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 $Z_2$-graded identities of $Z_2$-graded matrix algebra $M^{(m,k)}(F)$ when $\mathrm{char}\,F=0$. 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 $n \in N$ at which any of Capelli quasi-polynomials of even degree $2n$ belongs to ideal of matrix algebra $M_m(F)$ identities.