# Formulae for Hypergeometric functions.

In this page, all formulae are written in the user language of Risa/Asir a computer algebra system with yang package. The "dx" means the Euler differential operator for x, that is, dx ≡ x*(∂/∂x).

## Hypergeometric functions with 2 variables in Horn's List

### 14 Hypergeometric functions with regular singularities.

• Appell's F1(a,b1,b2,c;x,y)
• Program: F1.rr
• differntial system (yang format):
```[yang.mul(dx,dx+dy+c-1) - x*yang.mul(dx+b1,dx+dy+a),
yang.mul(dy,dx+dy+c-1) - y*yang.mul(dy+b2,dx+dy+a)];
```
• Rank of differentail system: 3
• Coeffcient matrices of Pfaffian for [1,dx,dy]
```[ [ [0,(1)/(x),0],
[(-b1*a)/(x-1),((-a-b1)*x^2+((a+b1-b2)*y+c-1)*x+(-c+b2+1)*y)/(x^3+(-y-1)*x^2+y*x),(b1*y-b1)/(x^2+(-y-1)*x+y)],
[0,(b2*y)/(x^2-y*x),(-b1)/(x-y)] ],

[ [0,0,(1)/(y)],
[0,(b2)/(x-y),(-b1*x)/(y*x-y^2)],
[(-b2*a)/(y-1),(-b2*x+b2)/((y-1)*x-y^2+y),(((-a+b1-b2)*y+c-b1-1)*x+(a+b2)*y^2+(-c+1)*y)/((y^2-y)*x-y^3+y^2)] ] ];
```
• Appell's F2(a,b1,b2,c1,c2;x,y)
• Program: F2.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c1-1) - x*yang.mul(dx+dy+a, dx+b1),
yang.mul(dy,dy+c2-1) - y*yang.mul(dx+dy+a, dy+b2) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-b1*a)/(x-1),((-a-b1)*x+c1-1)/(x^2-x),(-b1)/(x-1),(-1)/(x-1)],
[0,0,0,(1)/(x)],
[(b2*b1*a*y)/(x^2+(y-2)*x-y+1),((b2*a+b2*b1+(-c1+1)*b2)*y)/(x^2+(y-2)*x-y+1),((-b1*a+(c2-1)*b1)*x+b2*b1*y+b1*a+(-c2+1)*b1)/(x^2+(y-2)*x-y+1),((-a-b1+c2-1)*x^2+((b2-c1+1)*y+a+b1+c1-c2)*x+(c1-1)*y-c1+1)/(x^3+(y-2)*x^2+(-y+1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-b2*a)/(y-1),(-b2)/(y-1),((-a-b2)*y+c2-1)/(y^2-y),(-1)/(y-1)],
[(b2*b1*a*x)/((y-1)*x+y^2-2*y+1),(b2*b1*x+(-b2*a+(c1-1)*b2)*y+b2*a+(-c1+1)*b2)/((y-1)*x+y^2-2*y+1),((b1*a+(b2-c2+1)*b1)*x)/((y-1)*x+y^2-2*y+1),(((b1-c2+1)*y+c2-1)*x+(-a-b2+c1-1)*y^2+(a+b2-c1+c2)*y-c2+1)/((y^2-y)*x+y^3-2*y^2+y)] ] ]
```
• Appell's F3(a1,a2,b1,b2,c;x,y)
• Program: F3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+c-1) - x*yang.mul(dx+a1,dx+b1),
yang.mul(dy,dx+dy+c-1) - y*yang.mul(dy+a2,dy+b2) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-a1*b1)/(x-1),((-b1-a1)*x+c-1)/(x^2-x),0,(1)/(x^2-x)],
[0,0,0,(1)/(x)],
[0,(-a2*b2*y)/((y-1)*x^2-y*x),(-a1*b1*y+a1*b1)/((y-1)*x-y),(((-b1-a1)*y+b1+a1)*x+(c-b2-a2-1)*y)/((y-1)*x^2-y*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-a2*b2)/(y-1),0,((-b2-a2)*y+c-1)/(y^2-y),(1)/(y^2-y)],
[0,(-a2*b2*x+a2*b2)/((y-1)*x-y),(-a1*b1*x)/((y^2-y)*x-y^2),(((-b2-a2)*y+c-b1-a1-1)*x+(b2+a2)*y)/((y^2-y)*x-y^2)] ] ]
```
• Appell's F4(a,b,c1,c2;x,y)
• Program: F4.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c1-1) - x*yang.mul(dx+dy+a, dx+dy+b),
yang.mul(dy,dy+c2-1) - y*yang.mul(dx+dy+a, dx+dy+b) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-b*a)/(x+y-1),((-a-b)*x+(-c1+1)*y+c1-1)/(x^2+(y-1)*x),(-a-b+c2-1)/(x+y-1),(-2)/(x+y-1)],
[0,0,0,(1)/(x)],
[((-b*a^2+(-b^2+(2*c1-c2-2)*b)*a)*y*x+(b*a^2+(b^2+(-c2+2)*b)*a)*y^2+(-b*a^2+(-b^2+(c2-2)*b)*a)*y)/(x^3+(-y-3)*x^2+(-y^2-2*y+3)*x+y^3-3*y^2+3*y-1),((-a^2+(c1-c2-1)*a-b^2+(c1-c2-1)*b+c2*c1-c2)*y*x+(a^2+(4*b-3*c1-c2+5)*a+b^2+(-3*c1-c2+5)*b+2*c1^2+(c2-6)*c1-c2+4)*y^2+(-a^2+(-4*b+3*c1+c2-5)*a-b^2+(3*c1+c2-5)*b-2*c1^2+(-c2+6)*c1+c2-4)*y)/(x^3+(-y-3)*x^2+(-y^2-2*y+3)*x+y^3-3*y^2+3*y-1),(((-b+c2-1)*a+(c2-1)*b-c2^2+2*c2-1)*x^2+((-a^2+(-4*b+2*c1+2*c2-5)*a-b^2+(2*c1+2*c2-5)*b+(-2*c2+2)*c1-c2^2+5*c2-4)*y+(2*b-2*c2+2)*a+(-2*c2+2)*b+2*c2^2-4*c2+2)*x+(a^2+(b-c2+2)*a+b^2+(-c2+2)*b-c2+1)*y^2+(-a^2-a-b^2-b+c2^2-c2)*y+(-b+c2-1)*a+(c2-1)*b-c2^2+2*c2-1)/(x^3+(-y-3)*x^2+(-y^2-2*y+3)*x+y^3-3*y^2+3*y-1),((-a-b+2*c2-2)*x^3+((-2*a-2*b+3*c1-5)*y+2*a+2*b+c1-4*c2+3)*x^2+((3*a+3*b-2*c1-2*c2+6)*y^2+(-2*a-2*b+4*c1-6)*y-a-b-2*c1+2*c2)*x+(-c1+1)*y^3+(3*c1-3)*y^2+(-3*c1+3)*y+c1-1)/(x^4+(-y-3)*x^3+(-y^2-2*y+3)*x^2+(y^3-3*y^2+3*y-1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-b*a)/(x+y-1),(-a-b+c1-1)/(x+y-1),((-c2+1)*x+(-a-b)*y+c2-1)/(y*x+y^2-y),(-2)/(x+y-1)],
[((b*a^2+(b^2+(-c1+2)*b)*a)*x^2+((-b*a^2+(-b^2+(-c1+2*c2-2)*b)*a)*y-b*a^2+(-b^2+(c1-2)*b)*a)*x)/(x^3+(-y-3)*x^2+(-y^2-2*y+3)*x+y^3-3*y^2+3*y-1),((a^2+(b-c1+2)*a+b^2+(-c1+2)*b-c1+1)*x^2+((-a^2+(-4*b+2*c1+2*c2-5)*a-b^2+(2*c1+2*c2-5)*b-c1^2+(-2*c2+5)*c1+2*c2-4)*y-a^2-a-b^2-b+c1^2-c1)*x+((-b+c1-1)*a+(c1-1)*b-c1^2+2*c1-1)*y^2+((2*b-2*c1+2)*a+(-2*c1+2)*b+2*c1^2-4*c1+2)*y+(-b+c1-1)*a+(c1-1)*b-c1^2+2*c1-1)/(x^3+(-y-3)*x^2+(-y^2-2*y+3)*x+y^3-3*y^2+3*y-1),((a^2+(4*b-c1-3*c2+5)*a+b^2+(-c1-3*c2+5)*b+(c2-1)*c1+2*c2^2-6*c2+4)*x^2+((-a^2+(-c1+c2-1)*a-b^2+(-c1+c2-1)*b+(c2-1)*c1)*y-a^2+(-4*b+c1+3*c2-5)*a-b^2+(c1+3*c2-5)*b+(-c2+1)*c1-2*c2^2+6*c2-4)*x)/(x^3+(-y-3)*x^2+(-y^2-2*y+3)*x+y^3-3*y^2+3*y-1),((-c2+1)*x^3+((3*a+3*b-2*c1-2*c2+6)*y+3*c2-3)*x^2+((-2*a-2*b+3*c2-5)*y^2+(-2*a-2*b+4*c2-6)*y-3*c2+3)*x+(-a-b+2*c1-2)*y^3+(2*a+2*b-4*c1+c2+3)*y^2+(-a-b+2*c1-2*c2)*y+c2-1)/(y*x^3+(-y^2-3*y)*x^2+(-y^3-2*y^2+3*y)*x+y^4-3*y^3+3*y^2-y)] ] ]
```
• G1(a,b1,b2;x,y)
• Program: G1.rr
• differntial system (yang format):
```[ yang.mul(dx,dx-dy-b1) + x*yang.mul(dx+dy+a,dx-dy+b2),
yang.mul(dy,dy-dx-b2) + y*yang.mul(dx+dy+a,dy-dx+b1) ];
```
• Rank of differentail system: 3
• Coeffcient matrices of Pfaffian for [1,dx,dy]
```[ [ [0,(1)/(x),0],
[(((-2*b1-2*b2)*a*y^2+(-a^2+(-3*b2-1)*a)*y)*x+(-a^2+(b1-1)*a)*y^2+(-a^2+(b1+b2-1)*a)*y+b2*a)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1),(((-2*b1-2*b2)*y^2+(-5*a-3*b2-1)*y)*x^2+((-3*a+3*b1-1)*y^2+(-2*a+4*b1+b2-1)*y+a+b2)*x-b1*y-b1)/(4*y*x^3+(4*y^2+4*y-1)*x^2+(-y-1)*x),(((-2*b1-2*b2)*y^2+(a-b2-1)*y)*x+(-a+b1-1)*y^2+(-2*a+b1-1)*y-a)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1)],
[(((2*b1+2*b2)*a*y^2+(-a^2+(b2-1)*a)*y)*x+(-a^2+(b1-1)*a)*y^2+(-a^2-a)*y)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1),(((2*b1+2*b2)*y^2+(-a+b2-1)*y)*x+(-3*a-b1-1)*y^2+(-2*a-b1-1)*y)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1),(((2*b1+2*b2)*y^2+(-3*a-b2-1)*y)*x+(-a+b1-1)*y^2+(-2*a-b2-1)*y)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1)] ],

[ [0,0,(1)/(y)],
[(((2*b1+2*b2)*a*y-a^2+(b2-1)*a)*x^2+((-a^2+(b1-1)*a)*y-a^2-a)*x)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1),(((2*b1+2*b2)*y-a+b2-1)*x^2+((-3*a-b1-1)*y-2*a-b1-1)*x)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1),(((2*b1+2*b2)*y-3*a-b2-1)*x^2+((-a+b1-1)*y-2*a-b2-1)*x)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1)],
[(((-2*b1-2*b2)*a*y-a^2+(b2-1)*a)*x^2+((-a^2+(-3*b1-1)*a)*y-a^2+(b1+b2-1)*a)*x+b1*a)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1),(((-2*b1-2*b2)*y-a+b2-1)*x^2+((a-b1-1)*y-2*a+b2-1)*x-a)/(4*y*x^2+(4*y^2+4*y-1)*x-y-1),(((-2*b1-2*b2)*y^2+(-3*a+3*b2-1)*y)*x^2+((-5*a-3*b1-1)*y^2+(-2*a+b1+4*b2-1)*y-b2)*x+(a+b1)*y-b2)/(4*y^2*x^2+(4*y^3+4*y^2-y)*x-y^2-y)] ] ]
```
• G2(a1,a2,b1,b2;x,y)
• Program: G2.rr
• differntial system (yang format):
```[ yang.mul(dx,dx-dy-b1) + x*yang.mul(dx+a1,dx-dy+b2),
yang.mul(dy,dy-dx-b2) + y*yang.mul(dy+a2,dy-dx+b1) ];
```
• Rank of differentail system: 3
• Coeffcient matrices of Pfaffian for [1,dx,dy]
```[ [ [0,(1)/(x),0],
[((-a1*b2-a2*a1)*y*x-a2*a1*y+a1*b2)/(y*x^2+(y-1)*x-1),((-b2-a1-a2)*y*x^2+((b1-a2)*y+b2+a1)*x-b1)/(y*x^3+(y-1)*x^2-x),(-a1*y-a1)/(y*x^2+(y-1)*x-1)],
[(-a2*a1*y)/(y*x-1),(-a2*y)/(y*x-1),(-a1*y)/(y*x-1)]],

[ [0,0,(1)/(y)],
[(-a2*a1*x)/(y*x-1),(-a2*x)/(y*x-1),(-a1*x)/(y*x-1)],
[(((-a2*b1-a2*a1)*y-a2*a1)*x+a2*b1)/((y^2+y)*x-y-1),(-a2*x-a2)/((y^2+y)*x-y-1),(((-b1-a1-a2)*y^2+(b2-a1)*y)*x+(b1+a2)*y-b2)/((y^3+y^2)*x-y^2-y)]]]
```
• G3(a1,a2;x,y)
• Program: G3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx-2*dy-a1)+x*yang.mul(2*dx-dy+a2,2*dx-dy+a2+1),
yang.mul(dy,dy-2*dx-a2)+y*yang.mul(2*dy-dx+a1,2*dy-dx+a1+1) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[((a1^2+a1-4*a2^2-4*a2)*y-a2^2-a2)/((15*y+4)*x+4*y+1),(((-2*a1-16*a2-9)*y-4*a2-2)*x+4*a1*y+a1)/((15*y+4)*x^2+(4*y+1)*x),((4*a1+8*a2+6)*y+a2+1)/((15*y+4)*x+4*y+1),((12*y+2)*x+8*y+2)/((15*y+4)*x^2+(4*y+1)*x)],
[0,0,0,(1)/(x)],
[(((-48*a1^3+(-144*a2-144)*a1^2+(-108*a2^2-252*a2-96)*a1-24*a2^3-90*a2^2-66*a2)*y^3+(-16*a1^3+(-60*a2-44)*a1^2+(-48*a2^2-108*a2-28)*a1-11*a2^3-43*a2^2-32*a2)*y^2+(-16/3*a2*a1^2+(-16/3*a2^2-32/3*a2)*a1-4/3*a2^3-16/3*a2^2-4*a2)*y)*x^2+((-16*a1^3+(-72*a2-62)*a1^2+(-96*a2^2-168*a2-46)*a1-32*a2^3-88*a2^2-56*a2)*y^3+(-14/3*a1^3+(-29*a2-50/3)*a1^2+(-48*a2^2-77*a2-12)*a1-52/3*a2^3-142/3*a2^2-30*a2)*y^2+(-8/3*a2*a1^2+(-6*a2^2-26/3*a2)*a1-7/3*a2^3-19/3*a2^2-4*a2)*y)*x+(-8/3*a1^3+(-32/3*a2-32/3)*a1^2+(-32/3*a2^2-64/3*a2-8)*a1)*y^3+(-2/3*a1^3+(-4*a2-8/3)*a1^2+(-16/3*a2^2-28/3*a2-2)*a1)*y^2+(-1/3*a2*a1^2+(-2/3*a2^2-a2)*a1)*y)/((405*y^3+108*y^2)*x^3+(108*y^3-243*y^2-132*y-16)*x^2+(-132*y^2-65*y-8)*x-16*y^2-8*y-1),(((-144*a1^2+(-144*a2-216)*a1-36*a2^2-108*a2-36)*y^3+(-72*a1^2+(-72*a2-128)*a1-18*a2^2-64*a2-36)*y^2+(-32/3*a1^2+(-32/3*a2-64/3)*a1-8/3*a2^2-32/3*a2-8)*y)*x^2+((-144*a1^2+(-216*a2-242)*a1-72*a2^2-160*a2-66)*y^3+(-78*a1^2+(-123*a2-434/3)*a1-42*a2^2-295/3*a2-48)*y^2+(-32/3*a1^2+(-52/3*a2-64/3)*a1-6*a2^2-44/3*a2-8)*y)*x+(8/3*a1^2+(32/3*a2+32/3)*a1+32/3*a2^2+64/3*a2+8)*y^3+(2/3*a1^2+(4*a2+8/3)*a1+16/3*a2^2+28/3*a2+2)*y^2+(1/3*a2*a1+2/3*a2^2+a2)*y)/((405*y^3+108*y^2)*x^3+(108*y^3-243*y^2-132*y-16)*x^2+(-132*y^2-65*y-8)*x-16*y^2-8*y-1),(((-117*a1^2+(-360*a2-297)*a1-252*a2^2-432*a2-126)*y^3+(-24*a1^2+(-96*a2-56)*a1-69*a2^2-109*a2-24)*y^2+(16/3*a1^2+(16/3*a2+32/3)*a1+4/3*a2^2+16/3*a2+4)*y)*x^2+((-24*a1^2+(-96*a2-80)*a1-96*a2^2-160*a2-36)*y^3+(2*a1^2+(-4*a2+34/3)*a1-16*a2^2-52/3*a2+6)*y^2+(8/3*a1^2+(6*a2+26/3)*a1+7/3*a2^2+19/3*a2+4)*y)*x+(-16/3*a1^2+(-64/3*a2-64/3)*a1-64/3*a2^2-128/3*a2-16)*y^3+((-8/3*a2-4/3)*a1-16/3*a2^2-32/3*a2-4)*y^2+(1/3*a1^2+(2/3*a2+1)*a1)*y)/((405*y^3+108*y^2)*x^3+(108*y^3-243*y^2-132*y-16)*x^2+(-132*y^2-65*y-8)*x-16*y^2-8*y-1),(((-486*a1-648*a2-567)*y^3+(-144*a1-180*a2-162)*y^2)*x^3+((-252*a1-288*a2-324)*y^3+(-9*a1+90*a2-108)*y^2+(36*a1+84*a2-6)*y+16/3*a1+32/3*a2)*x^2+((48*a1+96*a2+12)*y^2+(68/3*a1+136/3*a2+3)*y+8/3*a1+16/3*a2)*x+(16/3*a1+32/3*a2)*y^2+(8/3*a1+16/3*a2)*y+1/3*a1+2/3*a2)/((405*y^3+108*y^2)*x^4+(108*y^3-243*y^2-132*y-16)*x^3+(-132*y^2-65*y-8)*x^2+(-16*y^2-8*y-1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[((-4*a1^2-4*a1+a2^2+a2)*x-a1^2-a1)/((15*y+4)*x+4*y+1),((8*a1+4*a2+6)*x+a1+1)/((15*y+4)*x+4*y+1),(((-16*a1-2*a2-9)*y+4*a2)*x+(-4*a1-2)*y+a2)/((15*y^2+4*y)*x+4*y^2+y),((12*y+8)*x+2*y+2)/((15*y^2+4*y)*x+4*y^2+y)],
[(((-24*a1^3+(-108*a2-90)*a1^2+(-144*a2^2-252*a2-66)*a1-48*a2^3-144*a2^2-96*a2)*y^2+(-32*a1^3+(-96*a2-88)*a1^2+(-72*a2^2-168*a2-56)*a1-16*a2^3-62*a2^2-46*a2)*y-32/3*a2*a1^2+(-32/3*a2^2-64/3*a2)*a1-8/3*a2^3-32/3*a2^2-8*a2)*x^3+((-11*a1^3+(-48*a2-43)*a1^2+(-60*a2^2-108*a2-32)*a1-16*a2^3-44*a2^2-28*a2)*y^2+(-52/3*a1^3+(-48*a2-142/3)*a1^2+(-29*a2^2-77*a2-30)*a1-14/3*a2^3-50/3*a2^2-12*a2)*y-16/3*a2*a1^2+(-4*a2^2-28/3*a2)*a1-2/3*a2^3-8/3*a2^2-2*a2)*x^2+((-4/3*a1^3+(-16/3*a2-16/3)*a1^2+(-16/3*a2^2-32/3*a2-4)*a1)*y^2+(-7/3*a1^3+(-6*a2-19/3)*a1^2+(-8/3*a2^2-26/3*a2-4)*a1)*y-2/3*a2*a1^2+(-1/3*a2^2-a2)*a1)*x)/((405*y^3+108*y^2)*x^3+(108*y^3-243*y^2-132*y-16)*x^2+(-132*y^2-65*y-8)*x-16*y^2-8*y-1),(((-252*a1^2+(-360*a2-432)*a1-117*a2^2-297*a2-126)*y^2+(-96*a1^2+(-96*a2-160)*a1-24*a2^2-80*a2-36)*y-64/3*a1^2+(-64/3*a2-128/3)*a1-16/3*a2^2-64/3*a2-16)*x^3+((-69*a1^2+(-96*a2-109)*a1-24*a2^2-56*a2-24)*y^2+(-16*a1^2+(-4*a2-52/3)*a1+2*a2^2+34/3*a2+6)*y-16/3*a1^2+(-8/3*a2-32/3)*a1-4/3*a2-4)*x^2+((4/3*a1^2+(16/3*a2+16/3)*a1+16/3*a2^2+32/3*a2+4)*y^2+(7/3*a1^2+(6*a2+19/3)*a1+8/3*a2^2+26/3*a2+4)*y+2/3*a2*a1+1/3*a2^2+a2)*x)/((405*y^3+108*y^2)*x^3+(108*y^3-243*y^2-132*y-16)*x^2+(-132*y^2-65*y-8)*x-16*y^2-8*y-1),(((-36*a1^2+(-144*a2-108)*a1-144*a2^2-216*a2-36)*y^2+(-72*a1^2+(-216*a2-160)*a1-144*a2^2-242*a2-66)*y+32/3*a1^2+(32/3*a2+64/3)*a1+8/3*a2^2+32/3*a2+8)*x^3+((-18*a1^2+(-72*a2-64)*a1-72*a2^2-128*a2-36)*y^2+(-42*a1^2+(-123*a2-295/3)*a1-78*a2^2-434/3*a2-48)*y+16/3*a1^2+(4*a2+28/3)*a1+2/3*a2^2+8/3*a2+2)*x^2+((-8/3*a1^2+(-32/3*a2-32/3)*a1-32/3*a2^2-64/3*a2-8)*y^2+(-6*a1^2+(-52/3*a2-44/3)*a1-32/3*a2^2-64/3*a2-8)*y+2/3*a1^2+(1/3*a2+1)*a1)*x)/((405*y^3+108*y^2)*x^3+(108*y^3-243*y^2-132*y-16)*x^2+(-132*y^2-65*y-8)*x-16*y^2-8*y-1),(((-648*a1-486*a2-567)*y^3+(-288*a1-252*a2-324)*y^2)*x^3+((-180*a1-144*a2-162)*y^3+(90*a1-9*a2-108)*y^2+(96*a1+48*a2+12)*y+32/3*a1+16/3*a2)*x^2+((84*a1+36*a2-6)*y^2+(136/3*a1+68/3*a2+3)*y+16/3*a1+8/3*a2)*x+(32/3*a1+16/3*a2)*y^2+(16/3*a1+8/3*a2)*y+2/3*a1+1/3*a2)/((405*y^4+108*y^3)*x^3+(108*y^4-243*y^3-132*y^2-16*y)*x^2+(-132*y^3-65*y^2-8*y)*x-16*y^3-8*y^2-y)] ] ]
```
• H1(a,b,c,d;x,y)
• Program: H1.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+d-1)-x*yang.mul(dx-dy+a,dx+dy+b),
yang.mul(dy,dy-dx-a)+y*yang.mul(dx+dy+b,dy+c) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[((-b*a-c*b)*y-b*a)/((y+1)*x-y-1),(((-a-b-c)*y-a-b)*x+(d-1)*y+d-1)/((y+1)*x^2+(-y-1)*x),((-a-c)*y+b)/((y+1)*x-y-1),(-y+1)/((y+1)*x-y-1)],
[0,0,0,(1)/(x)],
[(((c*b*a-c*b^2+(2*c^2-c)*b)*y^2+(c*b*a-c*b^2-c*b)*y)*x+(-c*b*a-c^2*b)*y^3+(-c*b*a+c*b^2+(-c^2+c)*b)*y^2+(c*b^2+c*b)*y)/((4*y^2+4*y)*x^2+(-y^3-7*y^2-7*y-1)*x+y^3+3*y^2+3*y+1),(((c*a-c*b+2*c^2-c)*y^2+(c*a-c*b-c)*y)*x+(-c*a-c*b-c^2+(d-1)*c)*y^3+(-c*a+c*b-c^2+c)*y^2+(2*c*b+(-d+2)*c)*y)/((4*y^2+4*y)*x^2+(-y^3-7*y^2-7*y-1)*x+y^3+3*y^2+3*y+1),(((-a^2+(-2*b+c-1)*a-b^2+(-c-1)*b+2*c^2-c)*y^2+(-a^2+(-2*b-c-1)*a-b^2+(-3*c-1)*b-c)*y)*x+(-c*a-c^2)*y^3+(a^2+(2*b+1)*a+b^2+(2*c+1)*b-c^2+c)*y^2+(a^2+(2*b+c+1)*a+b^2+(2*c+1)*b+c)*y)/((4*y^2+4*y)*x^2+(-y^3-7*y^2-7*y-1)*x+y^3+3*y^2+3*y+1),(((-4*a-4*b-2)*y^2+(-4*a-4*b-4*c-2)*y)*x^2+((-c+d-1)*y^3+(4*a+4*b+2*c+3*d-1)*y^2+(4*a+4*b+3*c+3*d-1)*y+d-1)*x+(-d+1)*y^3+(-3*d+3)*y^2+(-3*d+3)*y-d+1)/((4*y^2+4*y)*x^3+(-y^3-7*y^2-7*y-1)*x^2+(y^3+3*y^2+3*y+1)*x)]],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-c*b)/(y+1),(-c)/(y+1),((-b-c)*y+a)/(y^2+y),(-y+1)/(y^2+y)],
[(((3*c*b*a+c*b^2+(2*c^2+c)*b)*y+c*b*a-c*b^2-c*b)*x)/((4*y^2+4*y)*x-y^3-3*y^2-3*y-1),(((3*c*a+c*b+2*c^2+c)*y+c*a-c*b-c)*x+(c*b+(-d+1)*c)*y^2+(2*c*b+(-2*d+2)*c)*y+c*b+(-d+1)*c)/((4*y^2+4*y)*x-y^3-3*y^2-3*y-1),(((a^2+(2*b+3*c+1)*a+b^2+(c+1)*b+2*c^2+c)*y-a^2+(-2*b-c-1)*a-b^2+(-3*c-1)*b-c)*x)/((4*y^2+4*y)*x-y^3-3*y^2-3*y-1),(((4*a+2)*y^2+(-4*b-4*c-2)*y)*x+(b+c-d+1)*y^3+(-a+2*b+2*c-d+1)*y^2+(-2*a+b+c+d-1)*y-a+d-1)/((4*y^3+4*y^2)*x-y^4-3*y^3-3*y^2-y)] ] ]
```
• H2(a,b,c,d,e;x,y)
• Program: H2.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+e-1)-x*yang.mul(dx-dy+a,dx+b),
yang.mul(dy,dy-dx-a)+y*yang.mul(dy+c,dy+d) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-b*a)/(x-1),((-a-b)*x+e-1)/(x^2-x),(b)/(x-1),(1)/(x-1)],
[0,0,0,(1)/(x)],
[(-d*c*b*y)/(y*x-y-1),(-d*c*y)/(y*x-y-1),((-b*a+(-c-d)*b)*y)/(y*x-y-1),((-a-b-c-d)*y*x+(e-1)*y+e-1)/(y*x^2+(-y-1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-d*c)/(y+1),0,((-c-d)*y+a)/(y^2+y),(1)/(y^2+y)],
[(-d*c*b*x)/((y^2+y)*x-y^2-2*y-1),(-d*c*x+d*c)/(y*x-y-1),((-b*a+(-c-d)*b)*x)/((y^2+y)*x-y^2-2*y-1),(((-c-d)*y^2+(-b-c-d)*y)*x+(c+d)*y^2+(-a+c+d+e-1)*y-a+e-1)/((y^3+y^2)*x-y^3-2*y^2-y)]]]
```
• H3(a,b,c;x,y)
• Program: H3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+c-1)-x*yang.mul(2*dx+dy+a,2*dx+dy+a+1),
yang.mul(dy,dx+dy+c-1)-y*yang.mul(2*dx+dy+a,dy+b) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[((-a^2+(b-1)*a)*y+a^2+a)/((4*y-4)*x-y+1),(((-4*a+2*b-2)*y+4*a+2)*x+(c-1)*y-c+1)/((4*y-4)*x^2+(-y+1)*x),((-a+b-1)*y+2*a-c+2)/((4*y-4)*x-y+1),((-2*y+3)*x+y-1)/((4*y-4)*x^2+(-y+1)*x)],
[0,0,0,(1)/(x)],
[(((-2*b*a^2+(-2*b^2+(4*c-6)*b)*a)*y^2+(2*b*a^2+(-4*c+6)*b*a)*y)*x+(b*a^2+(-b^2+b)*a)*y^3+(-b*a^2+(b^2-c*b)*a)*y^2+(c-1)*b*a*y)/((4*y-4)*x^2+(4*y^3-8*y^2+3*y+1)*x-y^3+2*y^2-y),(((-4*b*a-4*b^2+(8*c-12)*b)*y^2+(4*b*a+(-8*c+12)*b)*y)*x^2+((-2*b^2+(4*c-2)*b)*y^3+(3*b*a+2*b^2+(-12*c+10)*b)*y^2+(-3*b*a+(8*c-8)*b)*y)*x+(b*a+(-2*c+2)*b)*y^3+(-2*b*a+(4*c-4)*b)*y^2+(b*a+(-2*c+2)*b)*y)/((4*y-4)*x^3+(4*y^3-8*y^2+3*y+1)*x^2+(-y^3+2*y^2-y)*x),(((-2*b*a-2*b^2+(4*c-6)*b)*y^2+(-4*a^2+(8*c-12)*a+(-2*c+4)*b-4*c^2+12*c-8)*y+4*a^2+(-8*c+12)*a+4*c^2-12*c+8)*x+(b*a-b^2+b)*y^3+(-2*b*a+b^2-b)*y^2+(a^2+(b-2*c+3)*a+c^2-3*c+2)*y-a^2+(2*c-3)*a-c^2+3*c-2)/((4*y-4)*x^2+(4*y^3-8*y^2+3*y+1)*x-y^3+2*y^2-y),(((-8*a-2*b+8*c-12)*y+8*a-8*c+12)*x^2+((-4*a+2*b+4*c-4)*y^3+(8*a-6*b-8*c+8)*y^2+(-2*a+4*b+2*c-1)*y-2*a+2*c-3)*x+(a-b-c+1)*y^3+(-2*a+2*b+2*c-2)*y^2+(a-b-c+1)*y)/((4*y-4)*x^3+(4*y^3-8*y^2+3*y+1)*x^2+(-y^3+2*y^2-y)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-b*a)/(y-1),(-2*b)/(y-1),((-a-b)*y+c-1)/(y^2-y),(-2*y+1)/(y^2-y)],
[(((b*a^2+(b^2+(-2*c+3)*b)*a)*y+(c-1)*b*a)*x)/((y-1)*x+y^3-2*y^2+y),(((2*b*a+2*b^2+(-4*c+6)*b)*y+(2*c-2)*b)*x+(b*a+(-2*c+2)*b)*y^2+(-2*b*a+(4*c-4)*b)*y+b*a+(-2*c+2)*b)/((y-1)*x+y^3-2*y^2+y),(((b*a+b^2+(-2*c+3)*b)*y^2+(2*a^2+(b-4*c+6)*a+2*c^2-6*c+4)*y-a^2+(2*c-3)*a-c^2+3*c-2)*x)/((y^2-y)*x+y^4-2*y^3+y^2),(((3*a+b-4*c+6)*y-2*a+3*c-4)*x+(a-b-2*c+2)*y^3+(-2*a+2*b+4*c-4)*y^2+(a-b-2*c+2)*y)/((y^2-y)*x+y^4-2*y^3+y^2)] ] ]
```
• H4(a,b,c,d;x,y)
• Program: H4.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c-1)-x*yang.mul(2*dx+dy+a,2*dx+dy+a+1),
yang.mul(dy,dy+d-1)-y*yang.mul(2*dx+dy+a,dy+b) ];
```
(Remark: the system above differ from [Elderi: pp.235 (19)].)
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[((-a^2+(b-1)*a)*y+a^2+a)/((4*y-4)*x-y+1),(((-4*a+2*b-2)*y+4*a+2)*x+(c-1)*y-c+1)/((4*y-4)*x^2+(-y+1)*x),((-a+b-1)*y+2*a-d+2)/((4*y-4)*x-y+1),(-2*y+4)/((4*y-4)*x-y+1)],
[0,0,0,(1)/(x)],
[(((-4*b^2+(4*d-4)*b)*a*y^2+(-4*d+4)*b*a*y)*x+(-b*a^2+(b^2-b)*a)*y^3+(3*b*a^2+(-b^2+(-d+4)*b)*a)*y^2+(-2*b*a^2+(d-3)*b*a)*y)/((16*y-16)*x^2+(-4*y^3+12*y^2-16*y+8)*x+y^3-3*y^2+3*y-1),(((-8*b^2+(8*d-8)*b)*y^2+(-8*d+8)*b*y)*x+(-4*b*a+2*b^2+(4*c-6)*b)*y^3+(12*b*a-2*b^2+(-12*c-2*d+20)*b)*y^2+(-8*b*a+(8*c+2*d-14)*b)*y)/((16*y-16)*x^2+(-4*y^3+12*y^2-16*y+8)*x+y^3-3*y^2+3*y-1),(((-4*b^2+(4*d-4)*b)*y^2+(-4*a^2+(-4*b+8*d-12)*a-4*d^2+12*d-8)*y+4*a^2+(-8*d+12)*a+4*d^2-12*d+8)*x+(-b*a+b^2-b)*y^3+(4*b*a-b^2+(-2*d+5)*b)*y^2+(a^2+(-3*b-2*d+3)*a+(2*d-4)*b+d^2-3*d+2)*y-a^2+(2*d-3)*a-d^2+3*d-2)/((16*y-16)*x^2+(-4*y^3+12*y^2-16*y+8)*x+y^3-3*y^2+3*y-1),(((-16*a-8*b+16*d-24)*y+16*a-16*d+24)*x^2+((-2*b+4*c-4)*y^3+(8*b-12*c+12)*y^2+(4*a-6*b+12*c-4*d-6)*y-4*a-4*c+4*d-2)*x+(-c+1)*y^3+(3*c-3)*y^2+(-3*c+3)*y+c-1)/((16*y-16)*x^3+(-4*y^3+12*y^2-16*y+8)*x^2+(y^3-3*y^2+3*y-1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-b*a)/(y-1),(-2*b)/(y-1),((-a-b)*y+d-1)/(y^2-y),(-2)/(y-1)],
[(((2*b^2+(-2*d+2)*b)*a*y+2*b*a^2+2*b*a)*x)/((4*y-4)*x-y^3+3*y^2-3*y+1),(((4*b^2+(-4*d+4)*b)*y+4*b*a+4*b)*x+(b*a+(-2*c+2)*b)*y^2+(-2*b*a+(4*c-4)*b)*y+b*a+(-2*c+2)*b)/((4*y-4)*x-y^3+3*y^2-3*y+1),(((2*b^2+(-2*d+2)*b)*y+2*a^2+(4*b-4*d+6)*a+(-2*d+4)*b+2*d^2-6*d+4)*x)/((4*y-4)*x-y^3+3*y^2-3*y+1),(((4*a+4*b-8*d+12)*y+4*d-4)*x+(a+b-2*c+2)*y^3+(-2*a-2*b+4*c-d-3)*y^2+(a+b-2*c+2*d)*y-d+1)/((4*y^2-4*y)*x-y^4+3*y^3-3*y^2+y)] ] ]
```
• H5(a,b,c;x,y)
• Program: H5.rr
• differntial system (yang format):
```[ yang.mul(dx,dx-dy-b)+x*yang.mul(2*dx+dy+a,2*dx+dy+a+1),
yang.mul(dy,dy+c-1)-y*yang.mul(2*dx+dy+a,dy-dx+b) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[((-a^2+(b-1)*a)*y+a^2+a)/((6*y-4)*x+y-1),(((-5*a+2*b-2)*y+4*a+2)*x+b*y-b)/((6*y-4)*x^2+(y-1)*x),((-a+b-1)*y+2*a-c+2)/((6*y-4)*x+y-1),((-3*y+4)*x+y-1)/((6*y-4)*x^2+(y-1)*x)],
[0,0,0,(1)/(x)],
[(((-4*a^3+(-16*b+12*c-20)*a^2+(-16*b^2+(24*c-40)*b+12*c-16)*a)*y^2+((-8*c+8)*a^2+((-16*c+16)*b-8*c+8)*a)*y)*x^2+((9*a^3+(9*b+21)*a^2+(-18*b^2+6*b+12)*a)*y^3+(-21*a^3+(-34*b+2*c-51)*a^2+(16*b^2+(10*c-40)*b+2*c-30)*a)*y^2+(12*a^3+(24*b-2*c+30)*a^2+((-8*c+32)*b-2*c+18)*a)*y)*x+(-a^3+(2*b-3)*a^2+(-b^2+3*b-2)*a)*y^3+(2*a^3+(-4*b+6)*a^2+(b^2+(c-6)*b+4)*a)*y^2+(-a^3+(2*b-3)*a^2+((-c+3)*b-2)*a)*y)/((96*y-64)*x^3+(162*y^3-324*y^2+208*y-48)*x^2+(27*y^3-69*y^2+54*y-12)*x-y^2+2*y-1),(((-8*a^2+(-32*b+24*c-40)*a-32*b^2+(48*c-80)*b+24*c-32)*y^2+((-16*c+16)*a+(-32*c+32)*b-16*c+16)*y)*x^2+((27*a^2+(54*b+60)*a+48*b+24)*y^3+(-58*a^2+(-132*b-2*c-130)*a-32*b^2+(8*c-136)*b+4*c-60)*y^2+(32*a^2+(80*b+72)*a+32*b^2+(-8*c+88)*b-4*c+36)*y)*x+(-8*a^2+(-11*b-15)*a-8*b^2-12*b-4)*y^3+(16*a^2+(23*b-c+30)*a+16*b^2+24*b+8)*y^2+(-8*a^2+(-12*b+c-15)*a-8*b^2-12*b-4)*y)/((96*y-64)*x^3+(162*y^3-324*y^2+208*y-48)*x^2+(27*y^3-69*y^2+54*y-12)*x-y^2+2*y-1),(((-4*a^2+(-16*b+12*c-20)*a-16*b^2+(24*c-40)*b+12*c-16)*y^2+(-32*a^2+(-16*b+48*c-80)*a-24*c^2+72*c-48)*y+16*a^2+(-32*c+48)*a+16*c^2-48*c+32)*x^2+((9*a^2+(9*b+21)*a-18*b^2+6*b+12)*y^3+(-30*a^2+(-52*b+11*c-72)*a+16*b^2+(28*c-58)*b+14*c-42)*y^2+(12*a^2+(40*b+8*c+24)*a+(-24*c+48)*b-10*c^2+14*c+12)*y+8*a^2+(-16*c+24)*a+8*c^2-24*c+16)*x+(-a^2+(2*b-3)*a-b^2+3*b-2)*y^3+(3*a^2+(-5*b-c+9)*a+b^2+(2*c-7)*b-2*c+6)*y^2+(-3*a^2+(3*b+3*c-9)*a+(-2*c+4)*b-c^2+5*c-6)*y+a^2+(-2*c+3)*a+c^2-3*c+2)/((96*y-64)*x^3+(162*y^3-324*y^2+208*y-48)*x^2+(27*y^3-69*y^2+54*y-12)*x-y^2+2*y-1),(((-112*a-32*b+96*c-160)*y+64*a-64*c+96)*x^3+((-27*a+108*b+54)*y^3+(36*a-252*b-126)*y^2+(-40*a+160*b+16*c+40)*y+32*a-16*b-16*c+32)*x^2+((-27*a-27)*y^3+(58*a-10*b+6*c+49)*y^2+(-35*a+18*b-10*c-20)*y+4*a-8*b+4*c-2)*x+(-b+c-1)*y^2+(2*b-2*c+2)*y-b+c-1)/((96*y-64)*x^4+(162*y^3-324*y^2+208*y-48)*x^3+(27*y^3-69*y^2+54*y-12)*x^2+(-y^2+2*y-1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[((-2*a^2+(-4*b-2)*a)*x-b*a)/((6*y-4)*x+y-1),((-4*a-8*b-4)*x+a)/((6*y-4)*x+y-1),(((-8*a-4*b-2)*y+4*c-4)*x+(-a-b)*y+c-1)/((6*y^2-4*y)*x+y^2-y),(-12*x+1)/((6*y-4)*x+y-1)],
[(((8*a^3+(32*b-24*c+40)*a^2+(32*b^2+(-48*c+80)*b-24*c+32)*a)*y+16*a^3+(32*b+32)*a^2+(32*b+16)*a)*x^3+((-6*a^2+(6*b-6)*a)*y^2+(-2*a^3+(2*c+4)*a^2+(8*b^2+(-8*c+4)*b+2*c+6)*a)*y+(8*b-4)*a^2+(8*b-4)*a)*x^2+((-a^3+(2*b-3)*a^2+(-b^2+3*b-2)*a)*y^2+(2*a^3+(-3*b+6)*a^2+((c-5)*b+4)*a)*y-a^3-3*a^2-2*a)*x)/((96*y-64)*x^3+(162*y^3-324*y^2+208*y-48)*x^2+(27*y^3-69*y^2+54*y-12)*x-y^2+2*y-1),(((16*a^2+(64*b-48*c+80)*a+64*b^2+(-96*c+160)*b-48*c+64)*y+32*a^2+(64*b+64)*a+64*b+32)*x^3+((-18*a^2+(-72*b-48)*a-72*b^2-60*b-12)*y^2+(16*a^2+(80*b+16*c+40)*a+96*b^2+(8*c+72)*b+4*c+12)*y-16*a^2+(-32*b-32)*a-32*b^2-32*b-8)*x^2+((-8*a^2+(-11*b-15)*a-8*b^2-12*b-4)*y^2+(15*a^2+(24*b-c+29)*a+16*b^2+24*b+8)*y-6*a^2+(-12*b-12)*a-8*b^2-12*b-4)*x)/((96*y-64)*x^3+(162*y^3-324*y^2+208*y-48)*x^2+(27*y^3-69*y^2+54*y-12)*x-y^2+2*y-1),(((8*a^2+(32*b-24*c+40)*a+32*b^2+(-48*c+80)*b-24*c+32)*y+80*a^2+(64*b-112*c+208)*a+(-32*c+64)*b+48*c^2-160*c+128)*x^3+((-6*a+6*b-6)*y^2+(-2*a^2+(2*c+10)*a+8*b^2+(-8*c+4)*b-4*c+12)*y+8*a^2+(16*b-16*c+16)*a+(-8*c+16)*b+8*c^2-20*c+8)*x^2+((-a^2+(2*b-3)*a-b^2+3*b-2)*y^2+(3*a^2+(-4*b-c+9)*a+(2*c-6)*b-2*c+6)*y-3*a^2+(3*c-9)*a-c^2+5*c-6)*x)/((96*y-64)*x^3+(162*y^3-324*y^2+208*y-48)*x^2+(27*y^3-69*y^2+54*y-12)*x-y^2+2*y-1),(((128*a+64*b-192*c+320)*y+64*c-64)*x^3+((-108*a-216*b-108)*y^3+(144*a+288*b+108*c+36)*y^2+(-64*a-96*b-128*c+64)*y+48*c-48)*x^2+((-27*a-27*b-27)*y^3+(52*a+56*b+15*c+37)*y^2+(-24*a-28*b-28*c+4)*y+12*c-12)*x+(c-1)*y^2+(-2*c+2)*y+c-1)/((96*y^2-64*y)*x^3+(162*y^4-324*y^3+208*y^2-48*y)*x^2+(27*y^4-69*y^3+54*y^2-12*y)*x-y^3+2*y^2-y)] ] ]
```
• H6(a,b,c;x,y)
• Program: H6.rr
• differntial system (yang format):
```[ yang.mul(dx,dx-dy-b)+x*yang.mul(2*dx-dy+a,2*dx-dy+a+1),
yang.mul(dy,dy-2*dx-a)+y*yang.mul(dy-dx+b,dy+c) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[((-a^2-a+c*b)*y-a^2-a)/((4*y+4)*x+y+1),(((-4*a-c-2)*y-4*a-2)*x+b*y+b)/((4*y+4)*x^2+(y+1)*x),((2*a+b+c+1)*y+a+1)/((4*y+4)*x+y+1),((3*y+2)*x+y+1)/((4*y+4)*x^2+(y+1)*x)],
[0,0,0,(1)/(x)],
[(((-3*c*a^2+(-8*c*b-3*c)*a-4*c*b^2+(-c^2-4*c)*b)*y^3+(-5*c*a^2+(-12*c*b-5*c)*a-4*c*b^2+(-2*c^2-4*c)*b)*y^2+(-2*c*a^2+(-4*c*b-2*c)*a)*y)*x+(-c*a^2+(-2*c*b-c)*a-c*b^2-c*b)*y^3+(-2*c*a^2+(-3*c*b-2*c)*a-c*b^2-c*b)*y^2+(-c*a^2+(-c*b-c)*a)*y)/((4*y^3+4*y^2)*x^2+(y^3-3*y^2-8*y-4)*x-y^2-2*y-1),(((-4*c*a-8*c*b+c^2-2*c)*y^3+(-8*c*a-16*c*b+2*c^2-6*c)*y^2+(-4*c*a-8*c*b-4*c)*y)*x+(-2*c*a-3*c*b-c)*y^3+(-5*c*a-7*c*b-3*c)*y^2+(-3*c*a-4*c*b-2*c)*y)/((4*y^3+4*y^2)*x^2+(y^3-3*y^2-8*y-4)*x-y^2-2*y-1),(((-4*a^2+(-8*b-2*c-4)*a-4*b^2+(-c-4)*b-c^2-c)*y^3+(-4*a^2+(-8*b-c-4)*a-4*b^2+(2*c-4)*b-2*c^2+c)*y^2+(2*c*a+4*c*b+2*c)*y)*x+(-a^2+(-2*b-1)*a-b^2-b)*y^3+(-a^2+(-2*b+c-1)*a-b^2+(c-1)*b+c)*y^2+(c*a+c*b+c)*y)/((4*y^3+4*y^2)*x^2+(y^3-3*y^2-8*y-4)*x-y^2-2*y-1),(((-8*a-8*b-3*c-4)*y^3+(-8*a-8*b-4*c-4)*y^2)*x^2+((-2*a-2*b-c-1)*y^3+(2*a+2*b-2*c-1)*y^2+(8*a+8*b-c)*y+4*a+4*b)*x+(a+b)*y^2+(2*a+2*b)*y+a+b)/((4*y^3+4*y^2)*x^3+(y^3-3*y^2-8*y-4)*x^2+(-y^2-2*y-1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-c*b)/(y+1),(c)/(y+1),((-b-c)*y+a)/(y^2+y),(y+2)/(y^2+y)],
[(((-c*a^2+(-2*c*b-c)*a-c*b^2-c*b)*y^2+(-2*c*a^2+(-5*c*b-2*c)*a-2*c*b^2+(-c^2-2*c)*b)*y-c*a^2+(-2*c*b-c)*a)*x)/((y^3+y^2)*x-y^2-2*y-1),(((-2*c*a-3*c*b-c)*y^2+(-3*c*a-6*c*b+c^2-2*c)*y-2*c*a-4*c*b-2*c)*x)/((y^3+y^2)*x-y^2-2*y-1),(((-a^2+(-2*b-1)*a-b^2-b)*y^2+(-2*a^2+(-4*b-c-2)*a-2*b^2-2*b-c^2)*y+c*a+2*c*b+c)*x)/((y^3+y^2)*x-y^2-2*y-1),(((-2*a-3*b-c-1)*y^3+(-3*a-4*b-2*c-2)*y^2)*x+(a+2*b)*y^2+(2*a+4*b)*y+a+2*b)/((y^4+y^3)*x-y^3-2*y^2-y)] ] ]
```
• H7(a,b,c,d;x,y)
• Program: H7.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+d-1)-x*yang.mul(2*dx-dy+a,2*dx-dy+a+1),
yang.mul(dy,dy-2*dx-a)+y*yang.mul(dy+b,dy+c) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[((-a^2-a+c*b)*y-a^2-a)/((4*y+4)*x-y-1),((-4*a-2)*x+d-1)/(4*x^2-x),((2*a+b+c+1)*y+a+1)/((4*y+4)*x-y-1),(4*y+2)/((4*y+4)*x-y-1)],
[0,0,0,(1)/(x)],
[((-2*c*b*a-c*b^2+(-c^2-c)*b)*y^2-c*b*a*y)/(4*y^2*x-y^2-2*y-1),(-4*c*b*y^2-2*c*b*y)/(4*y^2*x-y^2-2*y-1),((-a^2+(-2*b-2*c-1)*a-b^2+(-c-1)*b-c^2-c)*y^2+c*b*y)/(4*y^2*x-y^2-2*y-1),((-4*a-4*b-4*c-2)*y^2*x+(d-1)*y^2+(2*d-2)*y+d-1)/(4*y^2*x^2+(-y^2-2*y-1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-c*b)/(y+1),0,((-b-c)*y+a)/(y^2+y),(2)/(y^2+y)],
[(((-4*c*b*a-2*c*b^2+(-2*c^2-2*c)*b)*y-2*c*b*a)*x)/((4*y^3+4*y^2)*x-y^3-3*y^2-3*y-1),((-4*c*b*y-4*c*b)*x+c*b*y+c*b)/(4*y^2*x-y^2-2*y-1),(((-2*a^2+(-4*b-4*c-2)*a-2*b^2+(-2*c-2)*b-2*c^2-2*c)*y+2*c*b)*x)/((4*y^3+4*y^2)*x-y^3-3*y^2-3*y-1),(((-4*b-4*c)*y^3+(-4*a-8*b-8*c-4)*y^2)*x+(b+c)*y^3+(-a+2*b+2*c+2*d-2)*y^2+(-2*a+b+c+4*d-4)*y-a+2*d-2)/((4*y^4+4*y^3)*x-y^4-3*y^3-3*y^2-y)] ] ]
```

### 20 Confluent Hypergeometric functions

• Φ1(a,b,c;x,y)
• Program: Phi1.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+c-1)-x*(dx+dy+a),
yang.mul(dy,dx+dy+c-1)-y*yang.mul(dx+dy+a,dy+b) ];
```
• Rank of differentail system: 3
• Coeffcient matrices of Pfaffian for [1,dx,dy]
```[ [ [0,(1)/(x),0],
[a,(x+b-c+1)/(x),(y-1)/(y)],
[0,(-b)/(x),(1)/(y)] ],

[ [0,0,(1)/(y)],
[0,(-b)/(y),(x)/(y^2)],
[(-b*a)/(y-1),(-b)/(y^2-y),((-y+1)*x+(-a-b)*y^2+(c-1)*y)/(y^3-y^2)] ] ]
```
• Φ2(b1,b2,c;x,y)
• Program: Phi2.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+c-1)-x*yang.mul(dx+b1,dx+b2),
yang.mul(dy,dx+dy+c-1)-y ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-b2*b1)/(x-1),((-b1-b2)*x+c-1)/(x^2-x),0,(1)/(x^2-x)],
[0,0,0,(1)/(x)],
[0,(y)/(x^2),(-b2*b1)/(x),(-b1-b2)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[1,0,(-c+1)/(y),(-1)/(y)],
[0,(x-1)/(x),(b2*b1)/(y),(-c+b1+b2+1)/(y)] ] ]
```
• Φ3(b,c;x,y)
• Program: Phi3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+c-1)-x*(dx+b),
yang.mul(dy,dx+dy+c-1)-y ];
```
• Rank of differentail system: 3
• Coeffcient matrices of Pfaffian for [1,dx,dy]
```[ [ [0,(1)/(x),0],
[b,(x^2+(-c+1)*x-y)/(x^2),(b)/(x)],
[0,(y)/(x^2),(-b)/(x)] ],

[ [0,0,(1)/(y)],
[0,(1)/(x),(-b)/(y)],
[1,(-1)/(x),(b-c+1)/(y)] ] ]
```
• Ψ1(a,b,c1,c2;x,y)
• Program: Psi1.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c1-1)-x*yang.mul(dx+dy+a,dx+b),
yang.mul(dy,dxyc2-1)-y*(dx+dy+a) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-b*a)/(x-1),((-a-b)*x+c1-1)/(x^2-x),(-b)/(x-1),(-1)/(x-1)],
[0,0,0,(1)/(x)],
[(b*a*y)/(x^2-2*x+1),((a+b-c1+1)*y)/(x^2-2*x+1),((-b*a+(c2-1)*b)*x+b*y+b*a+(-c2+1)*b)/(x^2-2*x+1),((-a-b+c2-1)*x^2+(y+a+b+c1-c2)*x-c1+1)/(x^3-2*x^2+x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[a,1,(y-c2+1)/(y),0],
[(-b*a*x)/(x-1),(-b*x-a+c1-1)/(x-1),(-b*x)/(x-1),((-c2+1)*x-y+c2-1)/(y*x-y)]] ]
```
• Ψ2(a,c1,c2;x,y)
• Program: Psi2.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c1-1)-x*(dx+dy+a),
yang.mul(dy,dy+c2-1)-y*(dx+dy+a) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[a,(x-c1+1)/(x),1,0],
[0,0,0,(1)/(x)],
[a*y,y,y+a-c2+1,(x-c1+1)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[a,1,(y-c2+1)/(y),0],
[a*x,x+a-c1+1,x,(y-c2+1)/(y)] ] ]
```
• Ξ1(a1,a2,b,c;x,y)
• Program: Xi1.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+c-1)-x*yang.mul(dx+a1,dx+b),
yang.mul(dy,dx+dy+c-1)-y*(dy+a2) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-a1*b)/(x-1),((-b-a1)*x+c-1)/(x^2-x),0,(1)/(x^2-x)],
[0,0,0,(1)/(x)],
[0,(a2*y)/(x^2),(-a1*b)/(x),((-b-a1)*x+y)/(x^2)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[a2,0,(y-c+1)/(y),(-1)/(y)],
[0,(a2*x-a2)/(x),(a1*b)/(y),((y+b-c+a1+1)*x-y)/(y*x)] ] ]
```
• Ξ2(a,b,c1,c2;x,y)
• Program: Xi2.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c1-1)-x*(dx+a),
yang.mul(dy,dy+c2-1)-y*(dy+b) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[a,(x-c1+1)/(x),0,0],
[0,0,0,(1)/(x)],
[0,0,a,(x-c1+1)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[b,0,(y-c2+1)/(y),0],
[0,b,0,(y-c2+1)/(y)] ] ]
```
• Γ1(a,b1,b2;x,y)
• Program: Gamma1.rr
• differntial system (yang format):
```[ yang.mul(dx,dx-dy-b1)+x*yang.mul(dx+a,dx-dy+b2),
yang.mul(dy,dy-dx-b2)+y*(dy-dx+b1) ];
```
• Rank of differentail system: 3
• Coeffcient matrices of Pfaffian for [1,dx,dy]
```[ [ [0,(1)/(x),0],
[(a*y*x+a*y-b2*a)/(x+1),(y*x^2+(y-a-b2)*x+b1)/(x^2+x),(a)/(x+1)],
[a*y,y,0] ],

[ [0,0,(1)/(y)],
[a*x,x,0],
[a*x-b1,x+1,(-y+b2)/(y)] ] ]
```
• Γ2(b1,b2;x,y)
• Program: Gamma2.rr
• differntial system (yang format):
```[ yang.mul(dx,dx-dy-b1)+x*(dx-dy+b2),
yang.mul(dy,dy-dx-b2)+y*(dy-dx+b1) ];
```
• Rank of differentail system: 3
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0],
[y-b2,(-x+b1)/(x),1],
[y,0,0] ],

[ [0,0,(1)/(y)],
[x,0,0],
[x-b1,1,(-y+b2)/(y)] ] ]
```
• Η1(a,b,c;x,y)
• Program: Eta1.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c-1)-x*yang.mul(dx-dy+a,dx+dy+b),
yang.mul(dy,dy-dx-a)+y*(dx+dy+b) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-b*y-b*a)/(x-1),((-y-a-b)*x+c-1)/(x^2-x),(-y+b)/(x-1),(1)/(x-1)],
[0,0,0,(1)/(x)],
[((-2*b*y^2+(-b*a+b^2+b)*y)*x+b*y^2+(-b^2-b)*y)/(x-1),((-2*y^2+(-a+b+1)*y)*x+y^2+(-2*b+c-2)*y)/(x-1),((-2*y^2+(a+3*b+1)*y)*x+y^2+(-a-2*b-1)*y)/(x-1),(4*y*x^2+(-3*y-c+1)*x+c-1)/(x^2-x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[-b,-1,(-y+a)/(y),(1)/(y)],
[(-2*b*y-b*a+b^2+b)*x,(-2*y-a+b+1)*x-b+c-1,(-2*y+a+3*b+1)*x,(4*y*x-y+a-c+1)/(y)] ] ]
```
• Η2(a,b,c,d;x,y)
• Program: Eta2.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+d-1)-x*yang.mul(dx-dy+a,dx+b),
yang.mul(dy,dy-dx-a)+y*(dy+c) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-b*a)/(x-1),((-a-b)*x+d-1)/(x^2-x),(b)/(x-1),(1)/(x-1)],
[0,0,0,(1)/(x)],
[c*b*y,c*y,b*y,(y*x-d+1)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[-c,0,(-y+a)/(y),(1)/(y)],
[c*b*x,c*x-c,b*x,(y*x-y+a-d+1)/(y)] ] ]
```
• Η3(a,b,d;x,y)
• Program: Eta3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+d-1)-x*yang.mul(dx-dy+a,dx+b),
yang.mul(dy,dy-dx-a)+y ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-b*a)/(x-1),((-a-b)*x+d-1)/(x^2-x),(b)/(x-1),(1)/(x-1)],
[0,0,0,(1)/(x)],
[b*y,y,0,(-d+1)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[-1,0,(a)/(y),(1)/(y)],
[b*x,x-1,0,(a-d+1)/(y)] ] ]
```
• Η4(a,c,d;x,y)
• Program: Eta4.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+d-1)-x*(dx-dy+a),
yang.mul(dy,dy-dx-a)+y*(dy+c) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[a,(x-d+1)/(x),-1,0],
[0,0,0,(1)/(x)],
[c*y,0,y,(-d+1)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[-c,0,(-y+a)/(y),(1)/(y)],
[c*x,-c,x,(-y+a-d+1)/(y)] ] ]
```
• Η5(a,d;x,y)
• Program: Eta5.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+d-1)-x*(dx-dy+a),
yang.mul(dy,dy-dx-a)+y ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[a,(x-d+1)/(x),-1,0],
[0,0,0,(1)/(x)],
[y,0,0,(-d+1)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[-1,0,(a)/(y),(1)/(y)],
[x,-1,0,(a-d+1)/(y)] ] ]
```
• Η6(a,c;x,y)
• Program: Eta6.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+c-1)-x*yang.mul(2*dx+dy+a,2*dx+dy+a+1),
yang.mul(dy,dx+dy+c-1)-y*(2*dx+dy+a) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-a*y-a^2-a)/(4*x-1),((-2*y-4*a-2)*x+c-1)/(4*x^2-x),(-y-2*a+c-2)/(4*x-1),(-3*x+1)/(4*x^2-x)],
[0,0,0,(1)/(x)],
[((2*a*y^2+(-2*a^2+(4*c-6)*a)*y)*x-a*y^2+(-c+1)*a*y)/(4*x^2-x),((4*y^2+(-4*a+8*c-12)*y)*x^2+(-2*y^2+(3*a-8*c+8)*y)*x+(-a+2*c-2)*y)/(4*x^3-x^2),((2*y^2+(2*c-4)*y-4*a^2+(8*c-12)*a-4*c^2+12*c-8)*x-y^2-a*y+a^2+(-2*c+3)*a+c^2-3*c+2)/(4*x^2-x),((2*y-8*a+8*c-12)*x^2+(-4*y+2*a-2*c+3)*x+y)/(4*x^3-x^2)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[a,2,(y-c+1)/(y),(-1)/(y)],
[-a*y+(-c+1)*a,((-2*y-2*c+2)*x-a+2*c-2)/(x),(-y^2-a*y+a^2+(-2*c+3)*a+c^2-3*c+2)/(y),((-y+2*a-3*c+4)*x+y)/(y*x)] ] ]
```
• Η7(a,c,d;x,y)
• Program: Eta7.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c-1)-x*yang.mul(2*dx+dy+a,2*dx+dy+a+1),
yang.mul(dy,dy+d-1)-y*(2*dx+dy+a) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(-a*y-a^2-a)/(4*x-1),((-2*y-4*a-2)*x+c-1)/(4*x^2-x),(-y-2*a+d-2)/(4*x-1),(-4)/(4*x-1)],
[0,0,0,(1)/(x)],
[((4*a*y^2+(4*d-4)*a*y)*x+a*y^2+(2*a^2+(-d+3)*a)*y)/(16*x^2-8*x+1),((8*y^2+(8*d-8)*y)*x+2*y^2+(8*a-8*c-2*d+14)*y)/(16*x^2-8*x+1),((4*y^2+4*a*y-4*a^2+(8*d-12)*a-4*d^2+12*d-8)*x+y^2+(3*a-2*d+4)*y+a^2+(-2*d+3)*a+d^2-3*d+2)/(16*x^2-8*x+1),((8*y-16*a+16*d-24)*x^2+(6*y+4*a+4*c-4*d+2)*x-c+1)/(16*x^3-8*x^2+x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[a,2,(y-d+1)/(y),0],
[((-2*a*y-2*a^2-2*a)*x)/(4*x-1),((-4*y-4*a-4)*x-a+2*c-2)/(4*x-1),((-2*y-4*a+2*d-4)*x)/(4*x-1),((-4*y-4*d+4)*x-y+d-1)/(4*y*x-y)] ] ]
```
• Η8(a,b;x,y)
• Program: Eta8.rr
• differntial system (yang format):
```[ yang.mul(dx,dy-dx+b)-x*yang.mul(2*dx-dy+a,2*dx-dy+a+1),
yang.mul(dy,2*dx-dy+a)-y*(dy-dx+b) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(b*y-a^2-a)/(4*x+1),((-y-4*a-2)*x+b)/(4*x^2+x),(y+a+1)/(4*x+1),(2*x+1)/(4*x^2+x)],
[0,0,0,(1)/(x)],
[((2*b*y^2+(2*a^2+(4*b+2)*a)*y)*x+(a^2+(b+1)*a)*y)/(4*x+1),((-2*y^2+(4*a+8*b+4)*y)*x+(3*a+4*b+2)*y)/(4*x+1),((2*y^2+(-2*a-4*b-2)*y)*x+(-a-b-1)*y)/(4*x+1),((y-4*a-4*b)*x-a-b)/(4*x^2+x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[-b,1,(-y+a)/(y),(2)/(y)],
[(b*y+a^2+(2*b+1)*a)*x,(-y+2*a+4*b+2)*x,(y-a-2*b-1)*x,(-a-2*b)/(y)] ] ]
```
• Η9(a,b,d;x,y)
• Program: Eta9.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c-1)-x*yang.mul(2*dx-dy+a,2*dx-dy+a+1),
yang.mul(dy,2*dx-dy+a)-y*(dy+b) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(b*y-a^2-a)/(4*x-1),((-4*a-2)*x+c-1)/(4*x^2-x),(y+a+1)/(4*x-1),(2)/(4*x-1)],
[0,0,0,(1)/(x)],
[b*y^2+b*a*y,2*b*y,y^2-b*y,(-c+1)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[-b,0,(-y+a)/(y),(2)/(y)],
[(2*b*y+2*b*a)*x,4*b*x-b,(2*y-2*b)*x,(-y+a-2*c+2)/(y)] ] ]
```
• Η10(a,d;x,y)
• Program: Eta10.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c-1)-x*yang.mul(2*dx-dy+a,2*dx-dy+a+1),
yang.mul(dy,2*dx-dy+a)-y ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[(y-a^2-a)/(4*x-1),((-4*a-2)*x+c-1)/(4*x^2-x),(a+1)/(4*x-1),(2)/(4*x-1)],
[0,0,0,(1)/(x)],
[a*y,2*y,-y,(-c+1)/(x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[-1,0,(a)/(y),(2)/(y)],
[2*a*x,4*x-1,-2*x,(a-2*c+2)/(y)] ] ]
```
• Η11(a,b,c,d;x,y)
• Program: Eta11.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+d-1)-x*(dx-dy+a),
yang.mul(dy,dx-dy+a)-y*yang.mul(dy+b,dy+c) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dx*dy]
```[ [ [0,(1)/(x),0,0],
[a,(x-d+1)/(x),-1,0],
[0,0,0,(1)/(x)],
[(c*b*y)/(y+1),0,((a+b+c)*y)/(y+1),(y*x+(-d+1)*y-d+1)/((y+1)*x)] ],

[ [0,0,(1)/(y),0],
[0,0,0,(1)/(y)],
[(-c*b)/(y+1),0,((-b-c)*y+a)/(y^2+y),(1)/(y^2+y)],
[(c*b*x)/(y^2+2*y+1),(-c*b)/(y+1),((a+b+c)*x)/(y^2+2*y+1),(y*x+(-b-c)*y^2+(a-b-c-d+1)*y+a-d+1)/(y^3+2*y^2+y)] ] ]
```

## Lauricella's Hypergeometric functions with 3 variables.

• Lauricella's FD
• Program: FD3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+dz+c-1) - x*yang.mul(dx+b1,dx+dy+dz+a),
yang.mul(dy,dx+dy+dz+c-1) - y*yang.mul(dy+b2,dx+dy+dz+a),
yang.mul(dz,dx+dy+dz+c-1) - z*yang.mul(dz+b3,dx+dy+dz+a) ];
```
• Rank of differentail system: 4
• Coeffcient matrices of Pfaffian for [1,dx,dy,dz]
```[ [ [0,(1)/(x),0,0],
[(-b1*a)/(x-1),((-a-b1)*x^3+((a+b1-b2)*y+(a+b1-b3)*z+e)*x^2+(((-a-b1+b2+b3)*z-e+b2)*y+(-e+b3)*z)*x+(e-b2-b3)*z*y)/(x^4+(-y-z-1)*x^3+((z+1)*y+z)*x^2-z*y*x),(b1*y-b1)/(x^2+(-y-1)*x+y),(b1*z-b1)/(x^2+(-z-1)*x+z)],
[0,(b2*y)/(x^2-y*x),(-b1)/(x-y),0],
[0,(b3*z)/(x^2-z*x),0,(-b1)/(x-z)] ],

[ [0,0,(1)/(y),0],
[0,(b2)/(x-y),(-b1*x)/(y*x-y^2),0],
[(-b2*a)/(y-1),(-b2*x+b2)/((y-1)*x-y^2+y),(((-a+b1-b2)*y^2+((a-b1+b2-b3)*z+e-b1)*y+(-e+b1+b3)*z)*x+(a+b2)*y^3+((-a-b2+b3)*z-e)*y^2+(e-b3)*z*y)/((y^3+(-z-1)*y^2+z*y)*x-y^4+(z+1)*y^3-z*y^2),(b2*z-b2)/(y^2+(-z-1)*y+z)],
[0,0,(b3*z)/(y^2-z*y),(-b2)/(y-z)] ],

[ [0,0,0,(1)/(z)],
[0,(b3)/(x-z),0,(-b1*x)/(z*x-z^2)],
[0,0,(b3)/(y-z),(-b2*y)/(z*y-z^2)],
[(-b3*a)/(z-1),(-b3*x+b3)/((z-1)*x-z^2+z),(-b3*y+b3)/((z-1)*y-z^2+z),((((-a+b1+b2-b3)*z+e-b1-b2)*y+(a-b1+b3)*z^2+(-e+b1)*z)*x+((a-b2+b3)*z^2+(-e+b2)*z)*y+(-a-b3)*z^3+e*z^2)/(((z^2-z)*y-z^3+z^2)*x+(-z^3+z^2)*y+z^4-z^3)] ] ]
```
• Lauricella's FA
• Program: FA3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c1-1) - x*yang.mul(dx+dy+dz+a, dx+b1),
yang.mul(dy,dy+c2-1) - y*yang.mul(dx+dy+dz+a, dy+b2),
yang.mul(dz,dz+c3-1) - z*yang.mul(dx+dy+dz+a, dz+b3) ];
```
• Rank of differentail system: 8
• Coeffcient matrices of Pfaffian for [1,dx,dy,dz,dx*dy,dx*dz,dy*dz,dx*dy*dz]
```[ [ [0,(1)/(x),0,0,0,0,0,0],
[(-b1*a)/(x-1),((-a-b1)*x+c1-1)/(x^2-x),(-b1)/(x-1),(-b1)/(x-1),(-1)/(x-1),(-1)/(x-1),0,0],
[0,0,0,0,(1)/(x),0,0,0],
[0,0,0,0,0,(1)/(x),0,0],
[(b2*b1*a*y)/(x^2+(y-2)*x-y+1),((b2*a+b2*b1+(-c1+1)*b2)*y)/(x^2+(y-2)*x-y+1),((-b1*a+(c2-1)*b1)*x+b2*b1*y+b1*a+(-c2+1)*b1)/(x^2+(y-2)*x-y+1),(b2*b1*y)/(x^2+(y-2)*x-y+1),((-a-b1+c2-1)*x^2+((b2-c1+1)*y+a+b1+c1-c2)*x+(c1-1)*y-c1+1)/(x^3+(y-2)*x^2+(-y+1)*x),(b2*y)/(x^2+(y-2)*x-y+1),(-b1)/(x+y-1),(-1)/(x+y-1)],
[(b3*b1*a*z)/(x^2+(z-2)*x-z+1),((b3*a+b3*b1+(-c1+1)*b3)*z)/(x^2+(z-2)*x-z+1),(b3*b1*z)/(x^2+(z-2)*x-z+1),((-b1*a+(c3-1)*b1)*x+b3*b1*z+b1*a+(-c3+1)*b1)/(x^2+(z-2)*x-z+1),(b3*z)/(x^2+(z-2)*x-z+1),((-a-b1+c3-1)*x^2+((b3-c1+1)*z+a+b1+c1-c3)*x+(c1-1)*z-c1+1)/(x^3+(z-2)*x^2+(-z+1)*x),(-b1)/(x+z-1),(-1)/(x+z-1)],
[0,0,0,0,0,0,0,(1)/(x)],
[(-2*b3*b2*b1*a*z*y*x-b3*b2*b1*a*z*y^2+(-b3*b2*b1*a*z^2+2*b3*b2*b1*a*z)*y)/(x^4+(2*y+2*z-4)*x^3+(y^2+(3*z-6)*y+z^2-6*z+6)*x^2+((z-2)*y^2+(z^2-6*z+6)*y-2*z^2+6*z-4)*x+(-z+1)*y^2+(-z^2+3*z-2)*y+z^2-2*z+1),((-2*b3*b2*a-2*b3*b2*b1+(2*c1-2)*b3*b2)*z*y*x+(-b3*b2*a-b3*b2*b1+(c1-1)*b3*b2)*z*y^2+((-b3*b2*a-b3*b2*b1+(c1-1)*b3*b2)*z^2+(2*b3*b2*a+2*b3*b2*b1+(-2*c1+2)*b3*b2)*z)*y)/(x^4+(2*y+2*z-4)*x^3+(y^2+(3*z-6)*y+z^2-6*z+6)*x^2+((z-2)*y^2+(z^2-6*z+6)*y-2*z^2+6*z-4)*x+(-z+1)*y^2+(-z^2+3*z-2)*y+z^2-2*z+1),((b3*b1*a+(-c2+1)*b3*b1)*z*x^2+(-2*b3*b2*b1*z*y+(b3*b1*a+(-c2+1)*b3*b1)*z^2+(-2*b3*b1*a+(2*c2-2)*b3*b1)*z)*x-b3*b2*b1*z*y^2+(-b3*b2*b1*z^2+2*b3*b2*b1*z)*y+(-b3*b1*a+(c2-1)*b3*b1)*z^2+(b3*b1*a+(-c2+1)*b3*b1)*z)/(x^4+(2*y+2*z-4)*x^3+(y^2+(3*z-6)*y+z^2-6*z+6)*x^2+((z-2)*y^2+(z^2-6*z+6)*y-2*z^2+6*z-4)*x+(-z+1)*y^2+(-z^2+3*z-2)*y+z^2-2*z+1),((b2*b1*a+(-c3+1)*b2*b1)*y*x^2+((b2*b1*a+(-c3+1)*b2*b1)*y^2+(-2*b3*b2*b1*z-2*b2*b1*a+(2*c3-2)*b2*b1)*y)*x+(-b3*b2*b1*z-b2*b1*a+(c3-1)*b2*b1)*y^2+(-b3*b2*b1*z^2+2*b3*b2*b1*z+b2*b1*a+(-c3+1)*b2*b1)*y)/(x^4+(2*y+2*z-4)*x^3+(y^2+(3*z-6)*y+z^2-6*z+6)*x^2+((z-2)*y^2+(z^2-6*z+6)*y-2*z^2+6*z-4)*x+(-z+1)*y^2+(-z^2+3*z-2)*y+z^2-2*z+1),((b3*a+b3*b1+(-c1-c2+2)*b3)*z*x^2+(-2*b3*b2*z*y+(b3*a+b3*b1+(-c1-c2+2)*b3)*z^2+(-2*b3*a-2*b3*b1+(2*c1+2*c2-4)*b3)*z)*x-b3*b2*z*y^2+(-b3*b2*z^2+2*b3*b2*z)*y+(-b3*a-b3*b1+(c1+c2-2)*b3)*z^2+(b3*a+b3*b1+(-c1-c2+2)*b3)*z)/(x^4+(2*y+2*z-4)*x^3+(y^2+(3*z-6)*y+z^2-6*z+6)*x^2+((z-2)*y^2+(z^2-6*z+6)*y-2*z^2+6*z-4)*x+(-z+1)*y^2+(-z^2+3*z-2)*y+z^2-2*z+1),((b2*a+b2*b1+(-c1-c3+2)*b2)*y*x^2+((b2*a+b2*b1+(-c1-c3+2)*b2)*y^2+(-2*b3*b2*z-2*b2*a-2*b2*b1+(2*c1+2*c3-4)*b2)*y)*x+(-b3*b2*z-b2*a-b2*b1+(c1+c3-2)*b2)*y^2+(-b3*b2*z^2+2*b3*b2*z+b2*a+b2*b1+(-c1-c3+2)*b2)*y)/(x^4+(2*y+2*z-4)*x^3+(y^2+(3*z-6)*y+z^2-6*z+6)*x^2+((z-2)*y^2+(z^2-6*z+6)*y-2*z^2+6*z-4)*x+(-z+1)*y^2+(-z^2+3*z-2)*y+z^2-2*z+1),((-b1*a+(c2+c3-2)*b1)*x^2+((-b1*a+(b2+c2+c3-2)*b1)*y+(-b1*a+(b3+c2+c3-2)*b1)*z+2*b1*a+(-2*c2-2*c3+4)*b1)*x+b2*b1*y^2+((-b1*a+(c2+c3-2)*b1)*z+b1*a+(-b2-c2-c3+2)*b1)*y+b3*b1*z^2+(b1*a+(-b3-c2-c3+2)*b1)*z-b1*a+(c2+c3-2)*b1)/(x^3+(2*y+2*z-3)*x^2+(y^2+(3*z-4)*y+z^2-4*z+3)*x+(z-1)*y^2+(z^2-3*z+2)*y-z^2+2*z-1),((-a-b1+c2+c3-2)*x^3+((-a-b1+b2-c1+c2+c3-1)*y+(-a-b1+b3-c1+c2+c3-1)*z+2*a+2*b1+c1-2*c2-2*c3+3)*x^2+((b2-c1+1)*y^2+((-a-b1-2*c1+c2+c3)*z+a+b1-b2+3*c1-c2-c3-1)*y+(b3-c1+1)*z^2+(a+b1-b3+3*c1-c2-c3-1)*z-a-b1-2*c1+c2+c3)*x+((-c1+1)*z+c1-1)*y^2+((-c1+1)*z^2+(3*c1-3)*z-2*c1+2)*y+(c1-1)*z^2+(-2*c1+2)*z+c1-1)/(x^4+(2*y+2*z-3)*x^3+(y^2+(3*z-4)*y+z^2-4*z+3)*x^2+((z-1)*y^2+(z^2-3*z+2)*y-z^2+2*z-1)*x)] ],

[ [0,0,(1)/(y),0,0,0,0,0],
[0,0,0,0,(1)/(y),0,0,0],
[(-b2*a)/(y-1),(-b2)/(y-1),((-a-b2)*y+c2-1)/(y^2-y),(-b2)/(y-1),(-1)/(y-1),0,(-1)/(y-1),0],
[0,0,0,0,0,0,(1)/(y),0],
[(b2*b1*a*x)/((y-1)*x+y^2-2*y+1),(b2*b1*x+(-b2*a+(c1-1)*b2)*y+b2*a+(-c1+1)*b2)/((y-1)*x+y^2-2*y+1),((b1*a+(b2-c2+1)*b1)*x)/((y-1)*x+y^2-2*y+1),(b2*b1*x)/((y-1)*x+y^2-2*y+1),(((b1-c2+1)*y+c2-1)*x+(-a-b2+c1-1)*y^2+(a+b2-c1+c2)*y-c2+1)/((y^2-y)*x+y^3-2*y^2+y),(-b2)/(x+y-1),(b1*x)/((y-1)*x+y^2-2*y+1),(-1)/(x+y-1)],
[0,0,0,0,0,0,0,(1)/(y)],
[(b3*b2*a*z)/(y^2+(z-2)*y-z+1),(b3*b2*z)/(y^2+(z-2)*y-z+1),((b3*a+b3*b2+(-c2+1)*b3)*z)/(y^2+(z-2)*y-z+1),((-b2*a+(c3-1)*b2)*y+b3*b2*z+b2*a+(-c3+1)*b2)/(y^2+(z-2)*y-z+1),(b3*z)/(y^2+(z-2)*y-z+1),(-b2)/(y+z-1),((-a-b2+c3-1)*y^2+((b3-c2+1)*z+a+b2+c2-c3)*y+(c2-1)*z-c2+1)/(y^3+(z-2)*y^2+(-z+1)*y),(-1)/(y+z-1)],
[(-b3*b2*b1*a*z*x^2+(-2*b3*b2*b1*a*z*y-b3*b2*b1*a*z^2+2*b3*b2*b1*a*z)*x)/((y^2+(z-2)*y-z+1)*x^2+(2*y^3+(3*z-6)*y^2+(z^2-6*z+6)*y-z^2+3*z-2)*x+y^4+(2*z-4)*y^3+(z^2-6*z+6)*y^2+(-2*z^2+6*z-4)*y+z^2-2*z+1),(-b3*b2*b1*z*x^2+(-2*b3*b2*b1*z*y-b3*b2*b1*z^2+2*b3*b2*b1*z)*x+(b3*b2*a+(-c1+1)*b3*b2)*z*y^2+((b3*b2*a+(-c1+1)*b3*b2)*z^2+(-2*b3*b2*a+(2*c1-2)*b3*b2)*z)*y+(-b3*b2*a+(c1-1)*b3*b2)*z^2+(b3*b2*a+(-c1+1)*b3*b2)*z)/((y^2+(z-2)*y-z+1)*x^2+(2*y^3+(3*z-6)*y^2+(z^2-6*z+6)*y-z^2+3*z-2)*x+y^4+(2*z-4)*y^3+(z^2-6*z+6)*y^2+(-2*z^2+6*z-4)*y+z^2-2*z+1),((-b3*b1*a+(-b3*b2+(c2-1)*b3)*b1)*z*x^2+((-2*b3*b1*a+(-2*b3*b2+(2*c2-2)*b3)*b1)*z*y+(-b3*b1*a+(-b3*b2+(c2-1)*b3)*b1)*z^2+(2*b3*b1*a+(2*b3*b2+(-2*c2+2)*b3)*b1)*z)*x)/((y^2+(z-2)*y-z+1)*x^2+(2*y^3+(3*z-6)*y^2+(z^2-6*z+6)*y-z^2+3*z-2)*x+y^4+(2*z-4)*y^3+(z^2-6*z+6)*y^2+(-2*z^2+6*z-4)*y+z^2-2*z+1),(((b2*b1*a+(-c3+1)*b2*b1)*y-b3*b2*b1*z-b2*b1*a+(c3-1)*b2*b1)*x^2+((b2*b1*a+(-c3+1)*b2*b1)*y^2+(-2*b3*b2*b1*z-2*b2*b1*a+(2*c3-2)*b2*b1)*y-b3*b2*b1*z^2+2*b3*b2*b1*z+b2*b1*a+(-c3+1)*b2*b1)*x)/((y^2+(z-2)*y-z+1)*x^2+(2*y^3+(3*z-6)*y^2+(z^2-6*z+6)*y-z^2+3*z-2)*x+y^4+(2*z-4)*y^3+(z^2-6*z+6)*y^2+(-2*z^2+6*z-4)*y+z^2-2*z+1),(-b3*b1*z*x^2+(-2*b3*b1*z*y-b3*b1*z^2+2*b3*b1*z)*x+(b3*a+b3*b2+(-c1-c2+2)*b3)*z*y^2+((b3*a+b3*b2+(-c1-c2+2)*b3)*z^2+(-2*b3*a-2*b3*b2+(2*c1+2*c2-4)*b3)*z)*y+(-b3*a-b3*b2+(c1+c2-2)*b3)*z^2+(b3*a+b3*b2+(-c1-c2+2)*b3)*z)/((y^2+(z-2)*y-z+1)*x^2+(2*y^3+(3*z-6)*y^2+(z^2-6*z+6)*y-z^2+3*z-2)*x+y^4+(2*z-4)*y^3+(z^2-6*z+6)*y^2+(-2*z^2+6*z-4)*y+z^2-2*z+1),(b2*b1*x^2+((-b2*a+b2*b1+(c1+c3-2)*b2)*y+(-b2*a+(c1+c3-2)*b2)*z+b2*a-b2*b1+(-c1-c3+2)*b2)*x+(-b2*a+(c1+c3-2)*b2)*y^2+((-b2*a+(b3+c1+c3-2)*b2)*z+2*b2*a+(-2*c1-2*c3+4)*b2)*y+b3*b2*z^2+(b2*a+(-b3-c1-c3+2)*b2)*z-b2*a+(c1+c3-2)*b2)/((y+z-1)*x^2+(2*y^2+(3*z-4)*y+z^2-3*z+2)*x+y^3+(2*z-3)*y^2+(z^2-4*z+3)*y-z^2+2*z-1),(((b1*a+(b2-c2-c3+2)*b1)*y-b3*b1*z-b1*a+(-b2+c2+c3-2)*b1)*x^2+((b1*a+(b2-c2-c3+2)*b1)*y^2+(-2*b3*b1*z-2*b1*a+(-2*b2+2*c2+2*c3-4)*b1)*y-b3*b1*z^2+2*b3*b1*z+b1*a+(b2-c2-c3+2)*b1)*x)/((y^2+(z-2)*y-z+1)*x^2+(2*y^3+(3*z-6)*y^2+(z^2-6*z+6)*y-z^2+3*z-2)*x+y^4+(2*z-4)*y^3+(z^2-6*z+6)*y^2+(-2*z^2+6*z-4)*y+z^2-2*z+1),(((b1-c2+1)*y+(-c2+1)*z+c2-1)*x^2+((-a+b1-b2+c1-c2+c3-1)*y^2+((-a-b2+c1-2*c2+c3)*z+a-b1+b2-c1+3*c2-c3-1)*y+(-c2+1)*z^2+(3*c2-3)*z-2*c2+2)*x+(-a-b2+c1+c3-2)*y^3+((-a-b2+b3+c1-c2+c3-1)*z+2*a+2*b2-2*c1+c2-2*c3+3)*y^2+((b3-c2+1)*z^2+(a+b2-b3-c1+3*c2-c3-1)*z-a-b2+c1-2*c2+c3)*y+(c2-1)*z^2+(-2*c2+2)*z+c2-1)/((y^2+(z-1)*y)*x^2+(2*y^3+(3*z-4)*y^2+(z^2-3*z+2)*y)*x+y^4+(2*z-3)*y^3+(z^2-4*z+3)*y^2+(-z^2+2*z-1)*y)] ],

[ [0,0,0,(1)/(z),0,0,0,0],
[0,0,0,0,0,(1)/(z),0,0],
[0,0,0,0,0,0,(1)/(z),0],
[(-b3*a)/(z-1),(-b3)/(z-1),(-b3)/(z-1),((-a-b3)*z+c3-1)/(z^2-z),0,(-1)/(z-1),(-1)/(z-1),0],
[0,0,0,0,0,0,0,(1)/(z)],
[(b3*b1*a*x)/((z-1)*x+z^2-2*z+1),(b3*b1*x+(-b3*a+(c1-1)*b3)*z+b3*a+(-c1+1)*b3)/((z-1)*x+z^2-2*z+1),(b3*b1*x)/((z-1)*x+z^2-2*z+1),((b1*a+(b3-c3+1)*b1)*x)/((z-1)*x+z^2-2*z+1),(-b3)/(x+z-1),(((b1-c3+1)*z+c3-1)*x+(-a-b3+c1-1)*z^2+(a+b3-c1+c3)*z-c3+1)/((z^2-z)*x+z^3-2*z^2+z),(b1*x)/((z-1)*x+z^2-2*z+1),(-1)/(x+z-1)],
[(b3*b2*a*y)/((z-1)*y+z^2-2*z+1),(b3*b2*y)/((z-1)*y+z^2-2*z+1),(b3*b2*y+(-b3*a+(c2-1)*b3)*z+b3*a+(-c2+1)*b3)/((z-1)*y+z^2-2*z+1),((b2*a+(b3-c3+1)*b2)*y)/((z-1)*y+z^2-2*z+1),(-b3)/(y+z-1),(b2*y)/((z-1)*y+z^2-2*z+1),(((b2-c3+1)*z+c3-1)*y+(-a-b3+c2-1)*z^2+(a+b3-c2+c3)*z-c3+1)/((z^2-z)*y+z^3-2*z^2+z),(-1)/(y+z-1)],
[(-b3*b2*b1*a*y*x^2+(-b3*b2*b1*a*y^2+(-2*b3*b2*b1*a*z+2*b3*b2*b1*a)*y)*x)/(((z-1)*y+z^2-2*z+1)*x^2+((z-1)*y^2+(3*z^2-6*z+3)*y+2*z^3-6*z^2+6*z-2)*x+(z^2-2*z+1)*y^2+(2*z^3-6*z^2+6*z-2)*y+z^4-4*z^3+6*z^2-4*z+1),(-b3*b2*b1*y*x^2+(-b3*b2*b1*y^2+(-2*b3*b2*b1*z+2*b3*b2*b1)*y)*x+((b3*b2*a+(-c1+1)*b3*b2)*z-b3*b2*a+(c1-1)*b3*b2)*y^2+((b3*b2*a+(-c1+1)*b3*b2)*z^2+(-2*b3*b2*a+(2*c1-2)*b3*b2)*z+b3*b2*a+(-c1+1)*b3*b2)*y)/(((z-1)*y+z^2-2*z+1)*x^2+((z-1)*y^2+(3*z^2-6*z+3)*y+2*z^3-6*z^2+6*z-2)*x+(z^2-2*z+1)*y^2+(2*z^3-6*z^2+6*z-2)*y+z^4-4*z^3+6*z^2-4*z+1),((-b3*b2*b1*y+(b3*b1*a+(-c2+1)*b3*b1)*z-b3*b1*a+(c2-1)*b3*b1)*x^2+(-b3*b2*b1*y^2+(-2*b3*b2*b1*z+2*b3*b2*b1)*y+(b3*b1*a+(-c2+1)*b3*b1)*z^2+(-2*b3*b1*a+(2*c2-2)*b3*b1)*z+b3*b1*a+(-c2+1)*b3*b1)*x)/(((z-1)*y+z^2-2*z+1)*x^2+((z-1)*y^2+(3*z^2-6*z+3)*y+2*z^3-6*z^2+6*z-2)*x+(z^2-2*z+1)*y^2+(2*z^3-6*z^2+6*z-2)*y+z^4-4*z^3+6*z^2-4*z+1),((-b2*b1*a+(-b3+c3-1)*b2*b1)*y*x^2+((-b2*b1*a+(-b3+c3-1)*b2*b1)*y^2+((-2*b2*b1*a+(-2*b3+2*c3-2)*b2*b1)*z+2*b2*b1*a+(2*b3-2*c3+2)*b2*b1)*y)*x)/(((z-1)*y+z^2-2*z+1)*x^2+((z-1)*y^2+(3*z^2-6*z+3)*y+2*z^3-6*z^2+6*z-2)*x+(z^2-2*z+1)*y^2+(2*z^3-6*z^2+6*z-2)*y+z^4-4*z^3+6*z^2-4*z+1),(b3*b1*x^2+((-b3*a+(c1+c2-2)*b3)*y+(-b3*a+b3*b1+(c1+c2-2)*b3)*z+b3*a-b3*b1+(-c1-c2+2)*b3)*x+b3*b2*y^2+((-b3*a+b3*b2+(c1+c2-2)*b3)*z+b3*a-b3*b2+(-c1-c2+2)*b3)*y+(-b3*a+(c1+c2-2)*b3)*z^2+(2*b3*a+(-2*c1-2*c2+4)*b3)*z-b3*a+(c1+c2-2)*b3)/((y+z-1)*x^2+(y^2+(3*z-3)*y+2*z^2-4*z+2)*x+(z-1)*y^2+(2*z^2-4*z+2)*y+z^3-3*z^2+3*z-1),(-b2*b1*y*x^2+(-b2*b1*y^2+(-2*b2*b1*z+2*b2*b1)*y)*x+((b2*a+(b3-c1-c3+2)*b2)*z-b2*a+(-b3+c1+c3-2)*b2)*y^2+((b2*a+(b3-c1-c3+2)*b2)*z^2+(-2*b2*a+(-2*b3+2*c1+2*c3-4)*b2)*z+b2*a+(b3-c1-c3+2)*b2)*y)/(((z-1)*y+z^2-2*z+1)*x^2+((z-1)*y^2+(3*z^2-6*z+3)*y+2*z^3-6*z^2+6*z-2)*x+(z^2-2*z+1)*y^2+(2*z^3-6*z^2+6*z-2)*y+z^4-4*z^3+6*z^2-4*z+1),((-b2*b1*y+(b1*a+(b3-c2-c3+2)*b1)*z-b1*a+(-b3+c2+c3-2)*b1)*x^2+(-b2*b1*y^2+(-2*b2*b1*z+2*b2*b1)*y+(b1*a+(b3-c2-c3+2)*b1)*z^2+(-2*b1*a+(-2*b3+2*c2+2*c3-4)*b1)*z+b1*a+(b3-c2-c3+2)*b1)*x)/(((z-1)*y+z^2-2*z+1)*x^2+((z-1)*y^2+(3*z^2-6*z+3)*y+2*z^3-6*z^2+6*z-2)*x+(z^2-2*z+1)*y^2+(2*z^3-6*z^2+6*z-2)*y+z^4-4*z^3+6*z^2-4*z+1),(((-c3+1)*y+(b1-c3+1)*z+c3-1)*x^2+((-c3+1)*y^2+((-a-b3+c1+c2-2*c3)*z+3*c3-3)*y+(-a+b1-b3+c1+c2-c3-1)*z^2+(a-b1+b3-c1-c2+3*c3-1)*z-2*c3+2)*x+((b2-c3+1)*z+c3-1)*y^2+((-a+b2-b3+c1+c2-c3-1)*z^2+(a-b2+b3-c1-c2+3*c3-1)*z-2*c3+2)*y+(-a-b3+c1+c2-2)*z^3+(2*a+2*b3-2*c1-2*c2+c3+3)*z^2+(-a-b3+c1+c2-2*c3)*z+c3-1)/((z*y+z^2-z)*x^2+(z*y^2+(3*z^2-3*z)*y+2*z^3-4*z^2+2*z)*x+(z^2-z)*y^2+(2*z^3-4*z^2+2*z)*y+z^4-3*z^3+3*z^2-z)] ] ]
```
• Lauricella's FB
• Program: FB3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+dy+dz+c-1) - x*yang.mul(dx+a1,dx+b1),
yang.mul(dy,dx+dy+dz+c-1) - y*yang.mul(dy+a2,dy+b2),
yang.mul(dz,dx+dy+dz+c-1) - z*yang.mul(dz+a3,dz+b3) ];
```
• Rank of differentail system: 8
• Coeffcient matrices of Pfaffian for [1,dx,dy,dz,dx*dy,dx*dz,dy*dz,dx*dy*dz]
```[ [ [0,(1)/(x),0,0,0,0,0,0],
[(-a1*b1)/(x-1),((-b1-a1)*x+c-1)/(x^2-x),0,0,(1)/(x^2-x),(1)/(x^2-x),0,0],
[0,0,0,0,(1)/(x),0,0,0],
[0,0,0,0,0,(1)/(x),0,0],
[0,(-a2*b2*y)/((y-1)*x^2-y*x),(-a1*b1*y+a1*b1)/((y-1)*x-y),0,(((-b1-a1)*y+b1+a1)*x+(c-b2-a2-1)*y)/((y-1)*x^2-y*x),0,0,(y)/((y-1)*x^2-y*x)],
[0,(-a3*b3*z)/((z-1)*x^2-z*x),0,(-a1*b1*z+a1*b1)/((z-1)*x-z),0,(((-b1-a1)*z+b1+a1)*x+(c-b3-a3-1)*z)/((z-1)*x^2-z*x),0,(z)/((z-1)*x^2-z*x)],
[0,0,0,0,0,0,0,(1)/(x)],
[0,0,0,0,(-a3*b3*z*y)/(((z-1)*y-z)*x^2-z*y*x),(-a2*b2*z*y)/(((z-1)*y-z)*x^2-z*y*x),((-a1*b1*z+a1*b1)*y+a1*b1*z)/(((z-1)*y-z)*x-z*y),((((-b1-a1)*z+b1+a1)*y+(b1+a1)*z)*x+(c-b2-b3-a2-a3-1)*z*y)/(((z-1)*y-z)*x^2-z*y*x)] ],

[ [0,0,(1)/(y),0,0,0,0,0],
[0,0,0,0,(1)/(y),0,0,0],
[(-a2*b2)/(y-1),0,((-b2-a2)*y+c-1)/(y^2-y),0,(1)/(y^2-y),0,(1)/(y^2-y),0],
[0,0,0,0,0,0,(1)/(y),0],
[0,(-a2*b2*x+a2*b2)/((y-1)*x-y),(-a1*b1*x)/((y^2-y)*x-y^2),0,(((-b2-a2)*y+c-b1-a1-1)*x+(b2+a2)*y)/((y^2-y)*x-y^2),0,0,(x)/((y^2-y)*x-y^2)],
[0,0,0,0,0,0,0,(1)/(y)],
[0,0,(-a3*b3*z)/((z-1)*y^2-z*y),(-a2*b2*z+a2*b2)/((z-1)*y-z),0,0,(((-b2-a2)*z+b2+a2)*y+(c-b3-a3-1)*z)/((z-1)*y^2-z*y),(z)/((z-1)*y^2-z*y)],
[0,0,0,0,(-a3*b3*z*x)/(((z-1)*y^2-z*y)*x-z*y^2),((-a2*b2*z+a2*b2)*x+a2*b2*z)/(((z-1)*y-z)*x-z*y),(-a1*b1*z*x)/(((z-1)*y^2-z*y)*x-z*y^2),((((-b2-a2)*z+b2+a2)*y+(c-b1-b3-a1-a3-1)*z)*x+(b2+a2)*z*y)/(((z-1)*y^2-z*y)*x-z*y^2)] ],

[ [0,0,0,(1)/(z),0,0,0,0],
[0,0,0,0,0,(1)/(z),0,0],
[0,0,0,0,0,0,(1)/(z),0],
[(-a3*b3)/(z-1),0,0,((-b3-a3)*z+c-1)/(z^2-z),0,(1)/(z^2-z),(1)/(z^2-z),0],
[0,0,0,0,0,0,0,(1)/(z)],
[0,(-a3*b3*x+a3*b3)/((z-1)*x-z),0,(-a1*b1*x)/((z^2-z)*x-z^2),0,(((-b3-a3)*z+c-b1-a1-1)*x+(b3+a3)*z)/((z^2-z)*x-z^2),0,(x)/((z^2-z)*x-z^2)],
[0,0,(-a3*b3*y+a3*b3)/((z-1)*y-z),(-a2*b2*y)/((z^2-z)*y-z^2),0,0,(((-b3-a3)*z+c-b2-a2-1)*y+(b3+a3)*z)/((z^2-z)*y-z^2),(y)/((z^2-z)*y-z^2)],
[0,0,0,0,((-a3*b3*y+a3*b3)*x+a3*b3*y)/(((z-1)*y-z)*x-z*y),(-a2*b2*y*x)/(((z^2-z)*y-z^2)*x-z^2*y),(-a1*b1*y*x)/(((z^2-z)*y-z^2)*x-z^2*y),((((-b3-a3)*z+c-b1-b2-a1-a2-1)*y+(b3+a3)*z)*x+(b3+a3)*z*y)/(((z^2-z)*y-z^2)*x-z^2*y)] ] ]
```
• Lauricella's FC
• Program: FC3.rr
• differntial system (yang format):
```[ yang.mul(dx,dx+c1-1) - x*yang.mul(dx+dy+dz+a, dx+dy+dz+b),
yang.mul(dy,dy+c2-1) - y*yang.mul(dx+dy+dz+a, dx+dy+dz+b),
yang.mul(dz,dz+c3-1) - z*yang.mul(dx+dy+dz+a, dx+dy+dz+b) ]
```
• Rank of differentail system: 8
• Coeffcient matrices of Pfaffian for [1,dx,dy,dz,dx*dy,dx*dz,dy*dz,dx*dy*dz]
```[ [ [0,(1)/(x),0,0,0,0,0,0],
[(-b*a)/(x+y+z-1),((-a-b)*x+(-c1+1)*y+(-c1+1)*z+c1-1)/(x^2+(y+z-1)*x),(-a-b+c2-1)/(x+y+z-1),(-a-b+c3-1)/(x+y+z-1),(-2)/(x+y+z-1),(-2)/(x+y+z-1),(-2)/(x+y+z-1),0],
[0,0,0,0,(1)/(x),0,0,0],
[0,0,0,0,0,(1)/(x),0,0],
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] ]
```

## Hypergeometric functions of type (k,n)

• type (3,6)
• Program: E36.rr
• differntial system (yang format):
```T  = a2+a3+a4-1+dx1+dx2+dx3+dx4;
T13= dx1+dx3 +1-a5;
T24= dx2+dx4 +1-a6;
T12= dx1+dx2 +a2;
T34= dx3+dx4 +a3;
[ yang.mul(T, dx1)         - x1*yang.mul(T13, T12),
yang.mul(T, dx2)         - x2*yang.mul(T24, T12),
yang.mul(T, dx3)         - x3*yang.mul(T13, T34),
yang.mul(T, dx4)         - x4*yang.mul(T24, T34),
x1*yang.mul(-T13, dx2)   - x2*yang.mul(-T24, dx1),
x3*yang.mul(-T13, dx4)   - x4*yang.mul(-T24, dx3),
x1*yang.mul(-T12, dx3)   - x3*yang.mul(-T34, dx1),
x2*yang.mul(-T12, dx4)   - x4*yang.mul(-T34, dx2),
x2*x3*yang.mul(dx1,dx4)  - x1*x4*yang.mul(dx2,dx3) ];
```
(See Theorem 1.6.1 (pp.47) of
[1] Matsumoto, Sasaki and Yoshida: "The monodromy group of the period map of a 4-parameter family of K3 surface and the hypergeometric function of type (3,6)", Internat. J. Math. 3(1992), 1--164)
• Rank of differentail system: 6
• Coeffcient matrices of Pfaffian for [1,dx1,dx2,dx3,dx4,dx1*dx4]
```[ [ [0,(1)/(x1),0,0,0,0]
[((a5-1)*a2)/(x1-1),((((-a2+a3+a5-a6)*x1+a2+a4+a6-2)*x2+(a2-a3-a5+1)*x1^2+(-a2-a4+1)*x1)*x3+((a2-a5+a6)*x1^2+(-a2-a3-a4-a6+2)*x1)*x2+(-a2+a5-1)*x1^3+(a2+a3+a4-1)*x1^2)/(((x1^2-x1)*x2-x1^3+x1^2)*x3+(-x1^3+x1^2)*x2+x1^4-x1^3),((a5-1)*x2-a5+1)/((x1-1)*x2-x1^2+x1),(-a2*x3+a2)/((x1-1)*x3-x1^2+x1),0,((((2*x1-1)*x2-x1^2)*x3-x1^2*x2+x1^2)*x4+(-x2^2+x2)*x3^2+(x2^2+(x1^2-2*x1)*x2)*x3)/((((x1^2-x1)*x2-x1^3+x1^2)*x3+(-x1^3+x1^2)*x2+x1^4-x1^3)*x4)]
[0,((a6-1)*x2)/(x1*x2-x1^2),(-a5+1)/(x2-x1),0,0,(-x2*x4+x2*x3)/((x1*x2-x1^2)*x4)]
[0,(-a3*x3)/(x1*x3-x1^2),0,(a2)/(x3-x1),0,(-x3*x4+x2*x3)/((x1*x3-x1^2)*x4)]
[0,0,0,0,0,(1)/(x1)]
[0,(((((2*a6-2)*a3*x1+(-a6+1)*a3)*x2+(-a6+1)*a3*x1^2)*x3+(-a6+1)*a3*x1^2*x2+(a6-1)*a3*x1^2)*x4^2+(((-a6+1)*a3*x2^2+(a6-1)*a3*x2)*x3^2+((a6-1)*a3*x2^2+((a6-1)*a3*x1^2+(-2*a6+2)*a3*x1)*x2)*x3)*x4)/((((x1^2-x1)*x2-x1^3+x1^2)*x3+(-x1^3+x1^2)*x2+x1^4-x1^3)*x4^2+(((-2*x1+1)*x2^2+2*x1^2*x2-x1^2)*x3^2+(2*x1^2*x2^2+(-2*x1^3-2*x1^2)*x2+2*x1^3)*x3-x1^2*x2^2+2*x1^3*x2-x1^4)*x4+(x2^3+(-x1-1)*x2^2+x1*x2)*x3^3+((-x1-1)*x2^3+(x1^2+3*x1)*x2^2-2*x1^2*x2)*x3^2+(x1*x2^3-2*x1^2*x2^2+x1^3*x2)*x3),(((-a5+1)*a3*x1*x2+(a5-1)*a3*x1)*x4^2+((a5-1)*a3*x1*x2+(-a5+1)*a3*x1)*x3*x4)/(((x1^2-x1)*x2-x1^3+x1^2)*x4^2+(((-2*x1+1)*x2^2+2*x1^2*x2-x1^2)*x3+x1*x2^2-2*x1^2*x2+x1^3)*x4+(x2^3+(-x1-1)*x2^2+x1*x2)*x3^2+(-x2^3+2*x1*x2^2-x1^2*x2)*x3),(((-a6+1)*a2*x1*x3+(a6-1)*a2*x1)*x4^2+((a6-1)*a2*x1*x2*x3+(-a6+1)*a2*x1*x2)*x4)/(((x1^2-x1)*x3-x1^3+x1^2)*x4^2+(((-2*x1+1)*x2+x1)*x3^2+(2*x1^2*x2-2*x1^2)*x3-x1^2*x2+x1^3)*x4+(x2^2-x2)*x3^3+((-x1-1)*x2^2+2*x1*x2)*x3^2+(x1*x2^2-x1^2*x2)*x3),((a5-1)*a2*x1*x4^2+(-a5+1)*a2*x1*x4)/((x1^2-x1)*x4^2+(((-2*x1+1)*x2+x1)*x3+x1*x2-x1^2)*x4+(x2^2-x2)*x3^2+(-x2^2+x1*x2)*x3),(((((-a2-a3+a5+a6-2)*x1^2+(a2+a3+a4-1)*x1)*x2+(a2-a5-a6+2)*x1^3+(-a2-a4+1)*x1^2)*x3+((a2+a3-a5+1)*x1^3+(-a2-a3-a4-a6+2)*x1^2)*x2+(-a2+a5-1)*x1^4+(a2+a4+a6-2)*x1^3)*x4^2+((((a2+a3-a5-a6+1)*x1+1)*x2^2+((-a2+a3+a5+a6-1)*x1^2+(-a2-2*a3-a4)*x1)*x2+(a2+a4-1)*x1^2)*x3^2+(((-a2-a3+a5-a6+1)*x1^2+(-a2-a3-a4+a6-1)*x1)*x2^2+((a2-a3-a5+a6-1)*x1^3+(3*a2+3*a3+2*a4-a5-a6+1)*x1^2)*x2+(-2*a2-a4+a5)*x1^3)*x3+(a2+a3+a4+a6-2)*x1^2*x2^2+(-2*a2-a3-a4+a5-a6+1)*x1^3*x2+(a2-a5+1)*x1^4)*x4+(x2^3+((-a3-1)*x1-1)*x2^2+(a3+1)*x1*x2)*x3^3+(((a6-2)*x1-1)*x2^3+((a3-a6+2)*x1^2+(a3+a4+a5+1)*x1)*x2^2+(-2*a3-a4-a5)*x1^2*x2)*x3^2+((-a6+2)*x1*x2^3+(-a3-a4-a5+a6-1)*x1^2*x2^2+(a3+a4+a5-1)*x1^3*x2)*x3)/((((x1^3-x1^2)*x2-x1^4+x1^3)*x3+(-x1^4+x1^3)*x2+x1^5-x1^4)*x4^2+(((-2*x1^2+x1)*x2^2+2*x1^3*x2-x1^3)*x3^2+(2*x1^3*x2^2+(-2*x1^4-2*x1^3)*x2+2*x1^4)*x3-x1^3*x2^2+2*x1^4*x2-x1^5)*x4+(x1*x2^3+(-x1^2-x1)*x2^2+x1^2*x2)*x3^3+((-x1^2-x1)*x2^3+(x1^3+3*x1^2)*x2^2-2*x1^3*x2)*x3^2+(x1^2*x2^3-2*x1^3*x2^2+x1^4*x2)*x3)] ],

[ [0,0,(1)/(x2),0,0,0]
[0,(a6-1)/(x2-x1),((-a5+1)*x1)/(x2^2-x1*x2),0,0,(-x4+x3)/((x2-x1)*x4)]
[((a6-1)*a2)/(x2-1),((-a6+1)*x1+a6-1)/(x2^2+(-x1-1)*x2+x1),(((-a2+a3+a6-1)*x2^2+((a2-a3+a5-a6)*x1+a2+a4-1)*x2+(-a2-a4-a5+2)*x1)*x4+(a2-a6+1)*x2^3+((-a2-a5+a6)*x1-a2-a3-a4+1)*x2^2+(a2+a3+a4+a5-2)*x1*x2)/((x2^3+(-x1-1)*x2^2+x1*x2)*x4-x2^4+(x1+1)*x2^3-x1*x2^2),0,(-a2*x4+a2)/((x2-1)*x4-x2^2+x2),((x1^2-x1)*x4^2+((x2^2-2*x1*x2+x1)*x3-x1*x2^2+2*x1*x2-x1^2)*x4+(x1-1)*x2^2*x3)/((x1*x2^2+(-x1^2-x1)*x2+x1^2)*x4^2+(-x1*x2^3+(x1^2+x1)*x2^2-x1^2*x2)*x4)]
[0,0,0,0,0,(x3)/(x1*x4)]
[0,0,(-a3*x4)/(x2*x4-x2^2),0,(a2)/(x4-x2),(-x3+x1)/(x1*x4-x1*x2)]
[0,(((-a6+1)*a3*x1^2+(a6-1)*a3*x1)*x4^2+((a6-1)*a3*x1^2+(-a6+1)*a3*x1)*x3*x4)/(((x1^2-x1)*x2-x1^3+x1^2)*x4^2+(((-2*x1+1)*x2^2+2*x1^2*x2-x1^2)*x3+x1*x2^2-2*x1^2*x2+x1^3)*x4+(x2^3+(-x1-1)*x2^2+x1*x2)*x3^2+(-x2^3+2*x1*x2^2-x1^2*x2)*x3),(((a5-1)*a3*x1^3+(-a5+1)*a3*x1^2)*x4^3+(((a5-1)*a3*x1*x2^2+(-2*a5+2)*a3*x1^2*x2+(a5-1)*a3*x1^2)*x3+(-a5+1)*a3*x1^2*x2^2+(2*a5-2)*a3*x1^2*x2+(-a5+1)*a3*x1^3)*x4^2+((a5-1)*a3*x1^2+(-a5+1)*a3*x1)*x2^2*x3*x4)/(((x1^2-x1)*x2^2+(-x1^3+x1^2)*x2)*x4^3+(((-2*x1+1)*x2^3+2*x1^2*x2^2-x1^2*x2)*x3+(-x1^2+2*x1)*x2^3+(x1^3-3*x1^2)*x2^2+x1^3*x2)*x4^2+((x2^4+(-x1-1)*x2^3+x1*x2^2)*x3^2+((2*x1-2)*x2^4+(-2*x1^2+2*x1)*x2^3)*x3-x1*x2^4+2*x1^2*x2^3-x1^3*x2^2)*x4+(-x2^5+(x1+1)*x2^4-x1*x2^3)*x3^2+(x2^5-2*x1*x2^4+x1^2*x2^3)*x3),(((a6-1)*a2*x1*x3+(-a6+1)*a2*x1)*x4)/((x1^2-x1)*x4^2+(((-2*x1+1)*x2+x1)*x3+x1*x2-x1^2)*x4+(x2^2-x2)*x3^2+(-x2^2+x1*x2)*x3),(((-a5+1)*a2*x1*x3+(a5-1)*a2*x1^2)*x4^2+((a5-1)*a2*x1*x3+(-a5+1)*a2*x1^2)*x4)/((x1^2-x1)*x4^3+(((-2*x1+1)*x2+x1)*x3+(-x1^2+2*x1)*x2-x1^2)*x4^2+((x2^2-x2)*x3^2+(2*x1-2)*x2^2*x3-x1*x2^2+x1^2*x2)*x4+(-x2^3+x2^2)*x3^2+(x2^3-x1*x2^2)*x3),((a3*x1^2-a3*x1)*x4^3+((((a2-a3-a5-a6+3)*x1-a2-a4)*x2+(-a2-a3+a5+a6-3)*x1^2+(a2+2*a3+a4)*x1)*x3+((-a3+a5-1)*x1^2+(2*a3+a4+a6-2)*x1)*x2+(-a5+1)*x1^3+(-a3-a4-a6+2)*x1^2)*x4^2+(((-a2+a5+a6-3)*x2^2+((a2+a3-a5-a6+3)*x1+a2+a4)*x2+(-a2-a3-a4)*x1)*x3^2+(((-a2+a3-a5+a6-1)*x1+2*a2+a4-a6+2)*x2^2+((a2+a3+a5-a6+1)*x1^2+(-3*a2-3*a3-2*a4+a5+a6-3)*x1)*x2+(a2+a3+a4-a5+1)*x1^2)*x3+(-a3-a4-a6+2)*x1*x2^2+(a3+a4-a5+a6-1)*x1^2*x2+(a5-1)*x1^3)*x4+((a2-a6+2)*x2^3+((-a2-a3+a6-2)*x1-a2-a4-a5+1)*x2^2+(a2+a3+a4+a5-1)*x1*x2)*x3^2+((-a2+a6-2)*x2^3+(2*a2+a3+a4+a5-a6+1)*x1*x2^2+(-a2-a3-a4-a5+1)*x1^2*x2)*x3)/(((x1^2-x1)*x2-x1^3+x1^2)*x4^3+(((-2*x1+1)*x2^2+2*x1^2*x2-x1^2)*x3+(-x1^2+2*x1)*x2^2+(x1^3-3*x1^2)*x2+x1^3)*x4^2+((x2^3+(-x1-1)*x2^2+x1*x2)*x3^2+((2*x1-2)*x2^3+(-2*x1^2+2*x1)*x2^2)*x3-x1*x2^3+2*x1^2*x2^2-x1^3*x2)*x4+(-x2^4+(x1+1)*x2^3-x1*x2^2)*x3^2+(x2^4-2*x1*x2^3+x1^2*x2^2)*x3)] ],

[ [0,0,0,(1)/(x3),0,0]
[0,(-a3)/(x3-x1),0,(a2*x1)/(x3^2-x1*x3),0,(-x4+x2)/((x3-x1)*x4)]
[0,0,0,0,0,(x2)/(x1*x4)]
[((a5-1)*a3)/(x3-1),(a3*x1-a3)/(x3^2+(-x1-1)*x3+x1),0,(((-a3+a5-a6)*x3^2+((-a2+a3-a5+a6)*x1+a2+a3+a4+a6-2)*x3+(-a3-a4-a6+2)*x1)*x4+(a3-a5+1)*x3^3+((a2-a3+a5-1)*x1-a2-a3-a4+1)*x3^2+(a3+a4-1)*x1*x3)/((x3^3+(-x1-1)*x3^2+x1*x3)*x4-x3^4+(x1+1)*x3^3-x1*x3^2),((a5-1)*x4-a5+1)/((x3-1)*x4-x3^2+x3),((x1^2-x1)*x4^2+((x2-x1)*x3^2+(-2*x1*x2+2*x1)*x3+x1*x2-x1^2)*x4+(x1-1)*x2*x3^2)/((x1*x3^2+(-x1^2-x1)*x3+x1^2)*x4^2+(-x1*x3^3+(x1^2+x1)*x3^2-x1^2*x3)*x4)]
[0,0,0,((a6-1)*x4)/(x3*x4-x3^2),(-a5+1)/(x4-x3),(-x2+x1)/(x1*x4-x1*x3)]
[0,(((-a6+1)*a3*x1^2+(a6-1)*a3*x1)*x4^2+((a6-1)*a3*x1^2+(-a6+1)*a3*x1)*x2*x4)/(((x1^2-x1)*x3-x1^3+x1^2)*x4^2+(((-2*x1+1)*x2+x1)*x3^2+(2*x1^2*x2-2*x1^2)*x3-x1^2*x2+x1^3)*x4+(x2^2-x2)*x3^3+((-x1-1)*x2^2+2*x1*x2)*x3^2+(x1*x2^2-x1^2*x2)*x3),(((a5-1)*a3*x1*x2+(-a5+1)*a3*x1)*x4)/((x1^2-x1)*x4^2+(((-2*x1+1)*x2+x1)*x3+x1*x2-x1^2)*x4+(x2^2-x2)*x3^2+(-x2^2+x1*x2)*x3),(((a6-1)*a2*x1^3+(-a6+1)*a2*x1^2)*x4^3+(((a6-1)*a2*x1*x2+(-a6+1)*a2*x1^2)*x3^2+((-2*a6+2)*a2*x1^2*x2+(2*a6-2)*a2*x1^2)*x3+(a6-1)*a2*x1^2*x2+(-a6+1)*a2*x1^3)*x4^2+((a6-1)*a2*x1^2+(-a6+1)*a2*x1)*x2*x3^2*x4)/(((x1^2-x1)*x3^2+(-x1^3+x1^2)*x3)*x4^3+(((-2*x1+1)*x2-x1^2+2*x1)*x3^3+(2*x1^2*x2+x1^3-3*x1^2)*x3^2+(-x1^2*x2+x1^3)*x3)*x4^2+((x2^2+(2*x1-2)*x2-x1)*x3^4+((-x1-1)*x2^2+(-2*x1^2+2*x1)*x2+2*x1^2)*x3^3+(x1*x2^2-x1^3)*x3^2)*x4+(-x2^2+x2)*x3^5+((x1+1)*x2^2-2*x1*x2)*x3^4+(-x1*x2^2+x1^2*x2)*x3^3),(((-a5+1)*a2*x1*x2+(a5-1)*a2*x1^2)*x4^2+((a5-1)*a2*x1*x2+(-a5+1)*a2*x1^2)*x4)/((x1^2-x1)*x4^3+(((-2*x1+1)*x2-x1^2+2*x1)*x3+x1*x2-x1^2)*x4^2+((x2^2+(2*x1-2)*x2-x1)*x3^2+(-x2^2+x1^2)*x3)*x4+(-x2^2+x2)*x3^3+(x2^2-x1*x2)*x3^2),(((-a6+1)*x1^2+(a6-1)*x1)*x4^3+((((a2+a3-a5+a6+1)*x1-a2-a3-a4-a6+1)*x2+(-a2+a6-1)*x1^2+(a2+a4+a5-a6-1)*x1)*x3+((-a2-a3+a5+a6-3)*x1^2+(a2+a3+a4-a6+1)*x1)*x2+a2*x1^3+(-a2-a4-a5+2)*x1^2)*x4^2+(((-a2-a3+a5-2)*x2^2+((a2-a3+a5-a6-1)*x1+a2+2*a3+a4-a5+a6+1)*x2+(-a2-a4-a5+2)*x1)*x3^2+(((a2+a3-a5-a6+3)*x1+a2+a3+a4+a6-1)*x2^2+((-a2+a3-a5-a6+3)*x1^2+(-3*a2-3*a3-2*a4+a5+a6-3)*x1)*x2+(2*a2+a4+a5-2)*x1^2)*x3+(-a2-a3-a4)*x1*x2^2+(2*a2+a3+a4)*x1^2*x2-a2*x1^3)*x4+((a3-a5+2)*x2^2+(-a3+a5-2)*x2)*x3^3+(((-a3+a5+a6-3)*x1-a3-a4-a6+1)*x2^2+(2*a3+a4-a5+2)*x1*x2)*x3^2+((a3+a4)*x1*x2^2+(-a3-a4)*x1^2*x2)*x3)/(((x1^2-x1)*x3-x1^3+x1^2)*x4^3+(((-2*x1+1)*x2-x1^2+2*x1)*x3^2+(2*x1^2*x2+x1^3-3*x1^2)*x3-x1^2*x2+x1^3)*x4^2+((x2^2+(2*x1-2)*x2-x1)*x3^3+((-x1-1)*x2^2+(-2*x1^2+2*x1)*x2+2*x1^2)*x3^2+(x1*x2^2-x1^3)*x3)*x4+(-x2^2+x2)*x3^4+((x1+1)*x2^2-2*x1*x2)*x3^3+(-x1*x2^2+x1^2*x2)*x3^2)] ],

[ [0,0,0,0,(1)/(x4),0]
[0,0,0,0,0,(1)/(x4)]
[0,0,(-a3)/(x4-x2),0,(a2*x2)/(x4^2-x2*x4),(-x2*x3+x1*x2)/(x1*x4^2-x1*x2*x4)]
[0,0,0,(a6-1)/(x4-x3),((-a5+1)*x3)/(x4^2-x3*x4),((-x2+x1)*x3)/(x1*x4^2-x1*x3*x4)]
[((a6-1)*a3)/(x4-1),0,(a3*x2-a3)/(x4^2+(-x2-1)*x4+x2),((-a6+1)*x3+a6-1)/(x4^2+(-x3-1)*x4+x3),((-a3+a6-1)*x4^3+((a3+a5-a6)*x3+(-a2+a3-a6+1)*x2+a2+a3+a4-1)*x4^2+(((a2-a3-a5+a6)*x2-a2-a3-a4-a5+2)*x3+(-a3-a4+1)*x2)*x4+(a3+a4+a5-2)*x2*x3)/(x4^4+(-x3-x2-1)*x4^3+((x2+1)*x3+x2)*x4^2-x2*x3*x4),(((x2-x1)*x3-x1*x2+x1)*x4^2+(2*x1-2)*x2*x3*x4+(-x2^2+x2)*x3^2+(x2^2-x1*x2)*x3)/(x1*x4^4+(-x1*x3-x1*x2-x1)*x4^3+((x1*x2+x1)*x3+x1*x2)*x4^2-x1*x2*x3*x4)]
[0,(((a6-1)*a3*x1^2+(-a6+1)*a3*x1)*x4)/((x1^2-x1)*x4^2+(((-2*x1+1)*x2+x1)*x3+x1*x2-x1^2)*x4+(x2^2-x2)*x3^2+(-x2^2+x1*x2)*x3),((((-a5+1)*a3*x1*x2+(a5-1)*a3*x1)*x3+(a5-1)*a3*x1^2*x2+(-a5+1)*a3*x1^2)*x4)/((x1^2-x1)*x4^3+(((-2*x1+1)*x2+x1)*x3+(-x1^2+2*x1)*x2-x1^2)*x4^2+((x2^2-x2)*x3^2+(2*x1-2)*x2^2*x3-x1*x2^2+x1^2*x2)*x4+(-x2^3+x2^2)*x3^2+(x2^3-x1*x2^2)*x3),((((-a6+1)*a2*x1*x2+(a6-1)*a2*x1^2)*x3+(a6-1)*a2*x1*x2+(-a6+1)*a2*x1^2)*x4)/((x1^2-x1)*x4^3+(((-2*x1+1)*x2-x1^2+2*x1)*x3+x1*x2-x1^2)*x4^2+((x2^2+(2*x1-2)*x2-x1)*x3^2+(-x2^2+x1^2)*x3)*x4+(-x2^2+x2)*x3^3+(x2^2-x1*x2)*x3^2),((((a5-1)*a2*x1*x2+(-a5+1)*a2*x1^2)*x3+(-a5+1)*a2*x1^2*x2+(a5-1)*a2*x1^2)*x4^2+((2*a5-2)*a2*x1^2+(-2*a5+2)*a2*x1)*x2*x3*x4+((-a5+1)*a2*x1*x2^2+(a5-1)*a2*x1*x2)*x3^2+((a5-1)*a2*x1*x2^2+(-a5+1)*a2*x1^2*x2)*x3)/((x1^2-x1)*x4^4+(((-2*x1+1)*x2-x1^2+2*x1)*x3+(-x1^2+2*x1)*x2-x1^2)*x4^3+((x2^2+(2*x1-2)*x2-x1)*x3^2+((2*x1-2)*x2^2+(x1^2-2*x1)*x2+x1^2)*x3-x1*x2^2+x1^2*x2)*x4^2+((-x2^2+x2)*x3^3+(-x2^3+(-2*x1+3)*x2^2)*x3^2+(x2^3-x1^2*x2)*x3)*x4+(x2^3-x2^2)*x3^3+(-x2^3+x1*x2^2)*x3^2),(((-a3+a6-1)*x1^2+(a3-a6+1)*x1)*x4^4+((((-a2+a3+a5-a6-1)*x1+a2+a4+a6-1)*x2+(a2+a3-a6+1)*x1^2+(-a2-2*a3-a4-a5+a6+1)*x1)*x3+((a3-a5-a6+2)*x1^2+(-2*a3-a4+a6)*x1)*x2+(a3+a4+a5-2)*x1^2)*x4^3+(((a2-a5+2)*x2^2+((-a2-a3-a5+a6+1)*x1-a2-a4+a5-a6-1)*x2+(a2+a3+a4+a5-2)*x1)*x3^2+(((a2-a3+a5+a6-1)*x1-2*a2-a4-a6)*x2^2+((-a2-a3+a5+a6-2)*x1^2+(3*a2+3*a3+2*a4-a5-a6+1)*x1)*x2+(-a2-a3-a4-a5+2)*x1^2)*x3+(a3+a4-1)*x1*x2^2+(-a3-a4+1)*x1^2*x2)*x4^2+(((a5-2)*x2^2+(-a5+2)*x2)*x3^3+((-a2-1)*x2^3+((a2+a3-a5-a6+1)*x1+a2+a4+a6+1)*x2^2+(-a2-a3-a4+a5-1)*x1*x2)*x3^2+((a2+1)*x2^3+(-2*a2-a3-a4)*x1*x2^2+(a2+a3+a4-1)*x1^2*x2)*x3)*x4+(x2^3-x2^2)*x3^3+(-x2^3+x1*x2^2)*x3^2)/((x1^2-x1)*x4^5+(((-2*x1+1)*x2-x1^2+2*x1)*x3+(-x1^2+2*x1)*x2-x1^2)*x4^4+((x2^2+(2*x1-2)*x2-x1)*x3^2+((2*x1-2)*x2^2+(x1^2-2*x1)*x2+x1^2)*x3-x1*x2^2+x1^2*x2)*x4^3+((-x2^2+x2)*x3^3+(-x2^3+(-2*x1+3)*x2^2)*x3^2+(x2^3-x1^2*x2)*x3)*x4^2+((x2^3-x2^2)*x3^3+(-x2^3+x1*x2^2)*x3^2)*x4)] ] ]
```