A Horn system is a system of linear partial differential equations whose solutions include multivariate hypergeometric
series. It consists, for 
, ..., 
,
of equations of the form
 |
(1)
|
where 
and 
are nonzero polynomials,
, and 
(Sadykov 2002). A Horn
function is a particular series solution, whereas the Horn system is the differential-equation
object and can have a multidimensional local solution space. At a nonsingular point,
the dimension of this space is the holonomic rank
when it is finite.
For a Horn-type power series with neighboring-coefficient ratios
 |
(2)
|
the corresponding differential operators are, for 
, ..., 
,
 |
(3)
|
and satisfy 
. When this is a holonomic
system, it can be written as a first-order Pfaffian
system
 |
(4)
|
where 
, the components of 
are 
and finitely many of its derivatives, and the entries of the
connection matrices 
are rational functions. Restricting the system
to a one-dimensional path gives an ordinary
differential equation. Matching generalized power-series solutions by the Frobenius method along the path provides a numerical
analytic continuation from a defining-series
region (Banik and Bera 2026).
