Some Properties and Generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers

Qi, Feng; Guo, Bai‐Ni

doi:10.1002/9781119414421.ch5

Cited by 26 publications

(25 citation statements)

References 49 publications

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“…For more information and details on applications of the integral representations (2.28) and (3.5), please refer to [27,41,42,43,44,45,50,52,56,57,58] and the closely related references therein.…”

Section: Preprints (Wwwpreprintsorg) | Not Peer-reviewed | Postedmentioning

confidence: 99%

“…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%

See 1 more Smart Citation

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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Section: Preprints (Wwwpreprintsorg) | Not Peer-reviewed | Postedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Integral Representations of Catalan Numbers and Their Applications

Qi¹,

Guo²

2017

Preprint

Self Cite

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“…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

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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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“…For more information on many results, applications, and generalizations of the Motzkin numbers M n , please refer to the papers [3,7,8,21,22] and closely related references therein. For more information on many results, applications, and generalizations of the Catalan numbers C n , please refer to the monograph [6], the papers [9,14,15,20], the survey article [12], and closely related references therein.…”

Section: Introductionmentioning

confidence: 99%