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Generalised powers revisited. (English) Zbl 0831.05004

The counting of special kinds of configurations can lead to functions with interesting properties. For example, the counting of the arrangements of \(k\) (distinguishable) objects has led to the factorial \(k!\) and then (after extending the domain in a “natural way”) to the gamma function \(\Gamma (x)\). In this paper we show how the function obtained by counting the number of multichains of length \(m\) in the lattice of subspaces of a \(k\)-dimensional vector space over \(\text{GF} (b)\) (the Galois field with \(b\) elements) leads to the “generalised power” \(m^k_{(b)}\) and then (after extending the domains in “natural ways”) to the extended generalised power \(x^t_{(b)}\), where \(x,t\), and \(b\) are complex variables. Our development helps to motivate the study of special classes of polynomials (including the polynomials of binomial type of Rota-Mullin) and functions of several complex variables.
Reviewer: R.Scurr (Dunedin)

MSC:

05A10 Factorials, binomial coefficients, combinatorial functions
05A15 Exact enumeration problems, generating functions
05E99 Algebraic combinatorics
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