Holonomic function

In mathematics, 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 homogenous recurrence relation with polynomial coefficients, or equivalently a linear homogenous difference equation with polynomial coefficients, is holonomic.

Holonomic functions and sequences in one variable

Let $\mathbb {K}$ be a field of characteristic 0 (for example, $\mathbb {K} =\mathbb {Q}$ or $\mathbb {K} =\mathbb {C}$).

A function $f=f(x)$ is called D-finite (or holonomic) if there exist polynomials $a_{r}(x),a_{r-1}(x),\ldots ,a_{0}(x)\in \mathbb {K} [x]$ such that

$a_{r}(x)f^{(r)}(x)+a_{r-1}(x)f^{(r-1)}(x)+\ldots +a_{1}(x)f'(x)+a_{0}(x)f(x)=0$

holds for all x. This can also be written as $Af=0$ where

$A=\sum _{k=0}^{r}a_{k}D_{x}$

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 $\mathbb {K} [x][D_{x}]$, called the annihilator of $f$). The quantity r is called the order of the annihilating operator (by extension, the sequence c is said to have order r when an annihilating operator of such order exists).

A sequence $c=c_{0},c_{1},\ldots$ is called P-recursive (or holonomic) if there exist polynomials $a_{r}(n),a_{r-1}(n),\ldots ,a_{0}(n)\in \mathbb {K} [n]$ such that

$a_{r}(n)c_{n+r}+a_{r-1}(n)c_{n+r-1}+\ldots +a_{0}(n)c_{n}=0$

holds for all n. This can also be written as $Ac=0$ where

$A=\sum _{k=0}^{r}a_{k}S_{n}$

and $S_{n}$ the shift operator that maps $c_{0},c_{1},\ldots$ to $c_{1},c_{2},\ldots$. $A$ is called an annihilating operator of c (the annihilating operators of $c$ form an ideal in the ring $\mathbb {K} [n][S_{n}]$, called the annihilator of $c$). The quantity r is called the order of the annihilating operator (by extension, the sequence c is said to have 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

$f(x)=\sum _{n=0}^{\infty }c_{n}x^{n}$

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).

Closure properties

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 $f(x)=\sum _{n=0}^{\infty }f_{n}x^{n}$ and $g(x)=\sum _{n=0}^{\infty }g_{n}x^{n}$ are holonomic functions, then the following functions are also holonomic:

• $h(x)=\alpha f(x)+\beta g(x)$, where $\alpha$ and $\beta$ are constants
• $h(x)=f(x)g(x)$ (the Cauchy product of the sequences)
• $h(x)=\sum _{n=0}^{\infty }f_{n}g_{n}x^{n}$ (the Hadamard product of the sequences)
• $h(x)=\int _{0}^{x}f(t)dt$
• $h(x)=\sum _{n=0}^{\infty }(\sum _{k=0}^{n}f_{k})x^{n}$
• $h(x)=f(a(x))$, where $a(x)$ is any algebraic function. However, $a(f(x))$ is not generally 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

Examples of holonomic functions include all algebraic functions and some transcendental functions ($\sin(x)$, $\cos(x)$, $e^{x}$, $\log(x)$ among them). More generally, the generalized hypergeometric function ${}_{p}F_{q}(a_{1},\ldots ,a_{p},b_{1},\ldots ,b_{q},x)$ is holonomic, considered as a function of $x$ with all the parameters $a_{i}$, $b_{i}$ held fixed. As a conseqence, the following special functions are all holonomic with respect to $x$:

• The error function $\operatorname {erf} (x)$
• The Bessel functions $J_{n}(x)$, $Y_{n}(x)$, $I_{n}(x)$, $K_{n}(x)$
• The Airy functions $\operatorname {Ai} (x)$, $\operatorname {Bi} (x)$
• The classical orthogonal polynomials (including Legendre polynomials $P_{n}(x)$, Chebyshev polynomials $T_{n}(x)$ and $U_{n}(x)$, etc.)

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 $F_{n}$
• The sequence of factorials $n!$
• The binomial coefficients ${n \choose k}$ (as functions of either n or k)
• The harmonic numbers $H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}$, or more generally $H_{n,m}=\sum _{k=1}^{n}{\frac {1}{k^{m}}}$ for any integer m
• The Motzkin numbers

The following are examples of functions that are not holonomic:

• The function ${\frac {x}{e^{x}-1}}$ is not holonomic, and therefore the Bernoulli numbers are not a holonomic sequence. This follows from the fact that the function has infinitely many singularities, and therefore cannot satisfy a linear differential equation with polynomial coefficients, which necessarily has only finitely many singular points. More generally, the quotient of two holonomic functions is not necessarily holonomic (but can be, in special cases).
• The sequence $\log(n)$, the sequence $n^{\alpha }$ where $\alpha \not \in \mathbb {Z}$, and the sequence of prime numbers are not holonomic.

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

Further reading

• Flajolet, Philippe; Sedgewick, Robert. Analytic Combinatorics. Cambridge University Press. ISBN 0521898064.
• Kauers, Manuel; Paule, Peter. The Concrete Tetrahedron. Text and Monographs in Symbolic Computation. Springer. ISBN 978-3-7091-0444-6.
• Stanley, Richard P. (1999). Enumerative Combinatorics, Volume 2. Cambridge University Press. ISBN 0-521-56069-1.