# G6memo.txt   20221111
# define S10 as 
S10:= b0+1/3*s11-3*b10*s21+27/2*b16*s31-3*b13*s22+27/2*b22*s32-7/3*s13
    +(-2-3*b15)*s23+(-3/2+27/2*b25)*s33
    +b10*s11^2+1/3*s11*s13+b13*s12^2+b15*s13^2-9/2*b16*s11*s21
    -9/2*b22*s12*s22-1/3*s13*s21+1/3*s13*s22
    +(1/6-9/2*b25)*s13*s23+b16*s11^3+1/9*s11^2*s13+b22*s12^3
    -1/9*s12^2*s13 +1/3*s11*s23+b25*s13^3:
# T10(e,a) for the equation G6 is defined as    
T10:=subs(par,S10):
# where
  par:={b0=a0-5/2,b10=a1+1/6,b13=a2-1/6,b15=a3+1/6,
          b16=a4+1/27,b22=a5-1/27,b25=a6}:
# E6=subs({a0=0,a1=0,a2=0,a3=0,a4=0,a5=0,a6=0},G6):

Then the PQ operators are given as
  P123n:=(-1+x)*dx-r:
  Q123n:=(-1+x)*dx+3-r:                 
  P456n:=x*dx-r:                 
  Q456n:=3-r+x*dx:                 
  P16n79p:=dx:                 
  Q16n79p:=dx:
where 123n,456n,16n79p denote the shifts [-00], [0-0], [--+],respectively.
Svalues are given as
S123np:=r*(-1+r)*(r-2)*(-r+e6)*(-r+e5)*(-r+e4),
S123pn:=r*(-1+r)*(r+1)*(-r+e6-1)*(-r-1+e5)*(-r+e4-1),
S456np:=-r*(-1+r)*(r-2)*(-r+e3)*(-r+e2)*(-r+e1),
S456pn:=-r*(-1+r)*(r+1)*(-r+e3-1)*(-r+e2-1)*(-r-1+e1),
S16n79p:=r*(-1+r)*(r+1)*(e8-1)*(e7-1)*(e9-1),
S16p79n:=r*e7*e8*e9*(-1+r)*(r-2),
which are defined to be the composition of shifted P123p and P123n,
shifted P123n and P123p, shifted P456p and P456n,
shifted P456n and P456p, P16n79p and P16p79n, and, P16p79n and P16n79p,
respectively.

The inverse operators P123p,P456p,P16p79n are given in G6NPQSdata.txt.

Let P123p0 the constant term relative to a0,...,a6: then, we have
P123p= P123p0 +a0*P123pa0 +a1*P123pa1 +a2*P123pa2 +a3*P123pa3
          +a4*P123pa4 +a5*P123pa5 +a6*P123pa6
where
P123pa0:=x*(-1+x)^3*dx^2-(-1+r)*(-1+x)^3*dx:                         
P123pa1:=(-1+x)^3*x*(-e1*e2+e1^2-e2*e3+e3^2+e2^2-e1*e3)*dx^2
          -(-1+r)*(-1+x)^3*(-e1*e2+e1^2-e2*e3+e3^2+e2^2-e1*e3)*dx:  
P123pa2:=(-1+x)^3*x*(e5^2-e4*e5-e5*e6+e6^2+e4^2-e4*e6)*dx^2
          -(-1+r)*(-1+x)^3*(e5^2-e4*e5-e5*e6+e6^2+e4^2-e4*e6)*dx:  
P123pa3:=(-1+x)^3*x*(e8^2+e9^2+e7^2-e7*e8-e7*e9-e8*e9)*dx^2
          -(-1+r)*(-1+x)^3*(e8^2+e9^2+e7^2-e7*e8-e7*e9-e8*e9)*dx:  
P123pa4:=1/2*(-1+x)^3*x*(-2*e2+e1+e3)*(e2+e1-2*e3)*(-e2+2*e1-e3)*dx^2
          -1/2*(-1+r)*(-1+x)^3*(-2*e2+e1+e3)*(e2+e1-2*e3)*(-e2+2*e1-e3)*dx:  
P123pa5 =1/2*(-1+x)^3*x*(e5+e4-2*e6)*(e4-2*e5+e6)*(-e5+2*e4-e6)*dx^2
          -1/2*(-1+r)*(-1+x)^3*(e5+e4-2*e6)*(e4-2*e5+e6)*(-e5+2*e4-e6)*dx: 
P123pa6:=1/2*(-1+x)^3*x*(-2*e8+e7+e9)*(e8+e7-2*e9)*(2*e7-e8-e9)*dx^2
          -1/2*(-1+r)*(-1+x)^3*(-2*e8+e7+e9)*(e8+e7-2*e9)*(2*e7-e8-e9)*dx: 

Similar decompositions of P456p, P16p79n are given in G6NPQSdata.txt.

The inverse operator of P123p789n   is given in G6NPQSdata.txt
The value S123n789p, composition of shifted P123p789n and P123n789p, is
equal to 
  = e7*e8*e9*r^2*(r+1)*(r-2)*(r-1)^2*(r+1-e1)*(r+1-e2)*(r+1-e3).