In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and more specifically in
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, a holonomic function is a smooth
function of several variables that is a solution of a system of
linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of
D-module
In mathematics, a ''D''-module is a module over a ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has ...
s theory. More precisely, a holonomic function is an element of a
holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called ''holonomic''. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.
Holonomic functions and sequences in one variable
Definitions
Let
be a
field of
characteristic 0 (for example,
or
).
A function
is called ''D-finite'' (or ''holonomic'') if there exist polynomials