The Ramanujan Journal

, Volume 14, Issue 3, pp 389–404

Ramanujan and extensions and contractions of continued fractions

• J. Mc Laughlin
• Nancy J. Wyshinski
Article

Abstract

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:
1. (i)

Prove the extension converges and find its limit;

2. (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);

3. (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 Rogers-Ramanujan continued fraction, which was conjectured by Blecksmith and Brillhart.

Keywords

Continued fractions Convergence Extension

11A55

References

1. 1.
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999). xvi+664 pp.
2. 2.
Berndt, B.C.: Ramanujan’s Notebook’s, Part II. Springer, New York (1989) Google Scholar
3. 3.
Berndt, B.C., Yee, A.J.: On the generalized Rogers-Ramanujan continued fraction. Rankin memorial issues. Ramanujan J. 7(1–3), 321–331 (2003)
4. 4.
Bernoulli, D.: Disquisitiones ulteriores de indola fractionum continuarum, Novi Comm. Acad. Sci. Imper. Petropol. 20 (1775) Google Scholar
5. 5.
Brillhart, J.: Email to Bruce C. Berndt, 27 January 2002 Google Scholar
6. 6.
Hill, M.J.M.: On a formula for the sum of a finite number of terms of the hypergeometric series when the fourth element is equal to unity. Proc. Lond. Math. Soc. (2) 6, 339–348 Google Scholar
7. 7.
Jacobsen, L.: Composition of linear fractional transformations in terms of tail sequences. Proc. Am. Math. Soc. 97(1), 97–104 (1986)
8. 8.
Jacobsen, L.: Domains of validity for some of Ramanujan’s continued fraction formulas. J. Math. Anal. Appl. 143(2), 412–437 (1989)
9. 9.
Jones, W.B., Thron, W.J.: Continued Fractions Analytic Theory and Applications. Addison-Wesley, London (1980) Google Scholar
10. 10.
Khovanskii, A.N.: The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory, Translated by P. Wynn. P. Noordhoff N.V., Groningen (1963). xii + 212 pp. Google Scholar
11. 11.
12. 12.
Lagrange, J.L.: Sur l’usuage des fractions continues dans le calcul intégral. Nouveaux Mém, Acad. Sci. Berlin 7, 236–264 (1776); Oeuvres, 4 (J.A. Serret, ed.), Gauthier-Villars, Paris (1869), 301–322 Google Scholar
13. 13.
Lange, L.J.: On a family of twin convergence regions for continued fractions. Ill. J. Math. 10, 97–108 (1966)
14. 14.
Lorentzen, L., Waadeland, H.: Continued Fractions with Applications. North-Holland, Amsterdam (1992)
15. 15.
Seidel, L.: Bemerkungen über den Zusammenhang zwischen dem Bildungsgesetze eines Kettenbruches Art Fortgangs Näherungsbrüche. Abh. Kgl. Bayr. Akad. der Wiss., München, Zweite Klasse, 7(3), 559 (1855) Google Scholar
16. 16.
Waadland, H.: Tales about tails. Proc. Am. Math. Soc. 90(1), 57–64 (1984)
17. 17.
Wall, H.S.: Partially bounded continued fractions. Proc. Am. Math. Soc. 7, 1090–1093 (1956)