More on Hurwitz and Tasoev Continued Fractions

Authors

  • Takao Komatsu Graduate School of Science and Technology, Hirosaki University, Hirosaki, Japan

DOI:

https://doi.org/10.5644/SJM.04.2.02

Abstract

Several new types of Hurwitz continued fractions have been studied. Most basic Hurwitz continued fractions can be expressed by using confluent hypergeometric functions ${}_0F_1(;c;z)$. This expression enables us to find some more general Hurwitz continued fractions. A contrast between Tasoev continued fractions and Hurwitz ones yields some more general Tasoev continued fractions. Some Ramanujan continued fractions are also discussed.

 

1991 Mathematics Subject Classification. 11A55, 11J70, 11Y16, 33C10

Downloads

Download data is not yet available.

References

B. C. Berndt, Ramanujan’s notebooks. Part V, Springer, New York, 1998.

A. Chˆatelet, Contribution `a la th´eorie des fractions continues arithm´etiques, Bull. Soc. Math. France, 40 (1912), 1–25.

A. Hurwitz, Uber die Kettenbr¨uche, deren Teilnenner arithmetische Reihen bilden, Vierteljahrsschrift d. Naturforsch. Gesellschaft in Z¨urich, Jahrg. 41, 1896 = Mathematische Werke, Band II, Birkh¨auser, Basel, 1963, pp. 276–302.

W. B. Jones and W. J. Thron, Continued Fractions: Analytic Theory and Applications, (Encyclopedia of mathematics and its applications; vol. 11), Addison-Wesley, Reading, 1980.

T. Komatsu, On Hurwitzian and Tasoev’s continued fractions, Acta Arith., 107 (2003), 161–177.

T. Komatsu, Simple continued fraction expansions of some values of certain hypergeometric functions, Tsukuba J. Math., 27 (2003), 161–173.

T. Komatsu, Tasoev’s continued fractions and Rogers-Ramanujan continued fractions, J. Number Theory, 109 (2004), 27–40.

T. Komatsu, Hurwitz and Tasoev continued fractions, Monatsh. Math., 145 (2005), 47–60.

T. Komatsu, An algorithm of infinite sums representations and Tasoev continued fractions, Math. Comp., 74 (2005), 2081–2094.

T. Komatsu, Hurwitz continued fractions with confluent hypergeometric functions, Czech. Math. J., 57 (132) (2007), 919–932.

T. Komatsu, Tasoev continued fractions with long period, Far East J. Math. Sci. (FJMS), 28 (2008), 89–121.

J. Mc Laughlin and N. J. Wyshinski, Ramanujan and the regular continued fraction expansion of real numbers, Math. Proc. Camb. Philos. Soc., 138 (2005), 367–381.

O. Perron, Die Lehre von den Kettenbr¨uchen, Band I, Teubner, Stuttgart, 1954.

A. J. van der Poorten, Continued fraction expansions of values of the exponential function and related fun with continued fractions, Nieuw Arch. Wiskd., 14 (1996), 221–230.

G. N. Raney, On continued fractions and finite automata, Math. Ann., 206 (1973), 265–283.

L. J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge 1966.

H. S. Wall, Analytic theory of continued fractions, D. van Nostrand Company, Toronto, 1948.

Downloads

Published

11.06.2024

How to Cite

Komatsu, T. (2024). More on Hurwitz and Tasoev Continued Fractions. Sarajevo Journal of Mathematics, 4(2), 155–180. https://doi.org/10.5644/SJM.04.2.02

Issue

Section

Articles