Editing Generating function (section)

====Definitions====

A formal power series (or function) {{math|''F''(''z'')}} is said to be '''holonomic''' if it satisfies a linear differential equation of the form<ref>{{harvnb|Flajolet|Sedgewick|2009|loc=§B.4}}</ref>
<math display="block">c_0(z) F^{(r)}(z) + c_1(z) F^{(r-1)}(z) + \cdots + c_r(z) F(z) = 0, </math>

where the coefficients {{math|''c<sub>i</sub>''(''z'')}} are in the field of rational functions, <math>\mathbb{C}(z)</math>. Equivalently, <math>F(z)</math> is holonomic if the vector space over <math>\mathbb{C}(z)</math> spanned by the set of all of its derivatives is finite dimensional.

Since we can clear denominators if need be in the previous equation, we may assume that the functions, {{math|''c<sub>i</sub>''(''z'')}} are polynomials in {{mvar|z}}. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a '''{{mvar|P}}-recurrence''' of the form
<math display="block">\widehat{c}_s(n) f_{n+s} + \widehat{c}_{s-1}(n) f_{n+s-1} + \cdots + \widehat{c}_0(n) f_n = 0,</math>

for all large enough {{math|''n'' ≥ ''n''<sub>0</sub>}} and where the {{math|''ĉ<sub>i</sub>''(''n'')}} are fixed finite-degree polynomials in {{mvar|n}}. In other words, the properties that a sequence be ''{{mvar|P}}-recursive'' and have a holonomic generating function are equivalent. Holonomic functions are closed under the [[Generating function transformation#Hadamard products and diagonal generating functions|Hadamard product]] operation {{math|⊙}} on generating functions.