A TRUNCATED IDENTITY OF EULER AND RELATED -CONGRUENCES
Ji-Cai Liu, ZHONG-YU HUANG
Abstract
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.
Topics & Concepts
Congruence relationCatalan numberMathematicsIdentity (music)Euler's formulaCatalanPure mathematicsCombinatoricsDiscrete mathematicsAlgebra over a fieldMathematical analysisLinguisticsPhysicsAcousticsPhilosophyAdvanced Mathematical IdentitiesAdvanced Combinatorial MathematicsAnalytic Number Theory Research