Domains of validity for some of Ramanujan's continued fraction formulas

Jacobsen, Lisa

doi:10.1016/0022-247x(89)90049-8

Cited by 16 publications

(15 citation statements)

References 7 publications

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“…Our approach is similar to that in several recent papers [5], [6], [12], [13] on the subject where Pincherle's theorem [11] has been used to bring out the connection between several of Ramanujan's Chapter 12 entries and the general theory of hypergeometric orthogonal functions (Askey and Wilson [1], Wilson [18]). For other approaches to explaining some of Ramanujan's continued fraction entries see [3], [10], [19].…”

mentioning

confidence: 99%

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Watson’s Basic Analogue of Ramanujan’s Entry $40$ and Its Generalization

Gupta¹,

Masson²

1994

SIAM J. Math. Anal.

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mentioning

confidence: 99%

“…To the left side of (3.5) we can apply the 'parabola theorem' (see Jones and Thron [11, p. 99] and Jacobsen [10]), since from (2.6),…”

mentioning

confidence: 99%

Watson’s Basic Analogue of Ramanujan’s Entry $40$ and Its Generalization

Gupta¹,

Masson²

1994

SIAM J. Math. Anal.

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“…Remark Interestingly, this proof, deriving from extending the right side of (3.30), coincides at the finish with Jacobsen's proof [8], which uses a theorem, due to her [7] and Waadeland [16], on tails of continued fractions. Both proofs eventually rely on Hill's result from Theorem 6 applied to the same 2 F 1 function.…”

Section: Entry 9 ([2]mentioning

confidence: 71%

Ramanujan and extensions and contractions of continued fractions

Laughlin

Wyshinski

2007

Ramanujan J

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If a continued fraction K ∞ n=1 a n /b n is known to converge but its limit is not easy to determine, it may be easier to use an extension of K ∞ n=1 a n /b n to find the limit. By an extension of K ∞ n=1 a n /b n we mean a continued fraction K ∞ n=1 c n /d n whose odd or even part is K ∞ n=1 a n /b n . One can then possibly find the limit in one of three ways: (i) Prove the extension converges and find its limit; (ii) Prove the extension converges and find the limit of the other contraction (for example, the odd part, if K ∞ n=1 a n /b n is the even part); (iii) Find the limit of the other contraction and show that the odd and even parts of the extension tend to the same limit.We apply these ideas to derive new proofs of certain continued fraction identities of Ramanujan and to prove a generalization of an identity involving the RogersRamanujan continued fraction, which was conjectured by Blecksmith and Brillhart.

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“…By an argument of Jacobsen [3], we can extend the domains of convergence for , n, x to the case when , n are arbitrary complex numbers and x is complex with Re x > 0. This completes the proof of Corollary 3.2.…”

Section: Corollary 32 (Entry 39) Let and N Denote Arbitrary Complexmentioning

confidence: 99%

An entry of Ramanujan on continued fraction involving the gamma function in his notebooks

2012

Journal of Number Theory

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Domains of validity for some of Ramanujan's continued fraction formulas

Cited by 16 publications

References 7 publications

Watson’s Basic Analogue of Ramanujan’s Entry $40$ and Its Generalization

Watson’s Basic Analogue of Ramanujan’s Entry $40$ and Its Generalization

Ramanujan and extensions and contractions of continued fractions

An entry of Ramanujan on continued fraction involving the gamma function in his notebooks

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