In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polynomial coefficients and satisfies a suitable dimension condition in terms of D-modules theory. More precisely, a holonomic function is an element of a holonomic module of smooth functions. Holonomic functions can also be described as differentiably finite functions, also known as D-finite functions. When a power series in the variables is the Taylor expansion of a holonomic function, the sequence of its coefficients, in one or several indices, is also called holonomic. Holonomic sequences are also called P-recursive sequences: they are defined recursively by multivariate recurrences satisfied by the whole sequence and by suitable specializations of it. The situation simplifies in the univariate case: any univariate sequence that satisfies a linear homogeneous recurrence relation with polynomial coefficients, or equivalently a linear homogeneous difference equation with polynomial coefficients, is holonomic.
Holonomic functions and sequences in one variable
Definitions Let K {\displaystyle \mathbb {K} } be a field of characteristic 0 (for example, K = Q {\displaystyle \mathbb {K} =\mathbb {Q} } or K = C {\displaystyle \mathbb {K} =\mathbb {C} } ). A function f = f ( x ) {\displaystyle f=f(x)} is called D {\displaystyle D} -finite (or holonomic) if there exist polynomials 0 ≠ a r ( x ) , a r − 1 ( x ) , … , a 0 ( x ) ∈ K [ x ] {\displaystyle 0\neq a_{r}(x),a_{r-1}(x),\ldots ,a_{0}(x)\in \mathbb {K} [x]} such that
a r ( x ) f ( r ) ( x ) + a r − 1 ( x ) f ( r − 1 ) ( x ) + ⋯ + a 1 ( x ) f ′ ( x ) + a 0 ( x ) f ( x ) = 0 {\displaystyle a_{r}(x)f^{(r)}(x)+a_{r-1}(x)f^{(r-1)}(x)+\cdots +a_{1}(x)f'(x)+a_{0}(x)f(x)=0}
holds for all x {\displaystyle x} . This can also be written as A f = 0 {\displaystyle Af=0} where
A = ∑ k = 0 r a k D x k {\displaystyle A=\sum _{k=0}^{r}a_{k}D_{x}^{k}}
and D x {\displaystyle D_{x}} is the differential operator that maps f ( x ) {\displaystyle f(x)} to f ′ ( x ) {\displaystyle f'(x)} . A {\displaystyle A} is called an annihilating operator of f {\displaystyle f} (the annihilating operators of f {\displaystyle f} form an ideal in the ring K [ x ] [ D x ] {\displaystyle \mathbb {K} [x][D_{x}]} , called the annihilator of f {\displaystyle f} ). The quantity r {\displaystyle r} is called the order of the annihilating operator. By extension, the holonomic function f {\displaystyle f} is said to be of order r {\displaystyle r} when an annihilating operator of such order exists. A sequence c = c 0 , c 1 , … {\displaystyle c=c_{0},c_{1},\ldots } is called P {\displaystyle P} -recursive (or holonomic) if there exist polynomials a r ( n ) , a r − 1 ( n ) , … , a 0 ( n ) ∈ K [ n ] {\displaystyle a_{r}(n),a_{r-1}(n),\ldots ,a_{0}(n)\in \mathbb {K} [n]} such that
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