关于Touchard的连分数及其推广：无组合的自包含证明
On Touchard's continued fraction and extensions: combinatorics-free, self-contained proofs
摘要：
给出了Touchard连分数的一个直接而简单的证明，并给出了它的一个推广，将其转化为与Motzkin数和Schroeder数有关的类似的推广.另一个证明，然后只使用归纳法。我们使用这种机器上的两个例子，出现在最近的论文Josuat-Verges;与一个额外的参数，这两个可以同时处理。
英文摘要：
We give a direct and simple proof of Touchard's continued fraction, provide an extension of it, and transform it into similar expansions related to Motzkin and Schroeder numbers. Another proof is then given that uses only induction. We use this machinery on two examples that appear in recent papers of Josuat-Verges; with an additional parameter, these two can be treated simultaneously.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
