On the Two q-Analogue Logarithmic Functions
摘要
There is a simple, multi-sheet Riemann surface associated with e_q(z)'s inverse function ln_q(w) for 0< q < 1. A principal sheet for ln_q(w) can be defined. However, the topology of the Riemann surface for ln_q(w) changes each time "q" increases above the collision point of a pair of the turning points of e_q(x). There is also a power series representation for ln_q(1+w). An infinite-product representation for e_q(z) is used to obtain the ordinary natural logarithm ln{e_q(z)} and the values of sum rules for the zeros "z_i" of e_q(z). For |z|<|z_1|, e_q(z)=exp{b(z)} where b(z) is a simple, explicit power series in terms of values of these sum rules. The values of the sum rules for the q-trigonometric functions, sin_q(z) and cos_q(z), are q-deformations of the usual Bernoulli numbers.
引用
@article{arxiv.q-alg/9608015,
title = {On the Two q-Analogue Logarithmic Functions},
author = {Charles A. Nelson and Michael G. Gartley},
journal= {arXiv preprint arXiv:q-alg/9608015},
year = {2009}
}
备注
This is the final version to appear in J.Phys.A: Math. & General. Some explict formulas added, and to update the references