{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 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 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "01." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 244 "GradSchranke:=proc(A,B,C,k)\nlocal pol1,pol2, deg1,deg2,a,b;\npol1:=collect(A-B,k);\npol2:=collect(A+B,k);\nif pol1= 0 \n then deg1:=-1\n else deg1:=degree(pol1,k)\nend if;\nif pol2=0\n then deg2:=-1\n else deg2:=degree(pol2,k) \nend if;\nif deg1<=deg2 \+ \n" }{MPLTEXT 1 0 203 " then return degree(C,k)-deg2;\nend if;\na:=co eff(pol1,k,deg1);\nif deg2=0)\n then ret urn degree(C,k)-deg1+1;\n" }{MPLTEXT 1 0 166 " else return max(-2*b/a ,degree(C,k)-deg1+1);\nend if;\nend proc:\n\nPrimDispersion:=proc(q,r, k)\nlocal s,t,n,a,b,c,d,j;\ns:=collect(q,k);\nt:=collect(r,k);\nn:=deg ree(s,k);\n" }{MPLTEXT 1 0 189 "if n=0 or not(n=degree(t,k))\n then r eturn []\nend if;\na:=coeff(s,k,n);\nb:=coeff(s,k,n-1);\nc:=coeff(t,k, n);\nd:=coeff(t,k,n-1);\nj:=normal((b*c-a*d)/(a*c*n));\nif not(type(j, integer) and j>=0)\n" }{MPLTEXT 1 0 142 " then return []\nend if;\nif normal(c*s-a*subs(k=k+j,t))=0\n then return [j]\n else return []\ne nd if;\nend proc:\n\nDispersionsMenge:=proc(q,r,k)\n" }{MPLTEXT 1 0 154 "local f,g,m,n,i,j,result,tmp,op1,op2;\nf:=factor(q);\ng:=factor(r );\nif type(f,`*`) \n then m:=nops(f)\n else m:=1\nend if;\nif type( g,`*`) \n then n:=nops(g)\n" }{MPLTEXT 1 0 138 " else n:=1\nend if; \nresult:=[];\nfor i from 1 to m do\n if type(f,`*`) \n then op1:= op(i,f)\n else op1:=f\n end if;\n if type(op1,`^`)\n" }{MPLTEXT 1 0 178 " then op1:=op(1,op1)\n end if;\n for j from 1 to n do\n \+ if type(g,`*`) \n then op2:=op(j,g)\n else op2:=g\n en d if;\n if type(op2,`^`)\n then op2:=op(1,op2)\n" }{MPLTEXT 1 0 154 " end if;\n tmp:=PrimDispersion(op1,op2,k);\n if tmp<>[ ]\n then result:=[op(result),tmp[1]]\n end if;\n end do;\nend do;\n\{op(result)\}\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 462 "SumRekursion:=proc(F,k,n,S)\nlocal a,sigma,ratk,ratn,rat,nenn er,u,v,M,dis,h,j,J,deg,rec,sol,A,B,CC,GCD,alpha,g,RE;\nRE:=[];\nratk:= normal(expand(subs(k=k+1,F)/F));\nratn:=normal(expand(subs(n=n+1,F)/F) );\nfor J from 1 to 5 do\n nenner:=1+add(sigma[j]*mul(subs(n=n+i,ratn ),i=0..j-1),j=1..J);\n rat:=normal(ratk*subs(k=k+1,nenner)/nenner);\n u:=numer(rat);\n v:=denom(rat);\n if not(type(u,polynom(anything,k )) and type(v,polynom(anything,k)))\n then break\n end if;\n" } {MPLTEXT 1 0 247 " M:=DispersionsMenge(subs(k=k-1,u),v,k);\n dis:=ma x(op(M));\n h:=gcd(mul(subs(k=k-1-j,u),j=0..dis),\n mul(subs( k=k+j,v),j=0..dis));\n if dis=0\n then A:=normal(u/subs(k=k+1,h)); \n B:=-normal(v/h);\n CC:=v;\n else A:=h*u;\n" } {MPLTEXT 1 0 203 " B:=-subs(k=k+1,h)*v;\n CC:=h*subs(k =k+1,h)*v;\n end if; \n GCD:=gcd(gcd(A,B),CC);\n A:=normal(A/GCD); \n B:=normal(B/GCD);\n CC:=normal(CC/GCD);\n deg:=GradSchranke(A,B, CC,k);\n if deg>=0\n" }{MPLTEXT 1 0 396 " then g:=sum(alpha[j]*k^j ,j=0..deg);\n rec:=collect(expand(A*subs(k=k+1,g)+B*g-CC),k); \n sol:=solve(\{coeffs(rec,k)\},\{seq(alpha[j],j=0..deg)\} uni on \{seq(sigma[j],j=1..J)\});\n if sol<>NULL\n then \+ RE:=S(n)+add(subs(sol,sigma[j]*S(n+j)),j=1..J);\n RE:=n umer(normal(RE));\n RE:=collect(RE,S,factor);\n \+ break;\n end if:\n" }{MPLTEXT 1 0 256 " end if:\nend \+ do;\nif not(type(u,polynom(anything,k)) and type(v,polynom(anything,k) ))\n then return \"Eingabe ist kein hypergeometrischer Term\"\n else if RE=[]\n then \"Es gibt keine Rekursion der Ordnung 5\"\n \+ else RE=0\n end if: \nend if;\n" }{MPLTEXT 1 0 9 "end pro c:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "02." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 34 "SumRekursion(binomial(n,k),k,n,S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"#\"\"\"-%\"SG6#%\"nGF'F'-F)6#,&F+F' F'F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "03." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "SumRekursion(binomial(n,k)^2,k,n,S) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&*&\"\"%\"\"\"%\"nGF)F)\"\" #F)F)-%\"SG6#F*F)F)*&,&F*!\"\"F)F1F)-F-6#,&F*F)F)F)F)F)\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "04." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "SumRekursion((-1)^k*binomial(n,k)^2,k,n,S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&,&*&\"\"%\"\"\"%\"nGF)F)F(F)F)-%\"SG6#F *F)F)*&,&F*F)\"\"#F)F)-F,6#F/F)F)\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "05." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "SumReku rsion(binomial(n,k)^3,k,n,S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*( \"\")\"\"\"),&%\"nGF'F'F'\"\"#F'-%\"SG6#F*F'F'*&,(*&\"\"(F')F*F+F'F'*& \"#@F'F*F'F'\"#;F'F'-F-6#F)F'F'*&),&F*F'F+F'F+F'-F-6#F;F'!\"\"\"\"!" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "06." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 36 "SumRekursion(binomial(n,k)^4,k,n,S);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,(*,\"\"%\"\"\",&*&F&F'%\"nGF'F'\"\"&F'F',&*&F& F'F*F'F'\"\"$F'F',&F*F'F'F'F'-%\"SG6#F*F'F'**\"\"#F',&*&F4F'F*F'F'F.F' F',(*&F.F')F*F4F'F'*&\"\"*F'F*F'F'\"\"(F'F'-F16#F/F'F'*&),&F*F'F4F'F.F '-F16#FAF'!\"\"\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "07." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sumtools[sumrecursion](binom ial(n,k)^3,k,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\")\"\"\" ),&%\"nGF&F&!\"\"\"\"#F&-%\"SG6#,&F)F&F+F*F&F**&,(*&\"\"(F&)F)F+F&F&*& F3F&F)F&F*F+F&F&-F-6#F(F&F**&-F-6#F)F&F4F&F&" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 200 3 "08." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sumto ols[sumrecursion](binomial(n,k)^4,k,S(n));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*,\"\"%\"\"\",&\"\"&!\"\"*&F%F&%\"nGF&F&F&,&F+F&F&F)F &,&\"\"$F)*&F%F&F+F&F&F&-%\"SG6#,&F+F&\"\"#F)F&F)**F4F&,&*&F4F&F+F&F&F &F)F&,(*&F.F&)F+F4F&F&*&F.F&F+F&F)F&F&F&-F16#F,F&F)*&-F16#F+F&)F+F.F&F &" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "09." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "sumtools[sumrecursion](binomial(n,k)^5,k,S(n)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,***\"#K\"\"\",(*&\"#bF&)%\"nG\"\" #F&F&*&\"#xF&F+F&!\"\"\"#GF&F&),&F+F&F,F/\"\"%F&-%\"SG6#,&F+F&\"\"$F/F &F&*&,0*&\"&:%>F&)F+\"\"'F&F&*&\"'rO9F&)F+\"\"&F&F/*&\"'$eM%F&)F+F3F&F &*&\"'LmoF&)F+F8F&F/*&\"'IpfF&F*F&F&*&\"'/2FF&F+F&F/\"&w,&F&F&-F56#F2F &F/*&,0*&\"%b6F&F=F&F&*&\"%PiF&FAF&F/*&\"&GJ\"F&FEF&F&*&\"&dR\"F&FHF&F /*&\"%$>)F&F*F&F&*&\"%iDF&F+F&F/\"$O$F&F&-F56#,&F+F&F&F/F&F/*(-F56#F+F &FEF&,(*&F)F&F*F&F&*&\"$(=F&F+F&F/\"$g\"F&F&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "10." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "Su mRekursion(binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k,n,P);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*(%\"xG\"\"\",&*&\"\"#F'%\"nGF'F'\" \"$F'F'-%\"PG6#,&F+F'F'F'F'!\"\"*&,&F+F'F*F'F'-F.6#F3F'F'*&F0F'-F.6#F+ F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "11." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "SumRekursion((-1)^k/k!*binomial(n+a lpha,n-k)*x^k,k,n,L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&,(%\"nG \"\"\"F(F(%&alphaGF(F(-%\"LG6#F'F(F(*&,*F)!\"\"\"\"$F/*&\"\"#F(F'F(F/% \"xGF(F(-F+6#,&F'F(F(F(F(F(*&,&F'F(F2F(F(-F+6#F8F(F(\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }