#C(x,y)[Dx,Dy]-module of G1(a,b1,b2,x,y) with basis [G1, DxG1, DyG1], reason to choose this basis is it saves some computation when computing matrices in M.

M:= [[x, y], [Dx, Dy], [Matrix(3, 3, {(1, 1) = 0, (1, 2) = -(y-b2+y^2+x*y-b2*y-b1*y+a*y+3*b2*x*y-b1*y^2+a*y^2+a*y*x+2*b2*x*y^2+2*b1*y^2*x)*a/((4*x^2*y+4*x*y-x+4*x*y^2-1-y)*x), (1, 3) = -a*(x+1+y-2*b2*x*y-b2*x-2*b1*x*y-b1*y+a*x+a+a*y)/(4*x^2*y+4*x*y-x+4*x*y^2-1-y), (2, 1) = 1, (2, 2) = -(2*b1*x^2*y^2-3*b1*y^2*x-4*b1*x*y+b1+b1*y-y-b2*x+5*x^2*y-1-x-a*x+5*a*x^2*y+5*x*y+2*a*y*x+3*b2*x^2*y-b2*x*y+3*a*x*y^2+5*x*y^2+2*b2*x^2*y^2)/(x*(4*x^2*y+4*x*y-x+4*x*y^2-1-y)), (2, 3) = -x*(x+y+1-2*b2*x*y-b2*x-2*b1*x*y+b1+b1*y+a*x+3*a*y+2*a)/(4*x^2*y+4*x*y-x+4*x*y^2-1-y), (3, 1) = 0, (3, 2) = (-a-y-2*a*y+b1*y-x*y+a*y*x-b2*x*y-y^2-a*y^2+b1*y^2-2*b1*y^2*x-2*b2*x*y^2)*y/((1+y+x)*(4*x*y-1)*x), (3, 3) = -y*(x+y+1-2*b2*x*y+b2*x+b2-2*b1*x*y-b1*y+3*a*x+a*y+2*a)/(4*x^2*y+4*x*y-x+4*x*y^2-1-y)}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = -a*(x+1+y-2*b2*x*y-b2*x-2*b1*x*y-b1*y+a*x+a+a*y)/(4*x^2*y+4*x*y-x+4*x*y^2-1-y), (1, 3) = -(-b1+a*x-x*b1-b2*x+x+x^2+x^2*a-x^2*b2+x*y+a*y*x+3*b1*x*y+2*x^2*y*b1+2*b2*x^2*y)*a/(y*(4*x^2*y+4*x*y-x+4*x*y^2-1-y)), (2, 1) = 0, (2, 2) = -x*(x+y+1-2*b2*x*y-b2*x-2*b1*x*y+b1+b1*y+a*x+3*a*y+2*a)/(4*x^2*y+4*x*y-x+4*x*y^2-1-y), (2, 3) = -(x+a+x*y+x^2-b2*x+2*a*x-x^2*b2+b1*x*y-a*y*x+x^2*a+2*b2*x^2*y+2*x^2*y*b1)*x/(y*(4*x^2*y+4*x*y-x+4*x*y^2-1-y)), (3, 1) = 1, (3, 2) = -y*(x+y+1-2*b2*x*y+b2*x+b2-2*b1*x*y-b1*y+3*a*x+a*y+2*a)/(4*x^2*y+4*x*y-x+4*x*y^2-1-y), (3, 3) = -(-1-y-x+b2+5*x*y+b2*x-b1*y-a*y+5*x*y^2+5*x^2*y-4*b2*x*y-b1*x*y+2*a*y*x-3*b2*x^2*y+3*b1*y^2*x+5*a*x*y^2+3*a*x^2*y+2*b2*x^2*y^2+2*b1*x^2*y^2)/(y*(4*x^2*y+4*x*y-x+4*x*y^2-1-y))})]]:


# G1 = (a)m+n(b1)n-m(b2)m-n/m!n! x^m*y^n
# order = 3
# Arguments: [x,u*x,1,1,1],[-a-c,-b-c,c],[[1,0,1,2,0],[0,1,1,0,0],[0,0,-1,-1,1]]
# Notes on A-hypergeometric function by Frits Beukers. P7
# reducible: a or a+b1 or a+b2 or b1+b2 is an integer is an integer 
# A-hypergeometric function and theri Monodromy