  We use $q$-binomial theorem to prove three new polynomial identities
involving $q$-trinomial coefficients. We then use summation formulas for the
$q$-trinomial coefficients to convert our identities into another set of three
polynomial identities, which imply Capparelli's partition theorems when the
degree of the polynomial tends to infinity. This way we also obtain an
interesting new result for the sum of the Capparelli's products. We finish this
paper by proposing an infinite hierarchy of polynomial identities.

Number Theory;Combinatorics