Multiple gamma function

In mathematics, the multiple gamma function Γ_N is a generalization of the Euler Gamma function and the Barnes G-function. The double gamma function was studied Barnes (1901). At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied these further in Barnes (1904).

Double gamma functions Γ₂ are closely related to the q-gamma function, and triple gamma functions Γ₃ are related to the elliptic gamma function.

Definition

$\Gamma_N(w|a_1,...,a_N) = \exp\left(\frac{\partial}{\partial s}\zeta_N(s,w|a_1,...,a_N)|_{s=0}\right)$

where ζ_N is the Barnes zeta function. (This differs by a constant from Barnes's original definition.)

Properties

Considered as a meromorphic function of w, Γ_N(w|a₁,...) has no zeros, and has poles exactly at the values w=−n₁a₁+...+n_Na_N for non-negative integers n₁,..., which are simple poles unless some of these numbers coincide. Up to multiplication by the exponential of a polynomial, it is the unique meromorphic function of finite order with these zeros and poles.

• Γ₀(w|)= 1/w
• Γ₁(w|a)= a^{w/a − 1/2}Γ(w/a)/√(2π)
• $\Gamma_N(w|a_1,...,a_N)=\Gamma_{N-1}(w|a_1,...,a_{N-1})\Gamma_N(w+a_N|a_1,...,a_N)$

References

• Barnes, E. W. (1899), "The Genesis of the Double Gamma Functions", Proc. London Math. Soc., s1-31: 358–381, doi:10.1112/plms/s1-31.1.358
• Barnes, E. W. (1899), "The Theory of the Double Gamma Function. [Abstract]", Proceedings of the Royal Society of London (The Royal Society) 66: 265–268, doi:10.1098/rspl.1899.0101, ISSN 0370-1662, JSTOR 116064
• Barnes, E. W. (1901), "The Theory of the Double Gamma Function", Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character (The Royal Society) 196: 265–387, doi:10.1098/rsta.1901.0006, ISSN 0264-3952, JSTOR 90809
• Barnes, E. W. (1904), "On the theory of the multiple gamma function", Trans. Cambridge Philos. Soc. 19: 374–425
• Friedman, Eduardo; Ruijsenaars, Simon (2004), "Shintani–Barnes zeta and gamma functions", Advances in Mathematics 187 (2): 362–395, doi:10.1016/j.aim.2003.07.020, ISSN 0001-8708, MR 2078341
• Ruijsenaars, S. N. M. (2000), "On Barnes' multiple zeta and gamma functions", Advances in Mathematics 156 (1): 107–132, doi:10.1006/aima.2000.1946, ISSN 0001-8708, MR 1800255