A series involving Catalan numbers: Proofs and demonstrations
نویسندگان
چکیده
منابع مشابه
Combinatorial aspects of continued fractions
We show that the universal continued fraction of the Stieltjes-Jacobi type is equivalent to the characteristic series of labelled paths in the plane . The equivalence holds in the set of series in non-commutative indeterminates . Using it, we derive direct combinatorial proofs of continued fraction expansions for series involving known combinatorial quantities : the Catalan numbers, the Bell an...
متن کاملIntegral Representations of Catalan Numbers and Their Applications
In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.
متن کاملIntegral Representations of the Catalan Numbers and Their Applications
In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.
متن کامل9 M ay 2 00 8 IDENTITIES INVOLVING NARAYANA POLYNOMIALS AND CATALAN NUMBERS
Abstract. We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead t...
متن کاملSeveral Series Identities Involving the Catalan Numbers
In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemanian zeta function, and the alternative Hurwitz zeta function.
متن کامل