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5 Gamma FunctionProperties

§5.17 Barnes’ G-Function (Double Gamma Function)

5.17.1 G(z+1) =Γ(z)G(z),
G(1) =1,
5.17.2 G(n)=(n-2)!(n-3)!1!,
5.17.3 G(z+1)=(2π)z/2exp(-12z(z+1)-12γz2)k=1((1+zk)kexp(-z+z22k)).
5.17.4 LnG(z+1)=12zln(2π)-12z(z+1)+zLnΓ(z+1)-0zLnΓ(t+1)dt.

In this equation (and in (5.17.5) below), the Ln’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i).

When z in |phz|π-δ(<π),

5.17.5 LnG(z+1)14z2+zLnΓ(z+1)-(12z(z+1)+112)Lnz-lnA+k=1B2k+22k(2k+1)(2k+2)z2k.

For error bounds and an exponentially-improved extension, see Nemes (2014). Here B2k+2 is the Bernoulli number (§24.2(i)), and A is Glaisher’s constant, given by

5.17.6 A=eC=1.28242 71291 00622 63687,


5.17.7 C=limn(k=1nklnk-(12n2+12n+112)lnn+14n2)=γ+ln(2π)12-ζ(2)2π2=112-ζ(-1),

and ζ is the derivative of the zeta function (Chapter 25).

For Glaisher’s constant see also Greene and Knuth (1982, p. 100) and §2.10(i).