{VERSION 3 0 "SGI MIPS UNIX" "3.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 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 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "with(MS):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Consider again the Bernoulli numbers " } {XPPEDIT 18 0 "x/(exp(x)-1) = sum(b[i]*x^i/i!,i = 0 .. infinity)/sum(d [j]*x^j/j!,j = 0 .. infinity);" "6#/*&%\"xG\"\"\",&-%$expG6#F%F&\"\"\" !\"\"F,*&-%$sumG6$*&*&&%\"bG6#%\"iGF&)F%F6F&F&-%*factorialG6#F6F,/F6; \"\"!%)infinityGF&-F/6$*&*&&%\"dG6#%\"jGF&)F%FFF&F&-F96#FFF,/FF;F=F>F, " }{TEXT -1 234 ". Multisection this by 3 at 1, using the formula, as given in Lemma \\ref\{lem:RPE Ms\}. After this, this example will c alculate the 1-st, 4-th 7-th and 10-th Bernoulli number, using the for mula given in Theorem \\ref\{thm:Main Result\}." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "s[h](x) = x;" "6#/-&%\"sG6#%\"hG6#% \"xGF*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t[h](x) = exp(x)-1;" "6#/- &%\"tG6#%\"hG6#%\"xG,&-%$expG6#F*\"\"\"\"\"\"!\"\"" }{TEXT -1 16 ", an d solve for " }{XPPEDIT 18 0 "s(x);" "6#-%\"sG6#%\"xG" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "t(x);" "6#-%\"tG6#%\"xG" }{TEXT -1 16 " in the th eorem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "s[h] := x -> x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"sG6#%\"hGR6#%\"xG6\"6$%)operato rG%&arrowGF+9$F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "t[h ] := (x) -> exp(x)-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#%\"h GR6#%\"xG6\"6$%)operatorG%&arrowGF+,&-%$expG6#9$\"\"\"!\"\"F4F+F+F+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "omega[3] := exp(2*Pi*I/3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&omegaG6#\"\"$,&#!\"\"\"\"#\"\" \"*&%\"IGF,-%%sqrtG6#F'\"\"\"#F,F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "From Lemma \\ref\{lem:RPE Ms\} " }{XPPEDIT 18 0 "s(x) = s[h](x )*product(t[h]*omega[m]^i,i = 1 .. m-1);" "6#/-%\"sG6#%\"xG*&-&F%6#%\" hG6#F'\"\"\"-%(productG6$*&&%\"tG6#F,F.)&%&omegaG6#%\"mG%\"iGF./F;;\" \"\",&F:F.\"\"\"!\"\"F." }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "t(x) = p roduct(t[h](x*omega[m]^i),i = 0 .. m-1);" "6#/-%\"tG6#%\"xG-%(productG 6$-&F%6#%\"hG6#*&F'\"\"\")&%&omegaG6#%\"mG%\"iGF1/F7;\"\"!,&F6F1\"\"\" !\"\"" }{TEXT -1 37 ", which, for this particular case is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "S := s[h](x) * t[h](x*omega[3]) * t [h](x*omega[3]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG*(%\"xG\" \"\",&-%$expG6#*&F&\"\"\",&#!\"\"\"\"#F-*&%\"IGF--%%sqrtG6#\"\"$F'#F-F 1F-F-F0F-F-,&-F*6#*&F&F')F.F1F'F-F0F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T := t[h](x)*t[h](x*omega[3])*t[h](x*omega[3]^2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG*(,&-%$expG6#%\"xG\"\"\"!\"\"F+F +,&-F(6#*&F*F+,&#F,\"\"#F+*&%\"IGF+-%%sqrtG6#\"\"$\"\"\"#F+F3F+F+F,F+F +,&-F(6#*&F*F:)F1F3F:F+F,F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 " Now, determine what the linear recurrence for this would be." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "pe/ms(S,b,x,3,1);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6&/-%\"bG6#%\"xG,&-F%6#,&F'\"\"\"!#7F,! \"\"-F%6#,&F'F,!\"'F,\"\"#F%F'70/-F%6#\"\"!F8/-F%6#F,F8/-F%6#F3F8/-F%6 #\"\"$F8/-F%6#\"\"%F-/-F%6#\"\"&F8/-F%6#\"\"'F8/-F%6#\"\"(!\"(/-F%6#\" \")F8/-F%6#\"\"*F8/-F%6#\"#5!#I/-F%6#\"#6F8/-F%6#\"#7F8/-F%6#\"#8!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "So s^1_3(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] = b[i-12]+2*b[i-6];" "6#/&%\"bG6#%\"iG,& &F%6#,&F'\"\"\"\"#7!\"\"F,*&\"\"#F,&F%6#,&F'F,\"\"'F.F,F," }{TEXT -1 25 ", with initial values of " }{XPPEDIT 18 0 "b[4] = -12,b[7] = -7,b[ 10] = -30;" "6%/&%\"bG6#\"\"%,$\"#7!\"\"/&F%6#\"\"(,$\"\"(F*/&F%6#\"#5 ,$\"#IF*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[13] = -13;" "6#/&%\"bG 6#\"#8,$\"#8!\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "convert_egf(T,d,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6&/-%\"dG6#%\"xG-F%6#,&F'\"\"\"!\"'F+F%F'7(/-F%6#\"\"!F1/-F%6#F+F1/-F% 6#\"\"#F1/-F%6#\"\"$\"\"'/-F%6#\"\"%F1/-F%6#\"\"&F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "So the bottom linear recurrence relation t^0_3 (x) = " }{XPPEDIT 18 0 "sum(d[j]*x^i/j!,j = 0 .. infinity);" "6#-%$sum G6$*(&%\"dG6#%\"jG\"\"\")%\"xG%\"iGF+-%*factorialG6#F*!\"\"/F*;\"\"!%) infinityG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "d[j] = d[j-6];" "6#/ &%\"dG6#%\"jG&F%6#,&F'\"\"\"\"\"'!\"\"" }{TEXT -1 29 ", and the initia l values are " }{XPPEDIT 18 0 "d[3] = 6;" "6#/&%\"dG6#\"\"$\"\"'" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "Equally easy the two bui lt in commands " }{TEXT -1 20 "could have been used" }{TEXT -1 33 " to do this in the naive fashion." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "top := top/ms/naive(x,exp(x)-1,b,x,3,1);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%$topG6&/-%\"bG6#%\"xG,&-F(6#,&F*\"\"\"!#7F/!\"\"-F( 6#,&F*F/!\"'F/\"\"#F(F*70/-F(6#\"\"!F;/-F(6#F/F;/-F(6#F6F;/-F(6#\"\"$F ;/-F(6#\"\"%F0/-F(6#\"\"&F;/-F(6#\"\"'F;/-F(6#\"\"(!\"(/-F(6#\"\")F;/- F(6#\"\"*F;/-F(6#\"#5!#I/-F(6#\"#6F;/-F(6#\"#7F;/-F(6#\"#8!#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "bot := bottom/ms/naive(exp (x)-1,d,x,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$botG6&/-%\"dG6#%\" xG-F(6#,&F*\"\"\"!\"'F.F(F*7(/-F(6#\"\"!F4/-F(6#F.F4/-F(6#\"\"#F4/-F(6 #\"\"$\"\"'/-F(6#\"\"%F4/-F(6#\"\"&F4" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "Now, to calculate the first few Bernoulli numbers, use t he formula as given in Theorem \\ref\{thm:Main Result\}, first noting \+ that " }{XPPEDIT 18 0 "s;" "6#%\"sG" }{TEXT -1 15 " is equal to 1." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Top := egf/makeproc(top): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Bot := egf/makeproc(b ot):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "s := 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "m := 3:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "k := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "q := 1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Bernoulli[m * k + q] := 1/binomial(m*(s + k) + q, m * s) / Bot(m * s) * " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 77 " (Top(m *(k + s) + q) - add(binomial (m *(k + s) + q, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 79 " m *(j + s)) * Bot(m * j) * Bernoulli[m * ( k+s-j) + q], " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 " \+ j = 1+s .. k+s));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*Bernoul liG6#\"\"\"#!\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k \+ := 1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Bernoulli[m * k + q] := 1/ binomial(m*(s + k) + q, m * s) / Bot(m * s) * " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 " (Top(m *(k + s) + q) - add(bi nomial (m *(k + s) + q, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 " \+ m *(j + s)) * Bot(m * (j + s)) * Bernoulli[m * (k-j) \+ + q], " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " j = 1 .. k));" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*BernoulliG6#\"\"%#!\"\"\"#I" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "k := 2:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "B ernoulli[m * k + q] := 1/binomial(m*(s + k) + q, m * s) / Bot(m * s) * " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 " (Top(m *(k + s) + q) - add(binomial (m *(k + s) + q, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 " m *(j + s)) * Bot(m * (j + s) ) * Bernoulli[m * (k-j) + q], " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " \+ j = 1 .. k));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%*BernoulliG6#\"\"(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k := 3:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Bernoulli[m * k + q ] := 1/binomial(m*(s + k) + q, m * s) / Bot(m * s) * " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 " (Top(m *(k + s) + q) - a dd(binomial (m* (k + s) + q, " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 " \+ m *(j + s)) * Bot(m * (j + s)) * Bernoulli[m * ( k-j) + q], " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 " \+ j = 1 .. k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%*BernoulliG6# \"#5#\"\"&\"#m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "There is automated code to get the same result." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "A := calc ul/normal(10, Top, Bot, 3, 1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "seq(A[3 * i + 1], i = 0 ..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&#!\"\"\"\"##F\$\"#I\"\"!#\"\"&\"#m" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 2" 173 }{VIEWOPTS 1 1 0 1 1 1803 }