To Chang theorem. III

Various multilinear polynomials of Capelli type belonging to a free associative algebra $F\{X\cup Y\}$ over an arbitrary field $F$ generated by a countable set $X \cup Y$ are considered. The formulas expressing coefficients of polynomial Chang ${\mathcal R}(\bar x, \bar y \vert \bar w)$ are found. It is proved that if the characteristic of field $F$ is not equal two then polynomial ${\mathcal R}(\bar x, \bar y \vert \bar w)$ may be represented by different ways in the form of sum of two consequences of standard polynomial $S^-(\bar x)$. The decomposition of Chang polynomial ${\mathcal H}(\bar x, \bar y \vert \bar w)$ different from already known is given. Besides, the connection between polynomials ${\mathcal R}(\bar x, \bar y \vert \bar w)$ and ${\mathcal H}(\bar x, \bar y \vert \bar w)$ is found. Some consequences of standard polynomial being of great interest for algebras with polynomial identities are obtained. In particular, a new identity of minimal degree for odd component of $Z_2$-graded matrix algebra $M^{(m,m)}(F)$ is given.