Touchard’s and Koshy’s identities are beautiful identities about Catalan numbers. It is worth noting that combinatorial interpretations for extended Touchard’s identity and extended Koshy’s identity can intuitively reflect the equations. In this paper, we give a new combinatorial proof for the extended Touchard’s identity by means of Dyck Paths. The principle of inclusion–exclusion (sieve method) is employed to prove the extended Koshy’s identity. Meanwhile, as an new extension of the extended Koshy’s identity, a nice general identity is also provided.

中文翻译：

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关于扩展Touchard和Koshy身份的组合证明的注记

Touchard和Koshy的身份是关于加泰罗尼亚语数字的美丽身份。值得注意的是，扩展的Touchard身份和扩展的Koshy身份的组合解释可以直观地反映方程。在本文中，我们通过Dyck路径为扩展的Touchard身份提供了新的组合证明。包含-排除（筛分法）的原理被用来证明扩展的Koshy的身份。同时，作为扩展的Koshy身份的新扩展，还提供了一个不错的常规身份。