An Improper Integral with a Square Root

Qi, Feng

doi:10.20944/preprints201610.0089.v1

Cited by 5 publications

(4 citation statements)

References 14 publications

(10 reference statements)

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“…In 2012, Dana-Picard and Zeitoun [46] deduced an integral representation for the Catalan numbers C n , which was corrected and developed by Qi [47] as the following integral representations. …”

Section: Dana-picard-zeitoun-qi's Integral Representations In 2012 Anmentioning

confidence: 99%

Integral Representations of the Catalan Numbers and Their Applications

Guo

2017

Mathematics

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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: Dana-picard-zeitoun-qi's Integral Representations In 2012 Anmentioning

confidence: 99%

Integral Representations of the Catalan Numbers and Their Applications

Guo

2017

Mathematics

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“…The integral representation (2.4) for the Catalan numbers C k and its variant forms can be found in [4,5,6,7,18,19,23,35,51] and the closely related references therein.…”

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

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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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“…The integral representation (2.4) for the Catalan numbers C k and its variant forms can be found in [3,4,5,6,16,17,44] and the closely related references therein.…”

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

View full text Add to dashboard Cite

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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An Improper Integral with a Square Root

Abstract: Abstract. In the paper, the author presents explicit and unified expressions for a sequence of improper integrals in terms of the beta functions and the Wallis ratios. Hereafter, the author derives integral representations for the Catalan numbers originating from combinatorics.

Cited by 5 publications

References 14 publications

Integral Representations of the Catalan Numbers and Their Applications

Integral Representations of the Catalan Numbers and Their Applications

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

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