Consider the GKZ system representing the Gauss hypergeometric function as in [SST, Example 1.2.9].
i1 : D = makeWeylAlgebra(frac(QQ[a,b,c])[x_1..x_4]);
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i2 : I = ideal(dx_2*dx_3 - dx_1*dx_4,x_1*dx_1 - x_4*dx_4 + 1 - c,x_2*dx_2 + x_4*dx_4 + a,x_3*dx_3 + x_4*dx_4 + b);
o2 : Ideal of D
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i3 : assert(2 == holonomicRank I)
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i4 : standardMonomials I
o4 = {1, dx }
4
o4 : List
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[SST, Example 1.4.23] computes the connection matrices for this system with constants $a=1/2,b=1/2,c=1$. Using the connectionMatrices function, we can find the system for arbitrary constants.
i5 : A = connectionMatrices I;
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i6 : isIntegrable A
o6 = true
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i7 : netList(Boxes => false, VerticalSpace => 1, apply(4, i -> i+1 => A#i))
o7 = 1 => | (c-1)/x_1 x_4/x_1 |
| ab/(x_2x_3-x_1x_4) (x_4a+x_4b-x_4c+x_4)/(x_2x_3-x_1x_4) |
2 => | (-a)/x_2 (-x_4)/x_2 |
| (-x_1ab)/(x_2^2x_3-x_1x_2x_4) (-x_2x_3a-x_1x_4b+x_2x_3c-x_2x_3)/(x_2^2x_3-x_1x_2x_4) |
3 => | (-b)/x_3 (-x_4)/x_3 |
| (-x_1ab)/(x_2x_3^2-x_1x_3x_4) (-x_1x_4a-x_2x_3b+x_2x_3c-x_2x_3)/(x_2x_3^2-x_1x_3x_4) |
4 => | 0 1 |
| x_1ab/(x_2x_3x_4-x_1x_4^2) (x_1x_4a+x_1x_4b-x_2x_3c+x_1x_4)/(x_2x_3x_4-x_1x_4^2) |
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Substituting the constants, we note that example in [SST] contains a small misprint.
- $A_1=\left(\!\begin{array}{cc} 0&\frac{x_{4}}{x_{1}}\\ \frac{1}{4\,x_{2}x_{3}-4\,x_{1}x_{4}}&\frac{x_{4}}{x_{2}x_{3}-x_{1}x_{4}} \end{array}\!\right)$
- $A_2=\left(\!\begin{array}{cc} \frac{-1}{2\,x_{2}}&\frac{-x_{4}}{x_{2}}\\ \frac{-x_{1}}{4\,x_{2}^{2}x_{3}-4\,x_{1}x_{2}x_{4}}&\frac{-x_{2}x_{3}-x_{1}x_{4}}{2\,x_{2}^{2}x_{3}-2\,x_{1}x_{2}x_{4}} \end{array}\!\right)$
- $A_3=\left(\!\begin{array}{cc} \frac{-1}{2\,x_{3}}&\frac{-x_{4}}{x_{3}}\\ \frac{-x_{1}}{4\,x_{2}x_{3}^{2}-4\,x_{1}x_{3}x_{4}}&\frac{-x_{2}x_{3}-x_{1}x_{4}}{2\,x_{2}x_{3}^{2}-2\,x_{1}x_{3}x_{4}} \end{array}\!\right)$
- $A_4=\left(\!\begin{array}{cc} 0&1\\ \frac{x_{1}}{4\,x_{2}x_{3}x_{4}-4\,x_{1}x_{4}^{2}}&\frac{-x_{2}x_{3}+2\,x_{1}x_{4}}{x_{2}x_{3}x_{4}-x_{1}x_{4}^{2}} \end{array}\!\right)$