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Differential Operator

Theoperator representing the computation of aderivative,

(1)

sometimes also called the Newton-Leibniz operator. The second derivative is then denoted D^~^2 , the third D^~^3 , etc. Theintegral is denoted D^~^(-1) .

The differential operator satisfies the identity

(2)

where H_n(x) is aHermite polynomial (Arfken 1985, p. 718), where the first few cases are given explicitly by

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

The symbol theta can be used to denote the operator

(9)

(Bailey 1935, p. 8). A fundamental identity for this operator is given by

(10)

where S(n,k) is aStirling number of the second kind (Roman 1984, p. 144), giving

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

and so on (OEISA008277). Special cases include

			(15)
			(16)
			(17)

A shifted version of the identity is given by

[(x-a)D^~]^n=sum_(k=0)^nS(n,k)(x-a)^kD^~^k

(18)

(Roman 1984, p. 146).

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References

Arfken, G.Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Bailey, W. N.Generalised Hypergeometric Series. Cambridge, England: University Press, 1935.Roman, S.The Umbral Calculus. New York: Academic Press, pp. 59-63, 1984.Sloane, N. J. A. SequenceA008277 in "The On-Line Encyclopedia of Integer Sequences."

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Differential Operator

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Weisstein, Eric W. "Differential Operator."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/DifferentialOperator.html

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Differential Operator

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