AI 中文总结
研究涉及第一类贝塞尔函数正零点的无穷级数，针对现有计算方法，提出一种依赖米塔格 - 莱夫勒展开、黎卡提微分方程及泰勒级数系数的递归算法，可产生级数值，有助于处理类似特殊函数零点无穷级数问题。
AI 中文摘要
研究了一些涉及第一类贝塞尔函数正零点的无穷级数。这些级数的动机源于量子力学微扰问题，其中未微扰态的能量和矩阵元可用第一类贝塞尔函数的零点表示。Pedersen和Urbanowicz计算这些级数的现有方法涉及托马斯 - 赖歇 - 库恩求和规则、贝塞尔函数比幂的微分递归或洛梅尔多项式的应用。本文提供了一种替代方法：提出了一种递归算法，理论上可以产生所讨论的无穷级数值。我们的方法相对简单，依赖于三个主要要素：第一类贝塞尔函数商的米塔格 - 莱夫勒展开和黎卡提微分方程，以及这些比的泰勒级数系数。本文采用的技术可能有助于处理涉及特殊函数零点无穷级数的类似问题。
英文摘要
Some infinite series involving the positive zeros of Bessel functions of the first kind are investigated. The motivation behind these series lies in quantum mechanical perturbation problems, in which energies and matrix elements of unperturbed states are expressible in terms of zeros of Bessel functions of the first kind. The existing approach by Pedersen and Urbanowicz for calculating these series involves the Thomas-Reiche-Kuhn sum rule, differential recurrences for powers of Bessel function ratios or application of Lommel polynomials. In this paper an alternative approach is provided: a recursive algorithm is proposed that theoretically can produce the infinite series values in question. Our method is relatively simple and rely on three main ingredients: the Mittag-Leffler expansion and Riccati differential equation for the quotient of Bessel functions of the first kind, as well as the Taylor series coefficients of these ratios. The technique employed in the paper could be useful to treat similar problems where infinite series of zeros of special functions is involved.
Comments16 pages, 3 appendices
