Truncations of Gauss’ square exponent theorem

Abstract

We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer n

$$\sum\limits_{k = 0}^n {{{\left( { - 1} \right)}^k}\left[ {\matrix{{2n - k} \cr k \cr } } \right]\,\,{{\left( {q;{q^2}} \right)}_{n - k}}{q^{\left( {\matrix{{k + 1} \cr 2 \cr } } \right)}} = \sum\limits_{k = - n}^n {{{\left( { - 1} \right)}^k}{q^{{k^2}}},} } $$

where

$$\left[ {\matrix{n \cr m \cr } } \right] = \prod\limits_{k = 1}^m {{{1 - {q^{n - k + 1}}} \over {1 - {q^k}}}\,\,\,\,{\rm{and}}\,\,\,\,{{\left( {a;q} \right)}_n} = \prod\limits_{k = 0}^{n - 1} {\left( {1 - a{q^k}} \right).} } $$