An Application of MacMahon’s Master Theorem
An Application of MacMahon’s Master Theorem
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麦克马洪主定理的应用
DOI：
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发表时间：
1974
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影响因子：
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通讯作者：
L. Carlitz
中科院分区：
文献类型：
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作者：
L. Carlitz
MacMahon’s Master Theorem is applied to prove the following result: [ sumlimits_{m_1 , cdots ,m_n = 0}^infty {ar m_1^{m_1 } } cdots ar m_n^{m_n } frac{{u_1^{m_1 } cdots u_n^{m_n } }} {{m_1 ! cdots m_n !}} = {f Delta }(x_1 ,x_2 , cdots ,x_n ), ] where [ ar m_j = sumlimits_{i = 1}^n {m_i a_{ij} ,} quad j = 1,2, cdots ,n, ][ u_i = x_i exp left{ { - sumlimits_{j = 1}^n {a_{ij} x_j } }
ight},quad i = 1,2, cdots ,n, ] and [ {f Delta }left(x_1 ,x_2 , cdots ,x_n
ight) = left| {egin{array}{*{20}c} {1 - x_1 a_{11} } & { - x_1 a_{12} } & cdots & { - x_1 a_{1n} } \ { - x_2 a_{21} } & {1 - x_2 a_{22} } & cdots & { - x_2 a_{2n} } \ vdots & vdots & {} & vdots \ { - x_n a_{n1} } & { - x_n a_{n2} } & cdots & {1 - x_n a_{nn} } \ end{array} }
ight|^{ - 1} .]
MacMahon’s Master Theorem is applied to prove the following result: [ sumlimits_{m_1 , cdots ,m_n = 0}^infty {ar m_1^{m_1 } } cdots ar m_n^{m_n } frac{{u_1^{m_1 } cdots u_n^{m_n } }} {{m_1 ! cdots m_n !}} = {f Delta }(x_1 ,x_2 , cdots ,x_n ), ] where [ ar m_j = sumlimits_{i = 1}^n {m_i a_{ij} ,} quad j = 1,2, cdots ,n, ][ u_i = x_i exp left{ { - sumlimits_{j = 1}^n {a_{ij} x_j } }
ight},quad i = 1,2, cdots ,n, ] and [ {f Delta }left(x_1 ,x_2 , cdots ,x_n
ight) = left| {egin{array}{*{20}c} {1 - x_1 a_{11} } & { - x_1 a_{12} } & cdots & { - x_1 a_{1n} } \ { - x_2 a_{21} } & {1 - x_2 a_{22} } & cdots & { - x_2 a_{2n} } \ vdots & vdots & {} & vdots \ { - x_n a_{n1} } & { - x_n a_{n2} } & cdots & {1 - x_n a_{nn} } \ end{array} }
ight|^{ - 1} .]