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G-Function

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As defined by Erdélyi et al. (1981, p. 20), the G-function is given by

G(z)=psi_0(1/2+1/2z)-psi_0(1/2z),
(1)

where psi_0(z) is the digamma function. Integral representations are given by

G(z)=2int_0^1(t^(z-1))/(1+t)dt
(2)
=2int_0^infty(e^(-zt))/(1+e^(-t))dt
(3)

for R[z]>0. G(z) is also given by the series

G(z)=2sum_(n=0)^infty((-1)^n)/(z+n),
(4)

and in terms of the hypergeometric function by

G(z)=2z^(-1)_2F_1(1,z;1+z;-1).
(5)

It obeys the functional relations

G(1+z)=2z^(-1)-G(z)
(6)
G(1-z)=2picsc(piz)-G(z)
(7)
G(mz)={-2/msum_(r=0)^(m-1)(-1)^rpsi_0(z+r/m) for m even; 1/msum_(r=0)^(m-1)(-1)^rG(z+r/m) for m odd.
(8)

See also

Barnes G-Function, Digamma Function, Meijer G-Function, Ramanujan g- and G-Functions

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References

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Function G(z)." §1.8 in Higher Transcendental Functions, Vol. 1. New York: Krieger, pp. 20 and 44-46, 1981.

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G-Function

Cite this as:

Weisstein, Eric W. "G-Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/G-Function.html

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