The q-Pochhammer symbol, also called the q-shifted factorial, is a product defined in the mathematical field of combinatorics as (a;q) sub n equal to the product for k from 0 to n minus 1 of (1 minus a times q to the k), which expands to (1-a)(1-aq)(1-aq squared) and so on up to (1-a q to the n-1), with the convention that (a;q) sub 0 equals 1. It is the q-analog of the ordinary Pochhammer symbol used in classical analysis, and it converges back to that classical symbol as q approaches 1. Unlike the ordinary Pochhammer symbol, it extends naturally to an infinite product, which is an analytic function inside the unit disk and also makes sense as a formal power series in q; the special case of that infinite product taken at a equals q is Euler's function, important throughout combinatorics, number theory, and the theory of modular forms. The q-Pochhammer symbol is a basic building block for constructing q-analogs generally, especially within the theory of basic hypergeometric series.
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In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product ( a; q) n = ∏ k = 0 n − 1 ( 1 − a q k) = ( 1 − a) ( 1 − a q) ( 1 − a q 2) ⋯ ( 1 − a q n − 1), {\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1}),} with ( a
In Branch: Combinatorics, Lead sentence
In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product ( a ; q )
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