Differentiably finite function: Difference between revisions

Revision as of 19:02, 13 May 2013

Revision as of 20:27, 15 May 2013

Line 1:

{{Unreferenced|date=January 2012}}

In mathematics, a '''differentiably finite function''' of one variable, also referred to as a '''D-finiteD‑finite''' or '''holonomic''' '''function''', is a [[Function (mathematics)|function]] which is a solution of a [[linear differential equation]] with polynomial coefficients. A '''differentiably finite''' (or D‑finite, or holonomic) '''power series''' is a [[formal power series]] that satisfies a linear differential equation with polynomial coefficients.

== Formal definition ==

<!-- ground field, finite dim over K(x), formal vs analytic case -->

<!-- counterexamples & ways to prove that a function is ''not'' D‑finite?

-->

== P-recursive sequences ==

== Closure properties ==

<!-- ring operations, algebraic substitution, Hadamard product -->

== In combinatorics ==

<!-- asymptotics? -->

<!-- diagonals? we need an article on D-finite functions of several

variables! -->

== In computer algebra: differential equations as a data structure ==

<!-- special functions? -->

== Computation ==

<!-- coefficients: binary splitting, baby steps-giant steps; values:

"bit-burst" algorithm -->

== Further reading ==

* {{cite book |last1=Flajolet |first1=Philippe |last2=Sedgewick |first2=Robert |title=Analytic Combinatorics |publisher=Cambridge University Press |isbn=0521898064}}

* {{cite book |last1=Kauers |first1=Manuel |last2=Paule |first2=Peter |title=The Concrete Tetrahedron| series=Text and Monographs in Symbolic Computation |publisher=Springer |isbn=978-3-7091-0444-6}}

* {{cite book|last=Stanley|first=Richard P. |year=1999|title=Enumerative Combinatorics, Volume&nbsp;2|publisher=Cambridge University Press|isbn=0-521-56069-1}}

[[Category:Functions and mappings]]