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Gauss' hypergeometric function

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]);
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
i3 : assert(2 == holonomicRank I)
i4 : standardMonomials I

o4 = {1, dx }
           4

o4 : List

[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;
i6 : isIntegrable A

o6 = true
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) |    

Substituting the constants, we note that example in [SST] contains a small misprint.

References

[SST] M. Saito, B. Sturmfels, and N. Takayama. Gröbner Deformations of Hypergeometric Differential Equations. Volume 6 of Algorithms and Computation in Mathematics. Springer, 2000.

See also


The source of this document is in ConnectionMatrices/examples.m2:141:0.