Abstract
In this note, we develop transformation formulae and expansions for the log tangent integral, which are then used to derive series acceleration formulae for certain values of Dirichlet L-functions, such as Catalan's constant. The formulae are characterized by the presence of an infinite series whose general term consists of a linear recurrence damped by the central binomial coefficient and a certain quadratic polynomial. Typically, the series can be expressed in closed form as a rational linear combination of Catalan's constant and π times the logarithm of an algebraic unit.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, p. 807, 1972.
D.H. Bailey and H.R.P. Ferguson, “Numerical results on relations between numerical constants using a new algorithm,” Mathematics of Computation 53 (1989), 649–656.
D.H. Bailey and H.R.P. Ferguson, “A polynomial time, numerically stable integer relation algorithm,” RNR Technical Report, RNR-91-032.
B.C. Berndt, Ramanujan's Notebooks: Part I, Springer-Verlag, p. 289, 1985.
A. Borodin, R. Fagin, J.E. Hopcroft, and M. Tompa, “Decreasing the nesting depth of expressions involving square roots,” J.Symbolic Comp. 1(1985), 169–188.
J.M. Borwein and P.B. Borwein, Pi and the AGM, Wiley-Interscience, John Wiley & Sons, Toronto, p. 384 1987.
J.M. Borwein and D.M. Bradley, “Searching symbolically for Apéry-like formulae for values of the Riemann zeta function,” SIGSAM Bulletin of Symbolic and Algebraic Manipulation, 30(2) (1996), 2–7.
J.M. Borwein and D.M. Bradley, “Empirically determined Apéry-like formulae for .4n C 3/,” Experimental Mathematics 6(1997), 181–194.
H.R.P. Ferguson and R.W. Forcade, “Generalization of the Euclidean algorithm for real numbers to all dimensions higher than two,” Bulletin of the American Mathematical Society 1(1979), 912–914.
G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th edition, Oxford University Press, New York, p. 148, 1979.
J. Hastad, B. Just, J.C. Lagarias, and C.P. Schnorr, “Polynomial time algorithms for finding integer relations among real numbers,” SIAM Journal on Computing 18 (1988), 859–881.
A.K. Lenstra, H.W. Lenstra, and L. Lovasz, “Factoring polynomials with rational coefficients,” Math.Annalen, 261 (1982), 515–534.
L. Lewin, Polylogarithms and Associated Functions, Elsevier, North Holland, New York, 1981.
S. Ramanujan, “On the integral R x 0 tan ¡ 1 t t dt,” Journal of the Indian Mathematical Society 7(1915), 93–96.
N.J.A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995.
R. Zippel, “Simplifications of expressions involving radicals,” J.Symbolic Comp. 1(1985), 189–210.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bradley, D.M. A Class of Series Acceleration Formulae for Catalan's Constant. The Ramanujan Journal 3, 159–173 (1999). https://doi.org/10.1023/A:1006945407723
Issue Date:
DOI: https://doi.org/10.1023/A:1006945407723