Multiple gamma function

For derivatives of the log of the gamma function, see polygamma 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). A 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)
• Γ_N(w | a₁,...,a_N) = Γ_{N − 1}(w | a₁,...,a_{N − 1})Γ_N(w + a_N | a₁,...,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, 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, 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, MR2078341 
• 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, MR1800255