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Some expansions and convolution formulas related to MacMahon’s master theorem. (English) Zbl 0359.05003

The author applied MacMahon’s master theorem to obtain \[ \sum\limits_{m_i=0}^\infty(\overline{m}_1+\alpha_1)^{m_1}\dots(\overline{m}_n+\alpha_n){\frac {u_1^{m_1}\dots u_n^{m_n}}{m_1!\dots m_n!}}=e^{\alpha_1x_1+\dots+\alpha_nx_n}(\Delta(x_1,x_2,\dots,x_n))^{-1} \] where \(\overline m_j=\sum\limits_{i=1}^n m_ia_{ij} (j=1,2,\dots,n), \Delta (x_1,x_2,\dots,x_n) = \det (S_{ij} -x_ia_{ij})\) and \linebreak\(u_i= x_i\exp\{-\sum\limits_{j=1}^n a_{ij}x_j\} (i=1,2,\dots,n)\). In this article more results are obtained along the same lines. These include convolution formulas and the inverse of the expression above for \(u_i\) in terms of \(x_i\) and \`\` factorial” analogs of these results.

MSC:

05A15 Exact enumeration problems, generating functions
05A19 Combinatorial identities, bijective combinatorics
05A10 Factorials, binomial coefficients, combinatorial functions
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