By George Boole

This Elibron Classics publication is a facsimile reprint of a 1877 version by means of Macmillan and Co., London.

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Particularly interesting is the case s = 2 (corresponding to the meromorphic Weyl algebra), where the generalized Stirling numbers are given by Bessel numbers. These generalized Stirling numbers were studied in several papers [289, 290, 763, 765–767, 771–773]. © 2016 by Taylor & Francis Group, LLC 20 Commutation Relations, Normal Ordering, and Stirling Numbers Now that we have deﬁned the chief characters of the book, we can succintly describe its focus as follows: We discuss diﬀerent aspects of normal ordering in As;h|q and in several interesting specializations, like A2;−1|1 .

7, we considered the ﬁrst letters of the word. In that case, we could have just as easily focused on the ﬁrst letter, as each ﬁrst letter is either 1 or 2. Focusing on the ﬁrst letter (the last letter, the maximal letter, or the minimal letter) are common methods for obtaining small classes. The recurrence relation can be used to ﬁnd the value of a speciﬁc term of the sequence, say 20 , where all preceding values j for j = 0, 1, . . 5), unless an explicit formula can be derived. Later, we will discuss several methods for obtaining an explicit formula from a recurrence relation.

42) where q ∈ C is assumed to be generic. 39). 42) is given for h = 1 by V → X and U → Dq , where Dq denotes the Jackson derivative. The action of the Jackson derivative on a function f is deﬁned by (Dq f )(x) = f (x) − f (qx) . (1 − q)x Diaz and Pariguan [345] considered in 2005 the meromorphic Weyl algebra which results by considering X −1 and D (instead of X and D, as in the Weyl algebra). One ﬁnds that DX −1 − X −1 D = −(X −1 )2 , that is, abstractly, U V − V U = −V 2 . 43) One can consider diﬀerent combinatorial aspects in this algebra, for example deﬁne associated Stirling numbers as normal ordering coeﬃcients of (V U )n .