# Holonomic ideals and D-finite power series

I would like to understand the connection between the term D-finite power series (in n variables) and the term of a holonomic module over the Weyl algebra A_n.

A power series $f \in K[[x_1,...,x_n]]$ is called D-finite if all partial derivations of f lie in a finite-dimensional vector space over K(x). Equivalently f satisfies a system of partial differential equations with polynomial coefficient (one equation for every variable).

An left ideal I in the Weyl algebra $A_n$ (the algebra of partial differential operators in n variables with polynomial coefficients) is called holonomic if its Bernstein dimension equals n (the smallest possible value), i.e. the Bernstein dimension of the module $A_n/I$ equals n.

Now i've read the following definition of a holonomic function (e.g. a power series): f is called holonomic if the $A_n$-module $A_n/Ann_f$ is holonomic, where $Ann_f = \{P \in A_n: Pf = 0\}$, the annihilator of f.

My question: does the following hold (i guess so): a formal power series $f \in K[[x_1,...,x_n]]$ is D-finite iff it is holonomic (over A_n)? (does it has to do with the module-isomorphism $A_n/Ann_f \cong A_n \cdot f$?) I've also read that f holonomic implies that f satisfies "many" differential equations with polynomial coefficients. What does this mean precise and what is the intuitive idea behind the term holonomic module? Thanks.

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