Editing Linear differential equation (section)

== Holonomic functions==
{{Main|holonomic function}}
A [[holonomic function]], also called  a ''D-finite function'', is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients.

Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include [[polynomial]]s, [[algebraic function]]s, [[logarithm]], [[exponential function]], [[sine]], [[cosine]], [[hyperbolic sine]], [[hyperbolic cosine]], [[inverse trigonometric functions|inverse trigonometric]] and [[inverse hyperbolic functions]], and many [[special function]]s such as [[Bessel function]]s and [[hypergeometric function]]s.

Holonomic functions have several [[closure property|closure properties]]; in particular, sums, products, [[derivative]] and [[antiderivative|integrals]] of holonomic functions are holonomic. Moreover, these closure properties are effective, in the sense that there are [[algorithm]]s for computing the differential equation of the result of any of these operations, knowing the differential equations of the input.<ref name =zeilberger>Zeilberger, Doron. ''[https://www.sciencedirect.com/science/article/pii/037704279090042X/pdf?md5=8b21c545d20a52a50dffdf6808bba4a8&isDTMRedir=Y&pid=1-s2.0-037704279090042X-main.pdf A holonomic systems approach to special functions identities]''. Journal of computational and applied mathematics. 32.3 (1990): 321-368</ref>

Usefulness of the concept of holonomic functions results of Zeilberger's theorem, which follows.<ref name=zeilberger/>

A ''holonomic sequence'' is a sequence of numbers that may be generated by a [[recurrence relation]] with polynomial coefficients. The coefficients of the [[Taylor series]] at a point of a holonomic function form a holonomic sequence. Conversely, if the sequence of the coefficients of a [[power series]] is holonomic, then the series defines a holonomic function (even if the [[radius of convergence]] is zero). There are efficient algorithms for both conversions, that is for computing the recurrence relation from the differential equation, and ''vice versa''. 
<ref name=zeilberger/>

It follows that, if one represents (in a computer) holonomic functions by their defining differential equations and initial conditions, most [[calculus]] operations can be done automatically on these functions, such as [[derivative]], [[indefinite integral|indefinite]] and [[definite integral]], fast computation of Taylor series (thanks of the recurrence relation on its coefficients), evaluation to a high precision with certified bound of the approximation error, [[limit (mathematics)|limit]]s, localization of [[singularity (mathematics)|singularities]], [[asymptotic behavior]] at infinity and near singularities, proof of identities, etc.<ref>Benoit, A., Chyzak, F., Darrasse, A., Gerhold, S., Mezzarobba, M., & Salvy, B. (2010, September). ''[https://hal.inria.fr/docs/00/78/30/48/PDF/ddmf.pdf The dynamic dictionary of mathematical functions (DDMF)]''. In International Congress on Mathematical Software (pp. 35-41). Springer, Berlin, Heidelberg.</ref>