"\"&\"#6/-F
%6#\"\"'F8/-F%6#F3F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "This is \+
a relationship for the Lucas numbers, type I that is only concerned wi
th $b_1, b_5, b_9, ...$" }}}}{MARK "11 0 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 }