Difference between revisions of "Second q-shifted factorial"

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The $q$-shifted factorial $\langle a;q \rangle_n$ is given by
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The $q$-shifted factorial $\langle a;q \rangle_n$ (not to be confused with the [[q-Pochhammer|$q$-Pochhammer]] $(a;q)_n$) is given by
 
$$\langle a;q \rangle_n = \left\{ \begin{array}{ll}
 
$$\langle a;q \rangle_n = \left\{ \begin{array}{ll}
 
1, & n=0; \\
 
1, & n=0; \\
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If $(a)=(a_1,a_2,\ldots,a_m)$ is a vector then we define the notation
 
If $(a)=(a_1,a_2,\ldots,a_m)$ is a vector then we define the notation
 
$$\langle (a);q \rangle_n = \langle a_1,a_2,\ldots,a_m; q \rangle_n = \displaystyle\prod_{j=1}^m \langle a_j;q \rangle_n.$$
 
$$\langle (a);q \rangle_n = \langle a_1,a_2,\ldots,a_m; q \rangle_n = \displaystyle\prod_{j=1}^m \langle a_j;q \rangle_n.$$
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=Properties=
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=References=
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[[Category:SpecialFunction]]

Revision as of 19:24, 15 December 2016

The $q$-shifted factorial $\langle a;q \rangle_n$ (not to be confused with the $q$-Pochhammer $(a;q)_n$) is given by $$\langle a;q \rangle_n = \left\{ \begin{array}{ll} 1, & n=0; \\ \displaystyle\prod_{k=0}^{n-1} (1-q^{a+m}), & n=1,2,\ldots \end{array} \right.$$ If $(a)=(a_1,a_2,\ldots,a_m)$ is a vector then we define the notation $$\langle (a);q \rangle_n = \langle a_1,a_2,\ldots,a_m; q \rangle_n = \displaystyle\prod_{j=1}^m \langle a_j;q \rangle_n.$$

Properties

References