Quasi-holonomic modules in positive characteristic

Abstract

We study modules over the Carlitz ring, a counterpart of the Weyl algebra in analysis over local fields of positive characteristic. It is shown that some basic objects of function field arithmetic, like the Carlitz module, Thakur's hypergeometric polynomials, and analogs of binomial coefficients arising in the function field version of umbral calculus, generate quasi-holonomic modules. This class of modules is, in many respects, similar to the class of holonomic modules in the characteristic zero theory.

Evolution Equations and Functions of Hypergeometric Type over Fields of Positive Characteristic

“…By the uniqueness of the representation (3.1) [15], we find that σ q l−µ = σ, whenever a l,µ,i 1 ,...,in = 0. Since σ is arbitrary, that is possible if and only if l = µ.…”

“…A natural next step in developing analysis over K is to try to consider partial differential equations with Carlitz derivatives. However, here we encounter a serious difficulty noticed in [15]: the Carlitz derivatives with respect to different variables do not commute. Considering the Carlitz rings of "differential operators" in the above sense in [15], the author found a class of partial differential operators (acting on an appropriate class of functions of several variables), which nevertheless possess reasonable properties.…”

“…However, here we encounter a serious difficulty noticed in [15]: the Carlitz derivatives with respect to different variables do not commute. Considering the Carlitz rings of "differential operators" in the above sense in [15], the author found a class of partial differential operators (acting on an appropriate class of functions of several variables), which nevertheless possess reasonable properties. Operators from this class contain the derivative d in only one distinguished variable, and the linear operator ∆ in every other variable.…”

“…A module M is called quasi-holonomic if d(M) = n + 1 (in the survey papers [16,17] the term "holonomic" is used). As in the classical case, modules associated with basic special functions on K possess this property [15]. In general, the holonomic property is a kind of a "quality certificate" to distinguish objects (equations, functions etc) with a reasonable behavior.…”

“…The basic ingredients of the calculus of F q -linear functions [9] are the Frobenius operator τ u = u q , the difference operator ∆u(t) = u(xt) − xu(t) (1.1) introduced in [3], and the nonlinear (F q -linear) operator d = τ −1 ∆ called the Carlitz derivative. The latter appears, as an analog of the classical derivative, in the theory of ordinary differential equations for F q -linear functions, which has been developed both in the traditional analytic direction (the Cauchy problem [10,13], regular singularity [11], new special functions defined via differential equations [14]) and within various algebraic frameworks (analogs of the canonical commutation relations of quantum mechanics [8,9], umbral calculus [12], an analog of the Weyl algebra [10,15,2]). Note that in our situation the meaning of a polynomial coefficient in a differential equation is not a usual multiplication by a polynomial, but the action of a polynomial in the operator τ (similarly, for holomorphic coefficients).…”