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Differential Operator


The operator representing the computation of a derivative,

 D^~=d/(dx),
(1)

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

The differential operator satisfies the identity

 (2x-d/(dx))^n1=H_n(x),
(2)

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

H_1(x)=2x-(partial1)/(partialx)
(3)
=2x
(4)
H_2(x)=2x(2x)-(partial(2x))/(partialx)
(5)
=4x^2-2
(6)
H_3(x)=2x(4x^2-2)-(partial(4x^2-2))/(partialx)
(7)
=8x^3-12x.
(8)

The symbol theta can be used to denote the operator

 theta=xd/(dx)
(9)

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

 (xD^~)^n=sum_(k=0)^nS(n,k)x^kD^~^k,
(10)

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

(xD^~)^1=xD^~
(11)
(xD^~)^2=xD^~+x^2D^~^2
(12)
(xD^~)^3=xD^~+3x^2D^~^2+x^3D^~^3
(13)
(xD^~)^4=xD^~+7x^2D^~^2+6x^3D^~^3+x^4D^~^4
(14)

and so on (OEIS A008277). Special cases include

theta^ne^x=e^xsum_(k=0)^(n)S(n,k)x^k
(15)
theta^ncosx=cosxsum_(k=0)^(n)(-1)^kS(n,2k)x^(2k)+sinxsum_(k=1)^(n)(-1)^kS(n,2k-1)x^(2k-1)
(16)
theta^nsinx=cosxsum_(k=1)^(n)(-1)^(k+1)S(n,2k-1)x^(2k-1)+sinxsum_(k=0)^(n)(-1)^kS(n,2k)x^(2k).
(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).


See also

Convective Derivative, 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."

Referenced on Wolfram|Alpha

Differential Operator

Cite this as:

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

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