Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.View current documentation (Version 11.2)


gives the holonomic function , specified by the linear differential equation .

represents a pure holonomic function .


  • Mathematical function, suitable for both symbolic and numerical manipulation; also known as holonomic function and D-finite function.
  • The holonomic function defined by a DifferentialRoot function satisfies a holonomic differential equation with polynomial coefficients and initial values .
  • DifferentialRoot can be used like any other mathematical function.
  • FunctionExpand will attempt to convert DifferentialRoot functions in terms of special functions.
  • The functions representable by DifferentialRoot include a large number of special functions.
  • DifferentialRootReduce can convert most special functions to DifferentialRoot functions.
  • Holonomic functions are closed under many operations, including:
  • constant multiple, integer power
    sums and products
    composition with polynomial, rational, and algebraic functions
    derivatives and integrals
  • DifferentialRoot is automatically generated by functions such as Integrate, DSolve, and GeneratingFunction.
  • Functions such as Integrate, D, SeriesCoefficient, and DSolve work with DifferentialRoot inputs.
  • DifferentialRoot can be evaluated to arbitrary numerical precision.
  • DifferentialRoot automatically threads over lists.
  • DifferentialRoot[lde,pred] represents a solution restricted to avoid cuts in the complex plane defined by , where can contain equations and inequalities.
Introduced in 2008