read qEKHAD:
print(``):
print(`L( b(n,k)c(n,k) ) = c(n,k)B(n,k)-c(n,k-1)B(n,k-1) + H(n,k); where`):
print(`H(n,k)=y_1(n)b(n+1,k)G(n,k)+ y_2(n)b(n+2,k)[G(n,k)+G(n+1,k)] -F(n,k-1)B(n,k-1)`):
print(`( use: F(n.k)=c(n,k+1)-c(n,k), G(n,k)=c(n+1,k)-c(n,k) )`):   
print(`Thus to prove L( b(n,k)c(n,k) ) = A(n,k)-A(n,k-1), we denote the second term`):
print(`in A(n,k) by D(n,k) and show that H(n,k) = D(n,k) - D(n,k-1).`):
print(``):

common:=(-1)^k*q^(k*(k-1)/2)*qbin(n+2,k):

b:= q^(k*(k+1)/2)*qbin(n+k,k)*qbin(n,k): 
gu:=qzeil(b,N,k,n,[]):
y1:=coeff(gu[1],N,1):
y2:=coeff(gu[1],N,2)*denom(y1):
P:=gu[2]*denom(y1)/(-1+q^k*q^n*q)*(-q^k+q^n*q)/q^k:
y1:=coeff(gu[1],N,1)*denom(y1):
Q:=q^(n+1)/(1+q^(n+1)):

h1:=q^k*y1*Q*(1-q^(n-k+2))*(1+q^(n+1))/(1-q^(n+2)):

h2:=y2*q^k*((1-q^(n+k+2))*Q*(1+q^(n+1))/(1-q^(n+2))+subs(n=n+1,Q)*(1-q^(n+1))/(1-q^(n+2))):

h3:=subs(k=k-1,P)*(1-q^k)*(1+q^(n+1))/(1-q^(n+2)):

H:=common*simplify(h1+h2+h3):

print(`Regrouping the terms in H(n,k), we get:`):

print(`H(n,k) = `, H):
print(`may set the constant factor to 1`):
print(``):
gu1:=qzeilsp(H,0,N,k,n,[]):

print(`while D(n,k) =  -(q)_n+1/(q^2)_n+1`, gu1[2]*common/(q^(n+2)-1)/(1+q^n*q^2)/(1-q^(n+1))):

print(`may set the constant factor to 1`):
print(`This took`):
print(time()):
print(`Seconds of CPU time`):

quit: