In mathematics, and more specifically in analysis, 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-modules 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 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. A function is called D-finite if there exist polynomials such that holds for all x. This can also be written as where and is the differential operator that maps to. is called an annihilating operator of f. The quantity r is called the order of the annihilating operator. By extension, the holonomic function f is said to be of order r when an annihilating operator of such order exists. A sequence is called P-recursive if there exist polynomials such that holds for all n. This can also be written as where and the shift operator that maps to. is called an annihilating operator of c. The quantity r is called the order of the annihilating operator. By extension, the holonomic sequence c is said to be of order r when an annihilating operator of such order exists. Holonomic functions are precisely the generating functions of holonomic sequences: if is holonomic, then the coefficients in the power series expansion form a holonomic sequence. Conversely, for a given holonomic sequence, the function defined by the above sum is holonomic.
Closure properties
Holonomic functions satisfy several closure properties. In particular, holonomic functions form a ring. They are not closed under division, however, and therefore do not form a field. If and are holonomic functions, then the following functions are also holonomic:
A crucial property of holonomic functions is that the closure properties are effective: given annihilating operators for and, an annihilating operator for as defined using any of the above operations can be computed explicitly.
The class of holonomic functions is a strict superset of the class of hypergeometric functions. Examples of special functions that are holonomic but not hypergeometric include the Heun functions. Examples of holonomic sequences include:
the sequence of Fibonacci numbers, and more generally, all constant-recursive sequences
the sequence of factorials
the sequence of binomial coefficients
the sequence of harmonic numbers, and more generally for any integer m
Hypergeometric functions, Bessel functions, and classical orthogonal polynomials, in addition to being holonomic functions of their variable, are also holonomic sequences with respect to their parameters. For example, the Bessel functions and satisfy the second-order linear recurrence.
Examples of nonholonomic functions and sequences
Examples of nonholonomic functions include:
the function
the function tan + sec
the quotient of two holonomic functions is generally not holonomic.
Holonomic functions are a powerful tool in computer algebra. A holonomic function or sequence can be represented by a finite amount of data, namely an annihilating operator and a finite set of initial values, and the closure properties allow carrying out operations such as equality testing, summation and integration in an algorithmic fashion. In recent years, these techniques have allowed giving automated proofs of a large number of special function and combinatorial identities. Moreover, there exist fast algorithms for evaluating holonomic functions to arbitrary precision at any point in the complex plane, and for numerically computing any entry in a holonomic sequence. Software for working with holonomic functions includes:
The HolonomicFunctions package for Mathematica, developed by Christoph Koutschan, which supports computing closure properties and proving identities for univariate and multivariate holonomic functions
The algolib library for Maple, which includes the following packages:
* gfun, developed by Bruno Salvy, Paul Zimmermann and Eithne Murray, for univariate closure properties and proving
* mgfun, developed by Frédéric Chyzak, for multivariate closure properties and proving
