content_image/370041.jpg content_image/370040.jpg content_image/370039.jpg content_image/370038.jpg content_image/370037.jpg content_image/370034.jpg content_image/370036.jpg content_image/370035.jpg content_image/370033.jpg
lattice and look at the pictures. 

## II. DEFINITIONS, NOTATIONS, THE VERY BASICS 

The q -binomial coefficient 

is a natural extension of the standard binomial coefficient 

The Pochhammer symbol (or shifted factorial) is defined as 

<img src='content_image/370041.jpg'>

<img src='content_image/370040.jpg'>

<img src='content_image/370039.jpg'>

so that (1)$_{n}$ = n !. Its q -deformed relative, the q -Pochhammer symbol, is defined as 

The q -binomial coefficient can then be expressed as 

The q -numbers are defined for any real number a as 

and it is easy to show that 

Note that for integers n ≥ 1 we have 

<img src='content_image/370038.jpg'>

<img src='content_image/370037.jpg'>

<img src='content_image/370034.jpg'>

<img src='content_image/370036.jpg'>

<img src='content_image/370035.jpg'>

so that [ n ]$_{q}$ → n when q → 1. To continue, the q -number factorials are defined as 

<img src='content_image/370033.jpg'>

