In mathematics, and more specifically in analysis, a holonomic function is a smooth function in 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.
Let
K
be a field of characteristic 0 (for example,
K
=
Q
or
K
=
C
).
A function
f
=
f
(
x
)
is called D-finite (or holonomic) if there exist polynomials
a
r
(
x
)
,
a
r
−
1
(
x
)
,
…
,
a
0
(
x
)
∈
K
[
x
]
such that
a
r
(
x
)
f
(
r
)
(
x
)
+
a
r
−
1
(
x
)
f
(
r
−
1
)
(
x
)
+
…
+
a
1
(
x
)
f
′
(
x
)
+
a
0
(
x
)
f
(
x
)
=
0
holds for all x. This can also be written as
A
f
=
0
where
A
=
∑
k
=
0
r
a
k
D
x
k
and
D
x
is the differential operator that maps
f
(
x
)
to
f
′
(
x
)
.
A
is called an annihilating operator of f (the annihilating operators of
f
form an ideal in the ring
K
[
x
]
[
D
x
]
, called the annihilator 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
c
=
c
0
,
c
1
,
…
is called P-recursive (or holonomic) if there exist polynomials
a
r
(
n
)
,
a
r
−
1
(
n
)
,
…
,
a
0
(
n
)
∈
K
[
n
]
such that
a
r
(
n
)
c
n
+
r
+
a
r
−
1
(
n
)
c
n
+
r
−
1
+
…
+
a
0
(
n
)
c
n
=
0
holds for all n. This can also be written as
A
c
=
0
where
A
=
∑
k
=
0
r
a
k
S
n
and
S
n
the shift operator that maps
c
0
,
c
1
,
…
to
c
1
,
c
2
,
…
.
A
is called an annihilating operator of c (the annihilating operators of
c
form an ideal in the ring
K
[
n
]
[
S
n
]
, called the annihilator 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
f
(
x
)
is holonomic, then the coefficients
c
n
in the power series expansion
f
(
x
)
=
∑
n
=
0
∞
c
n
x
n
form a holonomic sequence. Conversely, for a given holonomic sequence
c
n
, the function defined by the above sum is holonomic (this is true in the sense of formal power series, even if the sum has a zero radius of convergence).
Holonomic functions (or sequences) satisfy several closure properties. In particular, holonomic functions (or sequences) form a ring. They are not closed under division, however, and therefore do not form a field.
If
f
(
x
)
=
∑
n
=
0
∞
f
n
x
n
and
g
(
x
)
=
∑
n
=
0
∞
g
n
x
n
are holonomic functions, then the following functions are also holonomic:
h
(
x
)
=
α
f
(
x
)
+
β
g
(
x
)
, where
α
and
β
are constants
h
(
x
)
=
f
(
x
)
g
(
x
)
(the Cauchy product of the sequences)
h
(
x
)
=
∑
n
=
0
∞
f
n
g
n
x
n
(the Hadamard product of the sequences)
h
(
x
)
=
∫
0
x
f
(
t
)
d
t
h
(
x
)
=
∑
n
=
0
∞
(
∑
k
=
0
n
f
k
)
x
n
h
(
x
)
=
f
(
a
(
x
)
)
, where
a
(
x
)
is any algebraic function. However,
a
(
f
(
x
)
)
is generally not holonomic.
A crucial property of holonomic functions is that the closure properties are effective: given annihilating operators for
f
and
g
, an annihilating operator for
h
as defined using any of the above operations can be computed explicitly.
Examples of holonomic functions and sequences
Examples of holonomic functions include:
all algebraic functions
some transcendental functions such as
sin
(
x
)
,
cos
(
x
)
,
e
x
, and
log
(
x
)
the generalized hypergeometric function
p
F
q
(
a
1
,
…
,
a
p
,
b
1
,
…
,
b
q
,
x
)
, considered as a function of
x
with all the parameters
a
i
,
b
i
held fixed
the error function
erf
(
x
)
the Bessel functions
J
n
(
x
)
,
Y
n
(
x
)
,
I
n
(
x
)
,
K
n
(
x
)
the Airy functions
Ai
(
x
)
,
Bi
(
x
)
all classical orthogonal polynomials, including the Legendre polynomials
P
n
(
x
)
and the Chebyshev polynomials
T
n
(
x
)
and
U
n
(
x
)
.
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
F
n
, and more generally, all constant-recursive sequences
the sequence of factorials
n
!
the sequence of binomial coefficients
(
n
k
)
(as functions of either n or k)
the sequence of harmonic numbers
H
n
=
∑
k
=
1
n
1
k
, and more generally
H
n
,
m
=
∑
k
=
1
n
1
k
m
for any integer m
the sequence of Catalan numbers
the sequence of Motzkin numbers.
the sequence of derangements.
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
J
n
and
Y
n
satisfy the second-order linear recurrence
x
(
f
n
+
1
+
f
n
−
1
)
=
2
n
f
n
.
Examples of nonholonomic functions and sequences
Examples of nonholonomic functions include:
the function
x
e
x
−
1
the function tan(x) + sec(x)
the quotient of two holonomic functions is generally not holonomic.
Examples of nonholonomic sequences include:
the Bernoulli numbers
the numbers of alternating permutations
the numbers of integer partitions
the numbers
log
(
n
)
the numbers
n
α
where
α
∉
Z
the prime numbers
the enumerations of irreducible and connected permutations.
Algorithms and software
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 [1] package for Mathematica, developed by Christoph Koutschan, which supports computing closure properties and proving identities for univariate and multivariate holonomic functions
The algolib [2] library for Maple, which includes the following packages:
gfun, developed by Bruno Salvy, Paul Zimmermann and Eithne Murray, for univariate closure properties and proving [3]
mgfun, developed by Frédéric Chyzak, for multivariate closure properties and proving [4]
numgfun, developed by Marc Mezzarobba, for numerical evaluation
