On Touchard's Identity: Generalizations and Related Results
Kunle Adegoke
TL;DR
This paper generalizes Touchard's identity related to Catalan numbers using Beta functions and Stirling numbers, and introduces new combinatorial identities.
Contribution
It presents novel generalizations of Touchard's identity and derives several new combinatorial identities through analytical and combinatorial methods.
Findings
Two new generalizations of Touchard's identity involving Catalan numbers.
Establishment of several new combinatorial identities.
Connections made with Beta functions and Stirling numbers of the second kind.
Abstract
Starting with a known polynomial identity, we derive two generalizations of Touchard's identity concerning Catalan numbers; one obtained using the Beta function and the other via a connection with Stirling numbers of the second kind. We subsequently establish several new combinatorial identities.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Advanced Mathematical Identities · Advanced Mathematical Theories and Applications