Several Series Identities Involving the Catalan Numbers

Yin, Li; Qi, Feng

doi:10.20944/preprints201703.0029.v1

Cited by 7 publications

(7 citation statements)

References 13 publications

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“…For more information on the Catalan numbers C k and their recent developments, please refer to the monographs [3,14,64], the papers [15,24,39,45,53,54,63,68,71,72,73] and the closely related references therein. The first six Chebyshev polynomials of the second kind U k (x) for 0 ≤ k ≤ 5 are U 0 (x) = 1, U 1 (x) = 2x, U 2 (x) = 4x 2 − 1, U 3 (x) = 8x 3 − 4x, U 4 (x) = 16x 4 − 12x 2 + 1, U 5 (x) = 32x 5 − 32x 3 + 6x.…”

Section: =0mentioning

confidence: 99%

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Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

Qi¹,

Zou²,

Guo³

2017

Preprint

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Abstract. In the paper, the authors establish two identities to express higher order derivatives and integer powers of the generating function of the Chebyshev polynomials of the second kind in terms of integer powers and higher order derivatives of the generating function of the Chebyshev polynomials of the second kind respectively, find an explicit formula and an identity for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, acquire a binomial inversion formula, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials of the second kind, the central Delannoy numbers, and the Fibonacci polynomials.

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Section: =0mentioning

confidence: 99%

“…In recent years, there are plenty of literature, such as [16,20,24,28,35,36,37,45,48,51,53,54,68,72,73], dedicated to generalizations of the Catalan numbers C n and to investigating their properties.…”

Section: Remarksmentioning

confidence: 99%

Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

Qi¹,

Zou²,

Guo³

2017

Preprint

Self Cite

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show abstract

“…For more information on the Catalan numbers C k and their recent developments, please refer to the monographs [2,12,55], the papers [13,25,34,40,46,47,54,59,61,62,63] and the closely related references therein. The first six Chebyshev polynomials of the second kind U k (x) for 0 ≤ k ≤ 5 are U 0 (x) = 1, U 1 (x) = 2x, U 2 (x) = 4x 2 − 1, U 3 (x) = 8x 3 − 4x, U 4 (x) = 16x 4 − 12x 2 + 1, U 5 (x) = 32x 5 − 32x 3 + 6x.…”

Section: =0mentioning

confidence: 99%

“…In recent years, there are plenty of literature, such as [14,18,25,27,31,32,40,41,44,46,47,59,62,63], dedicated to generalizations of the Catalan numbers C n and to investigating their properties.…”

Section: Remarksmentioning

confidence: 99%

Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

Qi¹,

Guo²

2017

Preprint

Self Cite

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Abstract. In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

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“…See [18,19,20,22,23,24,25,26,64]. In recent years, the Catalan numbers C n has been analytically generalized and studied in [21,27,40,41,42,43,44,45,50,52,56,57,58,61,67,69,70] and the closely related references therein. For more information on the Catalan numbers C n , please refer to the monographs [10,15,59,63] and the closely related references therein.…”

Section: Introductionmentioning

confidence: 99%

Integral Representations of Catalan Numbers and Their Applications

Qi¹,

Guo²

2017

Preprint

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Abstract. 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.

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Several Series Identities Involving the Catalan Numbers

Abstract: Abstract. 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.

Cited by 7 publications

References 13 publications

Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

Integral Representations of Catalan Numbers and Their Applications

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