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

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

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

The symbol can be used to denote the operator

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

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

and so on (OEIS A008277). Special cases include

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

(Roman 1984, p. 146).

See also

Convective Derivative, D-Module, Derivative, Fractional Derivative, Gradient

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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. Sequence A008277 in "The On-Line Encyclopedia of Integer Sequences."

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Differential Operator

Cite this as:

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

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