Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities

A holonomic function is a differentiable or generalized function which satisfies a holonomic system of linear partial or ordinary differential equations with polynomial coefficients. The main purpose of this paper is to present algorithms for computing a holonomic system for the definite integral of a holonomic function with parameters over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic differencedifferential system including parameters, then a holonomic differencedifferential system for the integral can also be computed. In the algorithms, holonomic distributions (generalized functions in the sense of L. Schwartz) are inevitably involved even if the integrand is a usual function. Introduction Holonomic systems of linear differential equations, which play a central role in the D-module theory, were introduced by Bernstein [2] in the algebraic setting, and by Sato et al. [20] in the analytic setting under the name of ‘maximally overdetermined systems’. We follow the formulation by Bernstein, which would be the more adapted to practical applications with computers. Hence, in the present paper, we mean by a holonomic function a function which satisfies a holonomic system of linear differential equations with polynomial coefficients. Two equivalent definitions of a holonomic system will be recalled in Section 1. Most of the special functions in one variable such as various hypergeometric functions and the Bessel function are holonomic by the definition. As an important class of holonomic functions in several variables, let us consider the multi-valued analytic function u = f1 1 · · · fm m with non-zero polynomials f1, . . . , fm in n variables. As a multi-valued analytic function, u is defined on {x ∈ C | f1(x) · · · fm(x) 6= 0} and is holonomic for any complex number λj . We can regard this function also as a distribution (f1) λ1 + · · · (fm) + defined on R in the sense of L. Schwartz if fj are real polynomials and (λ1, . . . , λm) avoids some exceptional set. Such a distribution was introduced by Gel’fand and Shilov [7] in some restricted cases. See e.g., [10] for a theoretical study on generalized functions including such a distribution. In particular, substituting zeros for the parameters yields (f1) 0 + · · · (fm)+ = Y (f1) · · ·Y (fm),