Published online by Cambridge University Press: 08 April 2020
We discuss a truncated identity of Euler and present a combinatorial proof of it. We also derive two finite identities as corollaries. As an application, we establish two related $q$ -congruences for sums of $q$ -Catalan numbers, one of which has been proved by Tauraso [‘ $q$ -Analogs of some congruences involving Catalan numbers’, Adv. Appl. Math. 48 (2012), 603–614] by a different method.
The first author was supported by the National Natural Science Foundation of China (grant 11801417).
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