We give a matrix generalization of the family of exponential polynomials in one variable phi(k)(x). Our generalization consists of a matrix of polynomials Phi(k)(X) = (Phi(i, j)((k))(X))(i, j = 1)(n) depending on a matrix of variables X = (x(i, j))(i, j = 1)(n). We prove some identities of the matrix exponential polynomials which generalize classical identities of the ordinary exponential polynomials. We also introduce matrix generalizations of the decreasing factorial (x)(k) = x(x - 1)(x - 2) ... (x - k + 1), the increasing factorial (x)((k)) = x(x + 1)(x + 2) ... (x + k - 1), and the Laguerre polynomials. These polynomials have interesting combinatorial interpretations in terms of different kinds of walks on directed graphs. (C) 2000 Academic Press. [References: 4]
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