scholarly journals Asymptotic series and Stieltjes continued fractions for a gamma function ratio

1978 ◽  
Vol 4 (2) ◽  
pp. 105-111 ◽  
Author(s):  
K.O. Bowman ◽  
L.R. Shenton
Keyword(s):  
Gamma Function ◽  
Continued Fractions ◽  
Asymptotic Series

Ramanujan's Continued Fraction and the Bauer-Muir Transformation

1961 ◽  
Vol 57 (1) ◽  
pp. 76-79 ◽  
Author(s):  
V. N. Singh
Keyword(s):  
Positive Integer ◽  
Gamma Function ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Gamma Functions ◽  
Image Position ◽  
The Right ◽  
Basic Analogue ◽  
Ramanujan's Continued Fraction

Ramanujan's Continued Fraction may be stated as follows: Let where there are eight gamma functions in each product and the ambiguous signs are so chosen that the argument of each gamma function contains one of the specified number of minus signs. Then where the products and the sums on the right range over the numbers α, β, γ, δ, ε: provided that one of the numbers β, γ, δ, ε is equal to ± ±n, where n is a positive integer. In 1935, Watson (3) proved the theorem by induction and also gave a basic analogue. In this paper we give a new proof of Ramanujan's Continued Fraction by using the transformation of Bauer and Muir in the theory of continued fractions (Perron (1), §7;(2), §2).

THE BURNSIDE APPROXIMATION OF GAMMA FUNCTION

2010 ◽  
Vol 08 (03) ◽  
pp. 315-322 ◽  
Author(s):  
XIQUAN SHI ◽  
FENGSHAN LIU ◽  
HONGMIN QU
Keyword(s):  
Gamma Function ◽  
Asymptotic Series ◽  
Stirling’S Formula ◽  
Completely Monotonic

Different from the famous Stirling's formula [Formula: see text], Burnside presented another formula [Formula: see text]. In this paper, some estimations and a convergent asymptotic series of b(s) are obtained. At the same time, it is proved that both -b(s) and [Formula: see text] are completely monotonic on the interval (½, ∞).

scholarly journals A new asymptotic series for the Gamma function

2006 ◽  
Vol 195 (1-2) ◽  
pp. 134-154 ◽  
Author(s):  
Xiquan Shi ◽  
Fengshan Liu ◽  
Minghan Hu
Keyword(s):  
Gamma Function ◽  
Asymptotic Series

scholarly journals Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal

2015 ◽  
Vol 145 (3) ◽  
pp. 571-596 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Error Estimates ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Asymptotic Series ◽  
Smooth Transition ◽  
Error Terms

In 1994 Boyd derived a resurgence representation for the gamma function, exploiting the 1991 reformulation of the method of steepest descents by Berry and Howls. Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities and asymptotics for the late coefficients. The main aim of this paper is to modify Boyd’s resurgence formula, making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.

Infinitely many Stokes smoothings in the gamma function

1991 ◽  
Vol 434 (1891) ◽  
pp. 465-472 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Gamma Function ◽  
Error Function ◽  
Asymptotic Series ◽  
Stokes Line ◽  
Separate Component ◽  
Negative Real Axis ◽  
Stokes Lines ◽  
The Right ◽  
Anti Stokes

The Stokes lines for Г( z ) are the positive and negative imaginary axes, where all terms in the divergent asymptotic expansion for In Г( z ) have the same phase. On crossing these lines from the right to the left half-plane, infinitely many subdominant exponentials appear, rather than the usual one. The exponentials increase in magnitude towards the negative real axis (anti-Stokes line), where they add to produce the poles of Г( z ). Corresponding to each small exponential is a separate component asymptotic series in the expansion for In Г( z ). If each is truncated near its least term, its exponential switches on smoothly across the Stokes lines according to the universal error-function law. By appropriate subtractions from In Г( z ), the switching-on of successively smaller exponentials can be revealed. The procedure is illustrated by numerical computations.

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