quality:correction
Definitions

A formal power series (or function) F (z) is said to be holonomic if it satisfies a linear differential equation of the form c 0 (z) F (r) (z) + c 1 (z) F (r − 1) (z) + ⋯ + c r (z) F (z) = 0, {\displaystyle c_{0}(z)F^{(r)}(z)+c_{1}(z)F^{(r-1)}(z)+\cdots +c_{r}(z)F(z)=0,}

where the coefficients c (z) are in the field of rational functions, C (z) {\displaystyle \mathbb {C} (z)}. Equivalently, F (z) {\displaystyle F(z)} is holonomic if the vector space over C (z) {\displaystyle \mathbb {C} (z)} 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, c (z) are polynomials in z. Thus we can see an equivalent condition that a generating function is holonomic if its coefficients satisfy a P -recurrence of the form c ^ s (n) f n + s + c ^ s − 1 (n) f n + s − 1 + ⋯ + c ^ 0 (n) f n = 0, {\displaystyle {\widehat {c}}_{s}(n)f_{n+s}+{\widehat {c}}_{s-1}(n)f_{n+s-1}+\cdots +{\widehat {c}}_{0}(n)f_{n}=0,}

for all large enough n ≥ n and where the ĉ (n) are fixed finite-degree polynomials in n. In other words, the properties that a sequence be P -recursive and have a holonomic generating function are equivalent. Holonomic functions are closed under the Hadamard product operation ⊙ on generating functions.