## 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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