The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A337981 Decimal expansion of Pi*BesselY(0,2)/(2*BesselJ(0,2)) - gamma, where BesselJ and BesselY are the Bessel functions of the first and second kind, respectively, and gamma is Euler's constant (A001620). 0
3, 0, 0, 3, 5, 3, 1, 3, 1, 7, 8, 8, 4, 9, 4, 9, 0, 3, 9, 0, 6, 8, 1, 5, 0, 6, 8, 0, 8, 1, 1, 0, 7, 6, 1, 8, 8, 1, 4, 0, 0, 7, 0, 5, 4, 5, 1, 7, 0, 6, 5, 5, 7, 5, 7, 9, 5, 6, 7, 5, 3, 3, 2, 4, 8, 6, 6, 4, 4, 5, 7, 6, 7, 3, 0, 1, 5, 2, 5, 9, 4, 4, 8, 1, 2, 0, 5 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This constant is transcendental (Mahler, 1968).
Mahler's proof was obtained as a result of his attempt to prove that Euler's constant gamma (A001620) is transcendental. For one week he believed that he had succeeded in proving the transcendence of both gamma and exp(gamma) (Mahler, 1982).
REFERENCES
Kurt Mahler, Lectures on Transcendental Numbers, Springer-Verlag, 1976. See. p. 184.
David Masser, Auxiliary Polynomials in Number Theory, Cambridge University Press, 2016. See p. 307.
LINKS
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020. See p. 5.
Sanoli Gun, V. Kumar Murty and Ekata Saha, Linear and algebraic independence of generalized Euler-Briggs constants, Journal of Number Theory, Vol. 166 (2016), pp. 117-136. See p. 118.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, Bulletin of the American Mathematical Society, Vol. 50, No. 4 (2013), pp. 527-628. See Theorem 3.16.1, p. 606.
Kurt Mahler, Applications of a Theorem by A. B. Shidlovski, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 305, No. 1481 (1968), pp. 149-173; alternative link.
Kurt Mahler, Fifty years as a mathematician, Journal of Number Theory, Vol. 14, No. 2 (1982), pp. 121-155. See p. 137.
M. Ram Murty and N. Saradhab, Transcendental values of the digamma function, Journal of Number Theory, Vol. 125, No. 2 (2007), pp. 298-318. See p. 302.
A. J. van der Poorten, Obituary: Kurt Mahler, 1903-1988, Vol. 51, No. 3 (1991), pp. 343-380. See p. 348.
Tanguy Rivoal, On the arithmetic nature of the values of the gamma function, Euler's constant, and Gompertz's constant, Michigan Mathematical Journal, Vol. 61, No. 2 (2012), pp. 239-254. See p. 252.
Eric Weisstein's World of Mathematics, Bessel Function of the First Kind.
Eric Weisstein's World of Mathematics, Bessel Function of the Second Kind.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Wikipedia, Bessel function.
FORMULA
Equals Sum_{k>=1} (-1)^(k+1) * H(k)/(k!)^2 / Sum_{k>=0} (-1)^k/(k!)^2, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
EXAMPLE
3.00353131788494903906815068081107618814007054517065...
MATHEMATICA
RealDigits[Pi*BesselY[0, 2]/BesselJ[0, 2]/2 - EulerGamma, 10, 100][[1]]
PROG
(PARI) Pi*bessely(0, 2)/2/besselj(0, 2)-Euler \\ Charles R Greathouse IV, Oct 23 2023
CROSSREFS
Sequence in context: A222522 A091921 A037285 * A341761 A181787 A318925
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Nov 17 2020
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 17 20:03 EDT 2024. Contains 373464 sequences. (Running on oeis4.)