The Hadamard product of hypergeometric series
According to the classical definition of Horn, a formal Laurent series is called hypergeometric if the quotients of its adjacent coefficients are rational functions of the summation indices. The general form of the coefficients of a hypergeometric series in \(n\) variables is given by a formula due to Ore and Sato. Any function having this form is called the Ore-Sato coefficient of a hypergeometric series. It determines a set of vectors \(\{A_1,\dots,A_p\}\), with \(A_j \in \mathbb Z^n\), called the principal symbol of the Ore-Sato coefficient. The Ore-Sato coefficient is said to be nonconfluent when \(\sum_{j=1}^p A_j=0\). If \(n=2\), the vectors \(\{A_1,\dots,A_p\}\) are then the outer normals of the sides of a convex polygon, called the polygon of the Ore-Sato coefficient. A hypergeometric series in \(n\) variables with Ore-Sato coefficient \(\varphi\) satisfies a system of \(n\) partial differential equations, known as the Horn hypergeometric system defined by \(\varphi\). The resultant of the principal symbols of the differential operators appearing in the Horn system is a polynomial called the resultant of \(\varphi\). The singular locus of the hypergeometric series is contained in the zero set of the resultant of \(\varphi\). Suppose that a nonconfluent Ore-Sato coefficient in two variables has no identically zero resultant. The main result of this paper proves that, under this assumption, the polygon of the coefficient coincides with the Newton polytope of its resultant. Furthermore, since multiplication of principal symbols of the Ore-Sato coefficients corresponds to the Hadamard (termwise) product of hypergeometric series, it is shown that the Newton polytope of the polynomial defining the singular locus of the Hadamard product of nonconfluent hypergeometric series in two variables is a Minkowski summand in the sum of the Newton polytopes of the resultants of the Ore-Sato coefficients of the factors. Finally, it is shown by example that these results cannot be extended to cases with more than two variables.
scientific article; zbMATH DE number 1751549
scientific article