This paper deals with the class of Capelli polynomials in free associative algebra F{Z} where F is an arbitrary field and Z is a countable set. The interest to these objects is initiated by assumption that the polynomials (Capelli quasi-polynomials) of some odd degree introduced will be contained in the basis ideal Z2 -graded identities of Z2 -graded matrix algebra M(m,k)(F) when char F = 0. In connection with this assumption the fundamental properties of Capelli quasi-polynomials have been given in the paper. In particularly, the decomposition of Capelli type polynomials have been given by the polynomials of the same type and some betweeness of their T-ideals have been shown. Besides, taking into account some properties of Capelli quasi-polynomials obtained and also the Chang theorem we show that all Capelli quasi-polynomials of even degree 2n (n > 1) are consequence of standard polynomial S−n in case when the characteristic of field F is not equal to two. At last we find the least n ∈ N at which any of Capelli quasi-polynomials of even degree 2n belongs to ideal of matrix algebra Mm(F) identities.