Hadjicostas's Formula

Hadjicostas's formula is a generalization of the unit square double integral

(1)

(Sondow 2003, 2005; Borwein et al. 2004, p. 49), where  is the Euler-Mascheroni constant. It states

(2)

for , where  is the gamma function and  is the Riemann zeta function (although care must be taken at  because of the removable singularity present there). It was conjectured by Hadjicostas (2004) and almost immediately proved by Chapman (2004). The special case  gives Beukers's integral for ,

(3)

(Beukers 1979). At , the formula is related to Beukers's integral for Apéry's constant , which is how interest in this class of integrals originally arose.

There is an analogous formula

(4)

for , due to Sondow (2005), where  is the Dirichlet eta function. This includes the special cases

			(5)
			(6)
			(7)

(OEIS A094640; Sondow 2005) and

			(8)
			(9)

REFERENCES:

Beukers, F. "A Note on the Irrationality of  and ." Bull. London Math. Soc. 11, 268-272, 1979.

Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.

Chapman, R. "A Proof of Hadjicostas's Conjecture." 15 Jun 2004. http://arxiv.org/abs/math/0405478.

Guillera, J. and Sondow, J. "Double Integrals and Infinite Products for Some Classical Constants Via Analytic Continuations of Lerch's Transcendent." 16 June 2005 http://arxiv.org/abs/math.NT/0506319.

Hadjicostas, P. "A Conjecture-Generalization of Sondow's Formula." 21 May 2004. http://www.arxiv.org/abs/math.NT/0405423/.

Sloane, N. J. A. Sequences A094640, A103130 in "The On-Line Encyclopedia of Integer Sequences."

Sondow, J. "Criteria for Irrationality of Euler's Constant." Proc. Amer. Math. Soc. 131, 3335-3344, 2003. http://arxiv.org/abs/math.NT/0209070.

Sondow, J. "Double Integrals for Euler's Constant and  and an Analog of Hadjicostas's Formula." Amer. Math. Monthly112, 61-65, 2005.