Geodesic
Quasi-polynomials of Capelli.~III
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 2, pp. 142-150

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In this paper polynomials of Capelli type (double and quasi-polynomials of Capelli) belonging to a free associative algebra $F\{X\cup Y\}$ considering over an arbitrary field $F$ and generated by two disjoint countable sets $X, Y$ are investigated. It is shown that double Capelli's polynomials $C_{4k,\{1\}}$, $C_{4k,\{2\}}$ are consequences of the standard polynomial $S^-_{2k}$. Moreover, it is proved that these polynomials equal to zero both for square and for rectangular matrices of corresponding sizes. In this paper it is also shown that all Capelli's quasi-polynomials of the $(4k+1)$ degree are minimal identities of odd component of $Z_2$-graded matrix algebra $M^{(m, k)}(F)$ for any $F$ and $m\ne k$. 
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     author = {S. Yu. Antonov and A. V. Antonova},
     title = {Quasi-polynomials of {Capelli.~III}},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
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     year = {2021},
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S. Yu. Antonov; A. V. Antonova. Quasi-polynomials of Capelli.~III. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 21 (2021) no. 2, pp. 142-150. http://geodesic.mathdoc.fr/item/ISU_2021_21_2_a0/