The b-transform

Abstract

The b-transform is used to convert entire functions into “primary b-functions” by replacing the powers and factorials in the Taylor series of the entire function with corresponding “generalized powers” (which arise from a polynomial function with combinatorial applications) and “generalized factorials.” The b-transform of the exponential function turns out to be a generalization of the Euler partition generating function, and partition generating functions play a key role in obtaining results for the b-transforms of the elementary entire transcendental functions. A variety of normal-looking results arise, including generalizations of Euler's formula and De Moivre's theorem. Applications to discrete probability and applied mathematics (i.e., damped harmonic motion) are indicated. Also, generalized derivatives are obtained by extending the concept of a b-transform.