The operator representing the computation of a derivative,
(1)
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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)
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where is a Hermite polynomial (Arfken 1985, p. 718), where the first few cases are given explicitly by
(3)
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(4)
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(5)
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(6)
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(7)
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(8)
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The symbol can be used to denote the operator
(9)
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(Bailey 1935, p. 8). A fundamental identity for this operator is given by
(10)
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where is a Stirling number of the second kind (Roman 1984, p. 144), giving
(11)
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(12)
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(13)
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(14)
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and so on (OEIS A008277). Special cases include
(15)
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(16)
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(17)
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A shifted version of the identity is given by
(18)
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(Roman 1984, p. 146).