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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 \mathbb be a field of characteristic 0 (for example, \mathbb = \mathbb or \mathbb = \mathbb). A function f = f(x) is called ''D-finite'' (or ''holonomic'') if there exist polynomials 0 \neq a_r(x), a_(x), \ldots, a_0(x) \in \mathbb /math> such that :a_r(x) f^(x) + a_(x) f^(x) + \cdots + 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 = \sum_^r a_k D_x^k and D_x is the
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
that maps f(x) to f'(x). A is called an ''annihilating operator'' of ''f'' (the annihilating operators of f form an
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
in the ring \mathbb 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, \ldots is called ''P-recursive'' (or ''holonomic'') if there exist polynomials a_r(n), a_(n), \ldots, a_0(n) \in \mathbb /math> such that :a_r(n) c_ + a_(n) c_ + \cdots + a_0(n) c_n = 0 holds for all ''n''. This can also be written as A c = 0 where :A = \sum_^r a_k S_n and S_n the
shift operator In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator. Shift ...
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 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 function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary serie ...
s of holonomic sequences: if f(x) is holonomic, then the coefficients c_n in the power series expansion :f(x) = \sum_^ 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 In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...
, even if the sum has a zero
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
).


Closure properties

Holonomic functions (or sequences) satisfy several
closure properties Closure may refer to: Conceptual Psychology * Closure (psychology), the state of experiencing an emotional conclusion to a difficult life event Computer science * Closure (computer programming), an abstraction binding a function to its scope * ...
. 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_^ f_n x^n and g(x) = \sum_^ 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 In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infini ...
of the sequences) * h(x) = \sum_^ f_n g_n x^n (the Hadamard product of the sequences) * h(x) = \int_0^x f(t) dt * h(x) = \sum_^ (\sum_^n f_k) x^n * h(x) = f(a(x)), where a(x) is any
algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
. 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 function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
s, including
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s and
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s * the sine and cosine functions (but ''not'' tangent, cotangent, secant, or cosecant) *the hyperbolic sine and cosine functions (but ''not'' hyperbolic tangent, cotangent, secant, or cosecant) *
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
s and
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
s (to any base) * the generalized hypergeometric function _pF_q(a_1,\ldots,a_p, b_1, \ldots, b_q, x), considered as a function of x with all the parameters a_i, b_i held fixed * the
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementa ...
\operatorname(x) * the
Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrar ...
s J_n(x), Y_n(x), I_n(x), K_n(x) * the
Airy function In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solut ...
s \operatorname(x), \operatorname(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 function In mathematics, the local Heun function H \ell (a,q;\alpha ,\beta, \gamma, \delta ; z) is the solution of Heun's differential equation that is holomorphic and 1 at the singular point ''z'' = 0. The local Heun function is called a Heun ...
s. Examples of holonomic sequences include: * the sequence of
Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
s F_n, and more generally, all constant-recursive sequences * the sequence of
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...
s n! * the sequence of
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
s (as functions of either ''n'' or ''k'') * the sequence of
harmonic number In mathematics, the -th harmonic number is the sum of the reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \do ...
s H_n = \sum_^n \frac, and more generally H_ = \sum_^n \frac for any integer ''m'' * the sequence of
Catalan number In combinatorial mathematics, the Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after the French-Belgian mathematician Eugène Charles Ca ...
s * the sequence of
Motzkin number In mathematics, the th Motzkin number is the number of different ways of drawing non-intersecting chords between points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have d ...
s. * the sequence of derangements. Hypergeometric functions, Bessel functions, and classical
orthogonal polynomials In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the class ...
, 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_ + f_) = 2 n f_n.


Examples of nonholonomic functions and sequences

Examples of nonholonomic functions include: * the function \frac * the function tan(''x'') + sec(''x'')See . * the quotient of two holonomic functions is generally not holonomic. Examples of nonholonomic sequences include: * the
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
s * the numbers of
alternating permutation In combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set is a permutation (arrangement) of those numbers so that each entry is alternately greater or less than the preceding entry. For example, the five alte ...
s * the numbers of integer partitions * the numbers \log(n) * the numbers n^ where \alpha \not\in \mathbb * the
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s * the enumerations of ''irreducible and connected permutations''.See .


Holonomic functions in several variables


Algorithms and software

Holonomic functions are a powerful tool in
computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions ...
. 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 Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimiza ...
, developed by Christoph Koutschan, which supports computing closure properties and proving identities for univariate and multivariate holonomic functions * The ''algolib'

library for Maple (software), Maple, which includes the following packages: ** ''gfun'', developed by Bruno Salvy, Paul Zimmermann and Eithne Murray, for univariate closure properties and provin

** ''mgfun'', developed by Frédéric Chyzak, for multivariate closure properties and provin

** ''numgfun'', developed by Marc Mezzarobba, for numerical evaluation


See also

Dynamic Dictionary of Mathematical functions
an online software, based on holonomic functions for automatically studying many classical and special functions (evaluation at a point, Taylor series and asymptotic expansion to any user-given precision, differential equation, recurrence for the coefficients of the Taylor series, derivative, indefinite integral, plotting, ...)


Notes


References

*. * * * (ITI Series preprint) * * * {{Differential equations topics Ordinary differential equations Special functions Types of functions