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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3829 "restart:\nwith(Pol
ynomialTools):\nwith(Statistics):\nw:=PDF(RandomVariable(Uniform(-1,1)
),x) assuming x>0, x<1:\nassume(n,integer,n>1,alpha,real,alpha>0):\n# \+
Legendre\nR:=n*Pn=(2*n-1)*xPn_1+0*Pn_1-(n-1)*Pn_2:\nnormF:=sqrt(1/(2*n
+1)):\nS:=subs(Pn=normF*Qn,Pn_1=subs(n=n-1,normF)*Qn_1,xPn_1=subs(n=n-
1,normF)*xQn_1,Pn_2=subs(n=n-2,normF)*Qn_2,R):\nmap(_x->factor(simplif
y(convert(_x,GAMMA),symbolic)),[coeff(rhs(S),xQn_1)/coeff(lhs(S),Qn),c
oeff(rhs(S),Qn_1)/coeff(lhs(S),Qn),coeff(rhs(S),Qn_2)/coeff(lhs(S),Qn)
]):\nC0:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=0,1/normF
*Vector([1]))):\nC1:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs
(n=1,1/normF*Vector([0,1]))):\nU:=C0[1]:V:=C1[1]+C1[2]*x:\nlegendreFac
tory:=evalf(U):\nlegendreFactory:=legendreFactory,evalf(V):\nfor i fro
m 2 to 9 do\n  W:=map(factor,collect(subs(n=i,xQn_1=x*V,Qn_1=V,Qn_2=U,
solve(S,Qn)),x)):\n  legendreFactory:=legendreFactory,evalf(W);\n  U:=
V:\n  V:=W:\nod:\nw:=PDF(RandomVariable(Normal(0,1)),x):\n# Hermite\nR
:=Pn=xPn_1-(n-1)*Pn_2:\nnormF:=sqrt(GAMMA(n+1)):\nS:=subs(Pn=normF*Qn,
Pn_1=subs(n=n-1,normF)*Qn_1,xPn_1=subs(n=n-1,normF)*xQn_1,Pn_2=subs(n=
n-2,normF)*Qn_2,R):\nmap(_x->factor(simplify(convert(_x,GAMMA),symboli
c)),[coeff(rhs(S),xQn_1)/coeff(lhs(S),Qn),coeff(rhs(S),Qn_1)/coeff(lhs
(S),Qn),coeff(rhs(S),Qn_2)/coeff(lhs(S),Qn)]):\nC0:=map(_x->simplify(c
onvert(_x,GAMMA),symbolic),subs(n=0,1/normF*Vector([1]))):\nC1:=map(_x
->simplify(convert(_x,GAMMA),symbolic),subs(n=1,1/normF*Vector([0,1]))
):\nU:=C0[1]:\nV:=C1[1]+C1[2]*x:\nhermiteFactory:=evalf(U):\nhermiteFa
ctory:=hermiteFactory,evalf(V):\nfor i from 2 to 9 do\n  W:=map(factor
,collect(subs(n=i,xQn_1=x*V,Qn_1=V,Qn_2=U,solve(S,Qn)),x)):\n  hermite
Factory:=hermiteFactory,evalf(W);\n  U:=V:\n  V:=W:\nod:\nw:=PDF(Rando
mVariable(Gamma(1,alpha+1)),x):\n# Laguerre\nR:=n*Pn=-xPn_1+(2*n-1+alp
ha)*Pn_1-(n-1+alpha)*Pn_2:\nnormF:=sqrt(GAMMA(n+alpha+1)/GAMMA(alpha+1
)/GAMMA(n+1)):\nS:=subs(Pn=normF*Qn,Pn_1=subs(n=n-1,normF)*Qn_1,xPn_1=
subs(n=n-1,normF)*xQn_1,Pn_2=subs(n=n-2,normF)*Qn_2,R):\nmap(_x->facto
r(simplify(convert(_x,GAMMA),symbolic)),[coeff(rhs(S),xQn_1)/coeff(lhs
(S),Qn),coeff(rhs(S),Qn_1)/coeff(lhs(S),Qn),coeff(rhs(S),Qn_2)/coeff(l
hs(S),Qn)]):\nC0:=map(_x->simplify(convert(_x,GAMMA),symbolic),subs(n=
0,1/normF*Vector([1]))):\nC1:=map(_x->simplify(convert(_x,GAMMA),symbo
lic),subs(n=1,1/normF*Vector([1+alpha,-1]))):\nalpha_num:=2.5:\nU:=C0[
1]:\nV:=C1[1]+C1[2]*x:\nlaguerreFactory:=evalf(subs(alpha=alpha_num,U)
):\nlaguerreFactory:=laguerreFactory,evalf(subs(alpha=alpha_num,V)):\n
for i from 2 to 9 do\n  W:=map(_x->factor(simplify(_x,symbolic)),colle
ct(subs(n=i,xQn_1=x*V,Qn_1=V,Qn_2=U,alpha=alpha_num,solve(S,Qn)),x)):
\n  laguerreFactory:=laguerreFactory,evalf(W);\n  U:=V:\n  V:=W:\nod:
\np:=<0.5|0.5|0.5>:\n# Expand the enumerate function\nevalf(subs(x=p[1
],[laguerreFactory][1])*subs(x=p[2],[legendreFactory][1])*subs(x=p[3],
[hermiteFactory][1]));\nevalf(subs(x=p[1],[laguerreFactory][2])*subs(x
=p[2],[legendreFactory][1])*subs(x=p[3],[hermiteFactory][1]));\nevalf(
subs(x=p[1],[laguerreFactory][1])*subs(x=p[2],[legendreFactory][2])*su
bs(x=p[3],[hermiteFactory][1]));\nevalf(subs(x=p[1],[laguerreFactory][
1])*subs(x=p[2],[legendreFactory][1])*subs(x=p[3],[hermiteFactory][2])
);\nevalf(subs(x=p[1],[laguerreFactory][3])*subs(x=p[2],[legendreFacto
ry][1])*subs(x=p[3],[hermiteFactory][1]));\nevalf(subs(x=p[1],[laguerr
eFactory][2])*subs(x=p[2],[legendreFactory][2])*subs(x=p[3],[hermiteFa
ctory][1]));\nevalf(subs(x=p[1],[laguerreFactory][2])*subs(x=p[2],[leg
endreFactory][1])*subs(x=p[3],[hermiteFactory][2]));\nevalf(subs(x=p[1
],[laguerreFactory][1])*subs(x=p[2],[legendreFactory][3])*subs(x=p[3],
[hermiteFactory][1]));\nevalf(subs(x=p[1],[laguerreFactory][1])*subs(x
=p[2],[legendreFactory][2])*subs(x=p[3],[hermiteFactory][2]));\nevalf(
subs(x=p[1],[laguerreFactory][1])*subs(x=p[2],[legendreFactory][1])*su
bs(x=p[3],[hermiteFactory][3]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
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