A337981 - OEIS

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).

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

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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

Table of n, a(n) for n=1..87.

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.

Wikipedia, Euler-Mascheroni constant.

Index entries for transcendental numbers

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...