Differential Operator
المؤلف:
Arfken, G
المصدر:
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.
الجزء والصفحة:
...
15-5-2018
2492
Differential Operator
The operator representing the computation of a derivative,
 |
(1)
|
sometimes also called the Newton-Leibniz operator. The second derivative is then denoted 
, the third 
, etc. The integral is denoted 
.
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
The symbol 
can be used to denote the operator
 |
(9)
|
(Bailey 1935, p. 8). A fundamental identity for this operator is given by
 |
(10)
|
where 
is a Stirling number of the second kind (Roman 1984, p. 144), giving
and so on (OEIS A008277). Special cases include
A shifted version of the identity is given by
![[(x-a)D^~]^n=sum_(k=0)^nS(n,k)(x-a)^kD^~^k](http://mathworld.wolfram.com/images/equations/DifferentialOperator/NumberedEquation5.gif) |
(18)
|
(Roman 1984, p. 146).
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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