\mnb150ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\froman\fcharset1 Times New Roman;}{\f4\froman\fcharset1 Times New Roman;}{\f5\fmodern\fprq1 Courier New;}} {\colortbl\red0\green0\blue0;\red255\green0\blue0;} \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 }