q-正弦函数的q-Bernoulli数与零点
q-Bernoulli Numbers and Zeros of q-Sine Function
摘要：
正弦函数的零点、Bernoulli数和黎曼ζ函数之间存在一个众所周知的关系。本文找到了q-正弦函数零点的一个类似关系。本文引入了Bernoulli数的一个新的q-扩张，其生成函数都是由Jackson的q-指数函数表示的.利用q-广义重积Leibnitz定则和对数导数的q-模拟，建立了q-sin x的零点与我们的q-Bernoulli数之间的精确关系.这些关系对于分析零点的近似和渐进公式以及求解q-Sturm-Liouville问题的边值问题非常有用。
英文摘要：
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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