Abstract
We prove approximation formulas for the logarithms of some infinite products, in particular, for Euler’s constant γ, log \(\frac{4}{\pi}\) and log σ, where σ is Somos’s quadratic recurrence constant, in terms of classical Legendre polynomials and partial sums of their series expansions. We also give conditional irrationality and linear independence criteria for these numbers. The main tools are Euler-type integrals, hypergeometric series, and Laplace method.
Similar content being viewed by others
References
Alladi K., Robinson M. (1980) Legendre polynomials and irrationality. J. Reine Angew. Math. 318, 137–155
Allouche, J.-P., Shallit, J., Sondow, J. Summation of series defined by counting blocks of digits. e-print. math. NT/0512399 (2005)
Andrews G.E., Askey R., Roy R. (1999) Special Functions. Cambridge University Press, Cambridge
Elsner C. (1995) On a sequence transformation with integral coefficients for Euler’s constant. Proc. Amer. Math. Soc. 123(5): 1537–1541
Fedoryuk M.V. (1987) Asymptotics: Integrals and Series. Mathematical Reference Library, Nauka, Moscow
Guillera, J., Sondow, J. Double integrals and infinite products for some classical constants via analytic continuations of Lerch’s transcendent. e-print. math.NT/0506319 (2005)
Hessami Pilehrood T., Hessami Pilehrood Kh. (2004) Criteria for irrationality of generalized Euler’s constant. J. Number Theory. 108, 169–185
Prévost, M. A family of criteria for irrationality of Euler’s constant. e-print. math. NT/0507231 (2005)
Rivoal, T. Polynômes de type Legendre et approximations de la constante d’Euler. (2005, notes); available at http://www-fourier.ujf-grenoble.fr/~rivoal/.
Sondow J. (2003) Criteria for irrationality of Euler’s constant. Proc. Amer. Math. Soc. 131, 3335–3344
Sondow J. (2005) Double integrals for Euler’s constant and ln \(\frac{4}{\pi}\) and an analog of Hadjicostas’s formula. Amer. Math. Monthly 112, 61–65
Sondow, J. New Vacca-type rational series for Euler’s constant and its “alternating” analog log \(\frac{4}{\pi}\). e-print. math. NT/0508042 (2005)
Sondow, J., Zudilin, W. Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper. Ramanujan J. (to appear). e-print. math. NT/0304021 (2003)
Vacca G. (1910) A new series for the Eulerian constant A new series for the Eulerian constant γ = .577 . . . . Quart. J. Pure Appl. Math. 41, 363–364
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pilehrood, K.H., Pilehrood, T.H. Arithmetical properties of some series with logarithmic coefficients. Math. Z. 255, 117–131 (2007). https://doi.org/10.1007/s00209-006-0015-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-006-0015-1