Some finite generalizations of Gauss's square exponent identity

Abstract

We obtain three finite generalizations of Gauss's square exponent identity. For example, we prove that, for any non-negative integer $m$, \[ \sum _{k=-m}^{m}(-1)^k \left [{3m-k+1}\atop {m+k}\right ] (-q;q)_{m-k}q^{k^2}=1, \] where \[ \left [{\vphantom {3}n}\atop {\vphantom {k}m}\right ] =\prod _{k=1}^m\frac {1-q^{n-k+1}}{1-q^k} \quad \mbox {and}\quad (a;q)_n=\prod _{k=0}^{n-1}(1-aq^k). \] These identities reduce to Gauss's famous identity $$\sum _{k=-\infty }^{\infty }(-1)^kq^{k^2}=\frac {(q;q)_{\infty }}{(-q;q)_{\infty }}$$ by letting $m\to \infty $.