论文标题
在无限的Borwein产品上提高到积极的真实力量
On the infinite Borwein product raised to a positive real power
论文作者
论文摘要
在本文中，我们研究了出现在$ q $ series扩展中的系数的属性。我们使用Hardy-Ramanujan--Rademacher圆方法为系数提供渐近公式。对于$ P = 3 $，我们对它们的增长进行了估计，使我们能够部分确认第一作者的早期猜想，即当指数$δ$在指定的正实数范围内时，该系数的符号模式。我们进一步建立了无限borwein乘积立方体系数的一些消失和分裂性能。最后，我们以一个附录提出了几种新的猜想，该猜想是针对无限产品的精确符号模式的，这些产品与我们在$ p = 3 $案例中做出的猜想相似。
In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^δ$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $δ$. We use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $δ$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p=3$ case.
