Some best approximation formulas and inequalities for the Wallis ratio

In this paper, we present a new formula for approximating the Wallis ratio. This new formula is very fast in comparison with other classical or recently discovered asymptotic series. Next, inequalities related to this new formula and asymptotic series are established. Finally, for demonstrating the superiority of our new series, some numerical computations are provided.Mathematics Subject Classification. 33B15, 11A55, 11Y65, 40A15.

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“…Recently, based on the formula (3), Qi and Mortici [8] provided some better approximations, one is (4) another is…”

Section: Introductionmentioning

confidence: 99%

Some Quicker Approximations and Inequalities of the Wallis Ratio by Continued Fraction

Liu

Song

2015

Results. Math.

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“…The Wallis ratio, or say, the ratio of two gamma functions, has been studied and applied by many mathematicians, see [12,36,37,46,47,48,49,51], for example, and plenty of literature therein. Now we are in a position to recall and to alternatively prove three integral representations of the Catalan-Qi function C(a, b; x) as follows.…”

Section: Preprints (Wwwpreprintsorg) | Not Peer-reviewed |mentioning

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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“…The Wallis ratio has been studied and applied by many mathematicians. For more information, please refer to [2,3,7,12,14], for example, and plenty of literature therein.…”

Section: Remarksmentioning

confidence: 99%