Difference between revisions of "Second q-shifted factorial"
From specialfunctionswiki
| (5 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | The $q$-shifted factorial $\langle a;q \rangle_n$ is given by | + | The $q$-shifted factorial $\langle a;q \rangle_n$ (not to be confused with the [[q-shifted factorial|$q$-shifted factorial]] $(a;q)_n$ or the [[q-factorial|$q$-factorial]] $[n]_q!$) 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; \\ | ||
| − | \displaystyle\prod_{k=0}^{n-1} (1-q^ | + | \displaystyle\prod_{k=0}^{n-1} (1-q^{a+m}), & n=1,2,\ldots |
\end{array} \right.$$ | \end{array} \right.$$ | ||
| + | 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= | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 20:27, 18 December 2016
The $q$-shifted factorial $\langle a;q \rangle_n$ (not to be confused with the $q$-shifted factorial $(a;q)_n$ or the $q$-factorial $[n]_q!$) 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 $$\langle (a);q \rangle_n = \langle a_1,a_2,\ldots,a_m; q \rangle_n = \displaystyle\prod_{j=1}^m \langle a_j;q \rangle_n.$$
