Continued Fraction Representations for Functions Related to the Gamma Function

2020 ◽  
pp. 233-279 ◽  
Author(s):  
L. J. Lange
Keyword(s):  
Gamma Function ◽  
Continued Fraction

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).

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