Differential Operator

A differential operator represents differentiation. The first derivative with respect to is represented by

(1)

and this first-order operator is sometimes called the Newton-Leibniz operator. The second derivative is then denoted , the third , etc. The notation is used formally for antiderivatives; it is not a two-sided inverse without conditions fixing the constants of integration.

For a nonnegative integer , the differential operator satisfies the identity

(2)

where is a Hermite polynomial (Arfken 1985, p. 718), where the first few cases are given explicitly by

			(3)
			(4)
			(5)
			(6)
			(7)
			(8)

The symbol can be used to denote the operator

(9)

(Bailey 1935, p. 8). For a nonnegative integer , a fundamental identity for this operator on -times differentiable functions is

(10)

where is a Stirling number of the second kind (Roman 1984, p. 144), giving

			(11)
			(12)
			(13)
			(14)

and so on (OEIS A008277). Special cases include

			(15)
			(16)
			(17)

For constant , a shifted version of the identity is given by

(18)

(Roman 1984, p. 146).