By George Salmon

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The acceptance of the methods and notation of the continentals was surprisingly quick in England, and it was here that the calculus of operations was extended in scope and its applications. The ﬁrst mathematicians in England who were responsible for this development were Babbage, Herschel, Peacock, and Woodhouse; see [681] for a discussion. In the next 30–40 years, from the late 1830s to the 1870s, many important results were achieved. In 1837 Murphy published a paper [855] in which a very clear and general account of the theory of linear operations was given, and in which he also noticed explicitly the diﬀerence between commuting and noncommuting operations.

For each such word its ﬁrst letters are either 12 or 2. It is customary to deﬁne the number of words of length 0 to be 1 (for the empty word). 6. 8 Let’s determine an , the number of words over the alphabet {0, 1} of length n which do not contain the substring 00 unless the word does not contain the letter 1. 7, we obtain a0 = 1, a1 = 2, and an = an−1 + an−2 + 1. 6. Sometimes the recurrence relations can be obtained easily whereas explicit formulas are very diﬃcult to derive directly. Recurrence relations can be obtained naturally and describe how the objects under consideration can be obtained from other (simpler) objects.

Antinormal ordered), if all powers of p stand to the right (resp. left) of the powers of q, they gave the following normal ordering formula pn q = qpn + n(−i )pn−1 , as well as the analogous antinormal ordering formula qn p = pqn − n(−i )qn−1 . More generally, they also mentioned that min(n,m) p q n m = k! 32) and gave the analogous antinormal ordering formula. 19) due to the common algebraic structure. In the subsequent paper [130] together © 2016 by Taylor & Francis Group, LLC 16 Commutation Relations, Normal Ordering, and Stirling Numbers with Heisenberg , they derived for any function f (q, p), which can be formally expressed as power series in p and q, the rule pf − f p = (−i ) ∂f , ∂q qf − f q = (−i ) ∂f .