Agnes Scott College

Gloria Olive

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


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 eq2 motivates our development; for it implies that Nhk(b) is a generalized binomial coefficient and therefore it is reasonable to suspect that Nhk(b) could be used to generate "generalized powers" [i.e., eq3 just as eq4 can be used to generate ordinary powers [i.e., eq5].

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.