{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 "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 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "01." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "F:=binomial(n,k)*x^k*y^(n-k)/(x+y)^n;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "02." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "R:=k*y/((k-n-1)*(x+y));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "03." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ratk:=n ormal(expand(subs(k=k+1,F)/F));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "04." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ratn:=normal(expa nd(subs(n=n+1,F)/F));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "05." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "normal(ratn-1-(subs(k=k+1,R) *ratk-R));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "06." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sum(subs(n=0,F),k=0..0);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "07." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "F:=k*binomial(n,k)/(n*2^(n-1));" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 200 3 "08." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 " R:=(k-1)/(2*(k-n-1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "09." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ratk:=normal(expand(subs(k=k +1,F)/F));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "10." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ratn:=normal(expand(subs(n=n+1,F)/F ));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "11." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "normal(ratn-1-(subs(k=k+1,R)*ratk-R));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "12." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sum(subs(n=1,F),k=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "13." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "F:=bino mial(n,k)^2/binomial(2*n,n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "1 4." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "R:=k^2*(2*k-3*n-3)/(2 *(k-n-1)^2*(2*n+1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "15." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ratk:=normal(expand(subs(k=k +1,F)/F));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "16." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ratn:=normal(expand(subs(n=n+1,F)/F ));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "17." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 38 "normal(ratn-1-(subs(k=k+1,R)*ratk-R));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "18." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eval(sum(subs(n=0,F),k=0..0));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "19." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "F: =sumtools[hyperterm]([-n,a],[b],1,k)*pochhammer(b,n)/pochhammer(b-a,n) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "20." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "R:=k*(b+k-1)/((k-n-1)*(-a+b+n));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "21." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ratk:=normal(expand(subs(k=k+1,F)/F));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "22." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "ratn:=normal(expand(subs(n=n+1,F)/F));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "23." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "normal(ratn-1-(subs(k=k+1,R)*ratk-R));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "24." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eval(sum(subs(n=0,F),k=0..0));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 7 "25./26." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "PrimDispersion:=proc(q,r,k)\n" }{MPLTEXT 1 0 23 "local s,t,n,a,b,c ,d,j;\n" }{MPLTEXT 1 0 17 "s:=collect(q,k);\n" }{MPLTEXT 1 0 17 "t:=co llect(r,k);\n" }{MPLTEXT 1 0 16 "n:=degree(s,k);\n" }{MPLTEXT 1 0 29 " if n=0 or not(n=degree(t,k))\n" }{MPLTEXT 1 0 17 " then return []\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 17 "a:=coeff(s,k,n);\n" } {MPLTEXT 1 0 19 "b:=coeff(s,k,n-1);\n" }{MPLTEXT 1 0 17 "c:=coeff(t,k, n);\n" }{MPLTEXT 1 0 19 "d:=coeff(t,k,n-1);\n" }{MPLTEXT 1 0 30 "j:=no rmal((b*c-a*d)/(a*c*n));\n" }{MPLTEXT 1 0 33 "if not(type(j,integer) a nd j>=0)\n" }{MPLTEXT 1 0 17 " then return []\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 33 "if normal(c*s-a*subs(k=k+j,t))=0\n" } {MPLTEXT 1 0 18 " then return [j]\n" }{MPLTEXT 1 0 17 " else return \+ []\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 10 "end proc:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "DispersionsMenge:=proc(q,r,k)\n " }{MPLTEXT 1 0 38 "local f,g,m,n,i,j,result,tmp,op1,op2;\n" } {MPLTEXT 1 0 14 "f:=factor(q);\n" }{MPLTEXT 1 0 14 "g:=factor(r);\n" } {MPLTEXT 1 0 16 "if type(f,`*`) \n" }{MPLTEXT 1 0 18 " then m:=nops(f )\n" }{MPLTEXT 1 0 12 " else m:=1\n" }{MPLTEXT 1 0 8 "end if;\n" } {MPLTEXT 1 0 16 "if type(g,`*`) \n" }{MPLTEXT 1 0 18 " then n:=nops(g )\n" }{MPLTEXT 1 0 12 " else n:=1\n" }{MPLTEXT 1 0 8 "end if;\n" } {MPLTEXT 1 0 12 "result:=[];\n" }{MPLTEXT 1 0 21 "for i from 1 to m do \n" }{MPLTEXT 1 0 18 " if type(f,`*`) \n" }{MPLTEXT 1 0 22 " then \+ op1:=op(i,f)\n" }{MPLTEXT 1 0 16 " else op1:=f\n" }{MPLTEXT 1 0 10 " end if;\n" }{MPLTEXT 1 0 19 " if type(op1,`^`)\n" }{MPLTEXT 1 0 24 " then op1:=op(1,op1)\n" }{MPLTEXT 1 0 10 " end if;\n" } {MPLTEXT 1 0 23 " for j from 1 to n do\n" }{MPLTEXT 1 0 20 " if ty pe(g,`*`) \n" }{MPLTEXT 1 0 24 " then op2:=op(j,g)\n" }{MPLTEXT 1 0 18 " else op2:=g\n" }{MPLTEXT 1 0 12 " end if;\n" } {MPLTEXT 1 0 21 " if type(op2,`^`)\n" }{MPLTEXT 1 0 26 " then \+ op2:=op(1,op2)\n" }{MPLTEXT 1 0 12 " end if;\n" }{MPLTEXT 1 0 36 " \+ tmp:=PrimDispersion(op1,op2,k);\n" }{MPLTEXT 1 0 15 " if tmp<>[] \n" }{MPLTEXT 1 0 39 " then result:=[op(result),tmp[1]]\n" } {MPLTEXT 1 0 12 " end if;\n" }{MPLTEXT 1 0 10 " end do;\n" } {MPLTEXT 1 0 8 "end do;\n" }{MPLTEXT 1 0 13 "\{op(result)\}\n" } {MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 28 "G radSchranke:=proc(A,B,C,k)\n" }{MPLTEXT 1 0 31 "local pol1,pol2,deg1,d eg2,a,b;\n" }{MPLTEXT 1 0 22 "pol1:=collect(A-B,k);\n" }{MPLTEXT 1 0 22 "pol2:=collect(A+B,k);\n" }{MPLTEXT 1 0 11 "if pol1=0 \n" } {MPLTEXT 1 0 16 " then deg1:=-1\n" }{MPLTEXT 1 0 28 " else deg1:=deg ree(pol1,k)\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 10 "if pol2=0 \n" }{MPLTEXT 1 0 16 " then deg2:=-1\n" }{MPLTEXT 1 0 29 " else deg2 :=degree(pol2,k) \n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 15 "if \+ deg1<=deg2 \n" }{MPLTEXT 1 0 32 " then return degree(C,k)-deg2;\n" } {MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 23 "a:=coeff(pol1,k,deg1);\n " }{MPLTEXT 1 0 15 "if deg2=0)\n " }{MPLTEXT 1 0 34 " then return degree(C,k)-deg1+1;\n" }{MPLTEXT 1 0 46 " else return max(-2*b/a,degree(C,k)-deg1+1);\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 23 "REtoPol:=proc(A,B,C,k)\n" }{MPLTEXT 1 0 32 "local deg ,g,a,j,rec,sol,result;\n" }{MPLTEXT 1 0 28 "deg:=GradSchranke(A,B,C,k) ;\n" }{MPLTEXT 1 0 9 "if deg<0\n" }{MPLTEXT 1 0 37 " then return \"k eine Polynoml\366sung\"\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 27 "g:=sum(a[j]*k^j,j=0..deg);\n" }{MPLTEXT 1 0 39 "rec:=collect(A*sub s(k=k+1,g)+B*g-C,k);\n" }{MPLTEXT 1 0 50 "sol:=solve(\{coeffs(rec,k)\} ,\{seq(a[j],j=0..deg)\});\n" }{MPLTEXT 1 0 12 "if sol=NULL\n" } {MPLTEXT 1 0 36 " then return \"keine Polynoml\366sung\"\n" } {MPLTEXT 1 0 56 " else result:=applyrule(a(m::anything)=0,subs(sol,g) );\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 10 "end proc:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 33 "DiskreteStammfunktion:=proc(a,k) \n" }{MPLTEXT 1 0 36 "local u,v,rat,M,dis,g,h,GCD,A,B,CC;\n" } {MPLTEXT 1 0 38 "rat:=normal(expand(subs(k=k+1,a)/a));\n" }{MPLTEXT 1 0 15 "u:=numer(rat);\n" }{MPLTEXT 1 0 15 "v:=denom(rat);\n" }{MPLTEXT 1 0 68 "if not(type(u,polynom(anything,k)) and type(v,polynom(anything ,k)))\n" }{MPLTEXT 1 0 59 " then return \"Eingabe ist kein hypergeo metrischer Term\"\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 40 "M:= DispersionsMenge(subs(k=k-1,u),v,k);\n" }{MPLTEXT 1 0 17 "dis:=max(op( M));\n" }{MPLTEXT 1 0 38 "h:=gcd(mul(subs(k=k-1-j,u),j=0..dis),\n" } {MPLTEXT 1 0 37 " mul(subs(k=k+j,v),j=0..dis));\n" }{MPLTEXT 1 0 9 "if dis=0\n" }{MPLTEXT 1 0 35 " then A:=normal(u/subs(k=k+1,h)); \n" }{MPLTEXT 1 0 24 " B:=-normal(v/h);\n" }{MPLTEXT 1 0 14 " \+ CC:=v;\n" }{MPLTEXT 1 0 15 " else A:=h*u;\n" }{MPLTEXT 1 0 28 " \+ B:=-subs(k=k+1,h)*v;\n" }{MPLTEXT 1 0 30 " CC:=h*subs(k=k+1 ,h)*v;\n" }{MPLTEXT 1 0 9 "end if; \n" }{MPLTEXT 1 0 23 "GCD:=gcd(gcd( A,B),CC);\n" }{MPLTEXT 1 0 18 "A:=normal(A/GCD);\n" }{MPLTEXT 1 0 18 " B:=normal(B/GCD);\n" }{MPLTEXT 1 0 20 "CC:=normal(CC/GCD);\n" } {MPLTEXT 1 0 3 "g:=" }{MPLTEXT 1 0 19 "REtoPol(A,B,CC,k);\n" } {MPLTEXT 1 0 27 "if g=\"keine Polynoml\366sung\"\n" }{MPLTEXT 1 0 51 " then return \"Eingabe ist nicht Gospersummierbar\"\n" }{MPLTEXT 1 0 14 " else g/h*a;\n" }{MPLTEXT 1 0 8 "end if;\n" }{MPLTEXT 1 0 10 "end proc:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 27 "WZCertificate:=proc(F ,k,n)\n" }{MPLTEXT 1 0 11 "local a,G;\n" }{MPLTEXT 1 0 20 "a:=subs(n=n +1,F)-F;\n" }{MPLTEXT 1 0 31 "G:=DiskreteStammfunktion(a,k);\n" } {MPLTEXT 1 0 21 "normal(expand(G/F));\n" }{MPLTEXT 1 0 9 "end proc:" } }}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "27." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "WZCertificate(binomial(n,k)*x^k*y^(n-k)/(x+y)^n,k,n );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "28." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 47 "WZCertificate(k*binomial(n,k)/(n*2^(n-1)),k,n) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "29." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "WZCertificate(binomial(n,k)^2/binomial(2*n,n),k, n);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 3 "30." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 89 "WZCertificate(sumtools[hyperterm]([-n,a],[b],1 ,k)*pochhammer(b,n)/pochhammer(b-a,n),k,n);" }}}}{MARK "1 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }