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26¡ 1 rmonde and Stirling matrices, and showed by using MAPLE that a stochastic matrix Tn
links together these matrices. In [2], the author raised that to generate the elements of the matrix
Tn for any arbitray n using only one or two recurrence relations is an open question. In this
paper, some relations between the Stirling matrix Sn and the Pascal matrix Pn are obtained,
−1
and a factorization of the matrix Tn is given by the using the Stirling matrix of the second
kind. Furthermore, a decomposition of the matrix Tn is given via the Stirling matrix of the first
kind, and a recurrence relation of the elements of the matrix Tn is obtained, so an open problem
proposed by EI-Mikkawy[2] is solved. As a consequence we obtain some combinatorial identities
related to the Stirling numbers.
2. Preliminary results
Let n, k be nonnegative integers and n ≥ k, the Stirling numbers of the first kind s(n, k) and
Received date: 2004-03-30
Foundation item: the NSF of Gansu Province of China (3ZS041-A25-007)34 Journal of Mathematical Research and Exposition
of the second kind S(n, k) can be defined as the coefficients in the following expansion of a variable