Holonomic function

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

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

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

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

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

and ${\displaystyle D_{x}}$ is the differential operator that maps ${\displaystyle f(x)}$ to ${\displaystyle f'(x)}$. ${\displaystyle A}$ is called an annihilating operator of f (the annihilating operators of ${\displaystyle f}$ form an ideal in the ring ${\displaystyle \mathbb {K} [x][D_{x}]}$, called the annihilator of ${\displaystyle f}$). 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 ${\displaystyle c=c_{0},c_{1},\ldots }$ is called P-recursive (or holonomic) if there exist polynomials ${\displaystyle a_{r}(n),a_{r-1}(n),\ldots ,a_{0}(n)\in \mathbb {K} [n]}$ such that

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

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

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

and ${\displaystyle S_{n}}$ the shift operator that maps ${\displaystyle c_{0},c_{1},\ldots }$ to ${\displaystyle c_{1},c_{2},\ldots }$. ${\displaystyle A}$ is called an annihilating operator of c (the annihilating operators of ${\displaystyle c}$ form an ideal in the ring ${\displaystyle \mathbb {K} [n][S_{n}]}$, called the annihilator of ${\displaystyle c}$). 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 ${\displaystyle f(x)}$ is holonomic, then the coefficients ${\displaystyle c_{n}}$ in the power series expansion

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

form a holonomic sequence. Conversely, for a given holonomic sequence ${\displaystyle c_{n}}$, 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 ${\displaystyle f(x)=\sum _{n=0}^{\infty }f_{n}x^{n}}$ and ${\displaystyle g(x)=\sum _{n=0}^{\infty }g_{n}x^{n}}$ are holonomic functions, then the following functions are also holonomic:

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

A crucial property of holonomic functions is that the closure properties are effective: given annihilating operators for ${\displaystyle f}$ and ${\displaystyle g}$, an annihilating operator for ${\displaystyle h}$ as defined using any of the above operations can be computed explicitly.

Examples of holonomic functions include:

• all algebraic functions
• some transcendental functions such as ${\displaystyle \sin(x)}$, ${\displaystyle \cos(x)}$, ${\displaystyle e^{x}}$, and ${\displaystyle \log(x)}$[2]
• the generalized hypergeometric function ${\displaystyle {}_{p}F_{q}(a_{1},\ldots ,a_{p},b_{1},\ldots ,b_{q},x)}$, considered as a function of ${\displaystyle x}$ with all the parameters ${\displaystyle a_{i}}$, ${\displaystyle b_{i}}$ held fixed
• the error function ${\displaystyle \operatorname {erf} (x)}$
• the Bessel functions ${\displaystyle J_{n}(x)}$, ${\displaystyle Y_{n}(x)}$, ${\displaystyle I_{n}(x)}$, ${\displaystyle K_{n}(x)}$
• the Airy functions ${\displaystyle \operatorname {Ai} (x)}$, ${\displaystyle \operatorname {Bi} (x)}$
• all classical orthogonal polynomials, including the Legendre polynomials ${\displaystyle P_{n}(x)}$ and the Chebyshev polynomials ${\displaystyle T_{n}(x)}$ and ${\displaystyle U_{n}(x)}$.

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 ${\displaystyle F_{n}}$, and more generally, all constant-recursive sequences
• the sequence of factorials ${\displaystyle n!}$
• the sequence of binomial coefficients ${\displaystyle {n \choose k}}$ (as functions of either n or k)
• the sequence of harmonic numbers ${\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k}}}$, and more generally ${\displaystyle H_{n,m}=\sum _{k=1}^{n}{\frac {1}{k^{m}}}}$ 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 ${\displaystyle J_{n}}$ and ${\displaystyle Y_{n}}$ satisfy the second-order linear recurrence ${\displaystyle x(f_{n+1}+f_{n-1})=2nf_{n}}$.

Examples of nonholonomic functions include:

• the function ${\displaystyle {\frac {x}{e^{x}-1}}}$[3]
• the function tan(x) + sec(x)[4]
• the quotient of two holonomic functions is generally not holonomic.

Examples of nonholonomic sequences include:

• the Bernoulli numbers
• the numbers of alternating permutations[5]
• the numbers of integer partitions[4]
• the numbers ${\displaystyle \log(n)}$[4]
• the numbers ${\displaystyle n^{\alpha }}$ where ${\displaystyle \alpha \not \in \mathbb {Z} }$[4]
• the prime numbers[4]
• the enumerations of irreducible and connected permutations.[6]