A short calculation of the multiple sum of Krivokolesko–Leinartas with linear constraints on summation indices

The method of integral representation and calculation of the combinatorial sums of various type (the method of coefficients) using the formal Laurent power series over $\mathbb C$, the theory of analitical functions and the theory of multiple residues in $\mathbb C^n$ were proposed by the author in the late seventies. This method was applied in various fields of mathematics. The method of coefficients is important for a difficult problem of calculation of the multiple sums with linear constraints on summation indices. Various combinatorial problems can be formulated in terms of such constraints. The calculation of the multiple sum with $q$-binomial coefficients and linear recurrent constraints on summation indices was published by the author in «The Bulletin of Irkutsk State University. Series Mathematics» in 2016. This problem appears at the enumeration of all own $t$-dimensional subspaces of the space $V_m$ over field $GF(q)$. V.P. Krivokolesko and E.K. Leinartas in «The Bulletin of Irkutsk State University. Series Mathematics» in 2012, using the Hadamard composition have proved the multiple identity with polynomial coefficients and various constraints on the limits of summation, containing the family of free parameters. This identity is generalisation of the identities studied earlier by several authors, since constructions of the Deubechies filters in the wavelets theory. Using the author's method of coefficients the short and simple calculation of Krivokolesko–Leinartas sum is carried out. These calculations also automatically provides an equivalent way of calculation of the specified sum by means of a traditional method of generation functions, using only the well-known operations over corresponding multiple formal Laurent power series.