q-Bernoulli Numbers and Zeros of q-Sine Function

2012/02/10 by Sengul Nalci, Oktay Pashaev, Nalci, Sengul +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #Quantum Algebra (math.QA) #math.NT #math.QA

paper · pdf · doi:10.48550/arxiv.1202.2265

13 pages

arxiv created 2012/02/10 · arxiv updated 2012/02/13

Abstract

There exists a well-known relation between the zeros of sine function, Bernoulli numbers and the Riemann Zeta function. In the present paper, we find a similar relation for zeros of q-sine function. We introduce a new q-extension of the Bernoulli numbers with generating function written in terms of both Jackson's q-exponential functions. By q-generalized multiple product Leibnitz rule and the q-analogue of logarithmic derivative we established exact relations between zeros of q-sin x and our q-Bernoulli numbers. These relations could be useful for analyzing approximate and asymptotic formulas for the zeros and solving BVP for q-Sturm-Liouville problems.

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