where ''m'' and ''r'' are non-negative integers. For {{nowrap|''r'' {{=}} 0}} the value is 1 since numerator and denominator are both [[empty product]]s. Although the formula in the first clause appears to involve a [[rational function]], it actually designates a polynomial, because the division is exact in '''Z'''<nowiki>[</nowiki>''q''<nowiki>]</nowiki>. Absent any proof for the foregoing assertion one reader finds, in fact, that the case for m=43, r=3 is, in fact, NOT exact in '''Z'''<nowiki>[</nowiki>''q''<nowiki>]</nowiki>. Note that the formula can be applied for {{nowrap|''r'' {{=}} ''m'' + 1}}, and gives 0 due to a factor {{nowrap|1 − ''q''<sup>0</sup> {{=}} 0}} in the numerator, in accordance with the second clause (for even larger ''r'' the factor 0 remains present in the numerator, but its further factors would involve negative powers of ''q'', whence explicitly stating the second clause is preferable). All of the factors in numerator and denominator are divisible by {{nowrap|1 − ''q''}}, with as quotient a [[Q-analog#Introductory examples|''q'' number]]:
