Agnes Scott College

Gloria Olive

Generalized Powers
The American Mathematical Monthly, Vol. 72, No. 6 (June-July, 1965), 619-627

Introduction

The function which we shall write as \[N_h^k(b)=\lim_{z \rightarrow b}\prod_{i=0}^{h-1}\frac{z^{k-i}-1}{z^{i+1}-1}\]can be traced back to Gauss. Various names have been given to it (e.g., "Gaussian expression" and "q-number" [when q replaces b]) and its properties have motivated several papers...

The identity \(N_h^k(1)={k\choose h}\) motivates our development; for it implies that \(N_h^k(b)\) is a generalized binomial coefficient and therefore it is reasonable to suspect that \(N_h^k(b)\) could be used to generate "generalized powers" [i.e., \(\displaystyle \sum_{h=0}^k N_h^k(b)c_{(b)}^h = (c+1)_{(b)}^k\) ] just as \(k \choose h\) can be used to generate ordinary powers [i.e., \(\displaystyle \sum_{h=0}^k {k \choose h}c^h = (c+1)^k\) ].

The purpose of this paper is to create and investigate the generalized power \(M_{(b)}^E\) (read "M to the E to the base b") with medial M, exponent E, and base b representing complex numbers.