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

Introduction

The function which we shall write as

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 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., just as can be used to generate ordinary powers [i.e., ].

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.