一个新的Rogers-Ramanujan型连分数
A New Rogers-Ramanujan-Type Continued Fraction
摘要: 连分数是数论的重要研究对象，Rogers-Ramanujan型连分数是连分数的一种特殊情况，连分数的构造及其收敛性是连分数理论中重要研究内容。我们通过用两个Rogers-Ramanujan型恒等式来定义两个幂级数 F( z ) 和 G( z ) ，构造幂级数列 { s k } 和数列 { a k } 以满足递归关系 s k+1 = s k−1 −( a k + b k z ) s k ，求出 s k 和 a k 的前五项，对辅助级数进行猜想并使用数学归纳法证明，由此得到一个新的Rogers-Ramanujan型连分数。
Abstract: Continued fraction is an important research object in number, and Rogers-Ramanujan-type continued fraction is a special case of continued fraction. The construction of continued fractions and its convergence are important contents in the theory of continued fractions. Two power series F( z ) and G( z ) are defined by two Rogers-Ramanujan-type identities, the power series { s k } and number series { a k } are constructed to satisfy the recursive relation s k+1 = s k−1 −( a k + b k z ) s k , and the first five terms of s k and a k are obtained. The auxiliary series is conjectural and proved by mathematical induction, thus a new continued fraction of Rogers-Ramanujan-type is obtained.
文章引用：陈洪莉. 一个新的Rogers-Ramanujan型连分数[J]. 理论数学, 2025, 15(10): 177-186. https://doi.org/10.12677/pm.2025.1510261

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