dorsal/arxiv
View SchemaOn the Two q-Analogue Logarithmic Functions
| Authors | Charles A. Nelson, Michael G. Gartley |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9608015 |
| URL | https://arxiv.org/abs/q-alg/9608015 |
| DOI | 10.1088/0305-4470/29/24/031 |
Abstract
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.
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"abstract": "There is a simple, multi-sheet Riemann surface associated with e_q(z)\u0027s\ninverse function ln_q(w) for 0\u003c q \u003c 1. A principal sheet for ln_q(w) can be\ndefined. However, the topology of the Riemann surface for ln_q(w) changes each\ntime \"q\" increases above the collision point of a pair of the turning points of\ne_q(x). There is also a power series representation for ln_q(1+w). An\ninfinite-product representation for e_q(z) is used to obtain the ordinary\nnatural logarithm ln{e_q(z)} and the values of sum rules for the zeros \"z_i\" of\ne_q(z). For |z|\u003c|z_1|, e_q(z)=exp{b(z)} where b(z) is a simple, explicit power\nseries in terms of values of these sum rules. The values of the sum rules for\nthe q-trigonometric functions, sin_q(z) and cos_q(z), are q-deformations of the\nusual Bernoulli numbers.",
"arxiv_id": "q-alg/9608015",
"authors": [
"Charles A. Nelson",
"Michael G. Gartley"
],
"categories": [
"q-alg",
"math.QA"
],
"doi": "10.1088/0305-4470/29/24/031",
"title": "On the Two q-Analogue Logarithmic Functions",
"url": "https://arxiv.org/abs/q-alg/9608015"
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