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\deflang1031\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}PrimDispersion:=proc(q,r,k)
\par \pard\li600\ri1\fi-300\plain\f5\fs22\cf1 local s,t,n,a,b,c,d,j;
\par begin
\par s:=collect(q,k);
\par t:=collect(r,k);
\par n:=degree(s,k);
\par if n=0 or not(n=degree(t,k))
\par then return([])
\par end_if;
\par a:=coeff(s,k,n);
\par b:=coeff(s,k,n-1);
\par c:=coeff(t,k,n);
\par d:=coeff(t,k,n-1);
\par j:=normal((b*c-a*d)/(a*c*n));
\par if not(testtype(j,DOM_INT) and j>=0)
\par then return([])
\par end_if;
\par if normal(c*s-a*subs(t,k=k+j))=0
\par then return([j])
\par else return([])
\par end_if;
\par end_proc:
\par
\par DispersionsMenge:=proc(q,r,k)
\par local f,g,m,n,i,j,result,tmp,op1,op2;
\par begin
\par f:=factor(q);
\par g:=factor(r);
\par m:=(nops(f)-1)/2;
\par n:=(nops(g)-1)/2;
\par result:=[];
\par for i from 1 to m do
\par if m>1
\par then op1:=op(f,2*i)
\par else op1:=op(f,2)
\par end_if;
\par if testtype(op1,"_power")
\par then op1:=op(op1,1)
\par end_if;
\par for j from 1 to n do
\par if n>1
\par then op2:=op(g,2*j)
\par else op2:=op(g,2)
\par end_if;
\par if testtype(op2,"_power")
\par then op2:=op(op2,1)
\par end_if;
\par tmp:=PrimDispersion(op1,op2,k);
\par if not(tmp=[])
\par then result:=[op(result),tmp[1]]
\par end_if;
\par end_for;
\par end_for;
\par \{op(result)\}
\par end_proc:
\par
\par GradSchranke:=proc(A,B,C,k)
\par local pol1,pol2,deg1,deg2,a,b;
\par begin
\par pol1:=collect(A-B,k);
\par pol2:=collect(A+B,k);
\par if pol1=0
\par then deg1:=-1
\par else deg1:=degree(pol1,k)
\par end_if;
\par if pol2=0
\par then deg2:=-1
\par else deg2:=degree(pol2,k)
\par end_if;
\par if deg1<=deg2
\par then return(degree(C,k)-deg2);
\par end_if;
\par a:=coeff(pol1,k,deg1);
\par if deg2=0)
\par then return(degree(C,k)-deg1+1);
\par else return(max(-2*b/a,degree(C,k)-deg1+1));
\par end_if;
\par end_proc:
\par
\par REtoPol:=proc(A,B,C,k)
\par local deg,g,j,rec,sol;
\par begin
\par deg:=GradSchranke(A,B,C,k);
\par if deg<0
\par then return("keine Polynoml\'f6sung")
\par end_if;
\par g:=_plus(aa[j]*k^j $ j=0..deg);
\par rec:=collect(A*subs(g,k=k+1)+B*g-C,k);
\par sol:=solve(\{coeff(rec,k,i) $ i=0..degree(rec,k)\},\{aa[j] $ j=0..deg\},IgnoreSpecialCases);
\par if sol=NIL or sol=\{\}
\par then return("keine Polynoml\'f6sung")
\par else subs(g,sol[1]);
\par end_if;
\par end_proc:
\par \pard\ri4\plain\f3\fs22\cf0 01.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}SumRekursion:=proc(F,k,n,S)
\par \pard\li600\ri1\fi-300\plain\f5\fs22\cf1 local a,ratk,ratn,rat,nenner,u,v,M,dis,
\par h,j,J,deg,rec,sol,A,B,CC,GCD,g,RE;
\par begin
\par RE:=[];
\par ratk:=normal(expand(subs(F,k=k+1)/F));
\par ratn:=normal(expand(subs(F,n=n+1)/F));
\par for J from 1 to 5 do
\par nenner:=1+_plus(sigma[j]*_mult(subs(ratn,n=n+i)
\par $ i=0..j-1) $ j=1..J);
\par rat:=normal(ratk*subs(nenner,k=k+1)/nenner);
\par u:=numer(rat);
\par v:=denom(rat);
\par if not(denom(u)=1 and denom(v)=1)
\par then break
\par end_if;
\par M:=DispersionsMenge(subs(u,k=k-1),v,k);
\par dis:=max(op(M));
\par h:=gcd(_mult(subs(u,k=k-1-j) $ j=0..dis),
\par _mult(subs(v,k=k+j) $ j=0..dis));
\par if dis=0
\par then A:=normal(u/subs(h,k=k+1));
\par B:=-normal(v/h);
\par CC:=v;
\par else A:=h*u;
\par B:=-subs(h,k=k+1)*v;
\par CC:=h*subs(h,k=k+1)*v;
\par end_if;
\par GCD:=gcd(gcd(A,B),CC);
\par A:=normal(A/GCD);
\par B:=normal(B/GCD);
\par CC:=normal(CC/GCD);
\par deg:=GradSchranke(A,B,CC,k);
\par if deg>=0
\par then g:=_plus(alpha[j]*k^j $ j=0..deg);
\par rec:=collect(expand(A*subs(g,k=k+1)+B*g-
\par CC),k);
\par sol:=solve(\{coeff(rec,k,i) $ i=
\par 0..degree(rec,k)\},\{alpha[j] $ j=
\par 0..deg\} union \{sigma[j] $ j=
\par 1..J\},IgnoreSpecialCases);
\par if not(sol=NIL) and not(sol=\{\})
\par then RE:=S(n)+_plus(subs(sigma[j]*
\par S(n+j),sol[1]) $ j=1..J);
\par RE:=numer(normal(RE));
\par RE:=collect(RE,[S(n+j) $ j=
\par 0..J],factor);
\par break;
\par end_if:
\par end_if:
\par end_for;
\par if not(denom(u)=1 and denom(v)=1)
\par then return("Eingabe ist kein hypergeometrischer
\par Term")
\par else if RE=[]
\par then "Es gibt keine Rekursion der Ordnung
\par 5"
\par else RE=0
\par end_if:
\par end_if;
\par end_proc:
\par \pard\ri4\plain\f3\fs22\cf0 02.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}SumRekursion(binomial(n,k),k,n,S)
\par \pard\ri4\plain\f3\fs22\cf0 03.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}SumRekursion(binomial(n,k)^2,k,n,S)
\par \pard\ri4\plain\f3\fs22\cf0 04.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}SumRekursion((-1)^k*binomial(n,k)^2,k,n,S)
\par \pard\ri4\plain\f3\fs22\cf0 05.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}SumRekursion(binomial(n,k)^3,k,n,S)
\par \pard\ri4\plain\f3\fs22\cf0 06.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}SumRekursion(binomial(n,k)^4,k,n,S)
\par \pard\ri4\plain\f3\fs22\cf0 07.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}
\par \pard\ri4\plain\f3\fs22\cf0 08.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}
\par \pard\ri4\plain\f4\fs22\cf0 09.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}
\par \pard\ri4\plain\f4\fs22\cf0 10.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}SumRekursion(binomial(n,k)*binomial(-n-1,k)*((1-x)/2)^k,k,n,P)
\par \pard\ri4\plain\f4\fs22\cf0 11.
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f5\fs22\cf1 {\pntext\f1\'b7\tab}SumRekursion((-1)^k/k!*binomial(n+a,n-k)*x^k,k,n,L)
\par }