In mathematics, the multiple gamma function is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by 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).
where is the Barnes zeta function. (This differs by a constant from Barnes's original definition.)
Propertiesedit
Considered as a meromorphic function of , has no zeros. It has poles at for non-negative integers . These poles are simple unless some of them coincide. Up to multiplication by the exponential of a polynomial, is the unique meromorphic function of finite order with these zeros and poles.
In the case of the double Gamma function, the asymptotic behaviour for is known, and the leading factor is[1]
Infinite product representationedit
The multiple gamma function has an infinite product representation that makes it manifest that it is meromorphic, and that also makes the positions of its poles manifest. In the case of the double gamma function, this representation is [2]
where we define the -independent coefficients
where is an -th order residue at .
Another representation as a product over leads to an algorithm for numerically computing the double Gamma function.[1]
Reduction to the Barnes G-functionedit
The double gamma function with parameters obeys the relations [2]
^ abSpreafico, Mauro (2009). "On the Barnes double zeta and gamma functions". Journal of Number Theory. 129 (9): 2035–2063. doi:10.1016/j.jnt.2009.03.005.
^Ponsot, B. Recent progress on Liouville Field Theory (Thesis). arXiv:hep-th/0301193. Bibcode:2003PhDT.......180P.
Further readingedit
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", Proceedings of the Royal Society of London, 66 (424–433): 265–268, doi:10.1098/rspl.1899.0101, ISSN 0370-1662, JSTOR 116064, S2CID 186213903
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, 196 (274–286): 265–387, Bibcode:1901RSPTA.196..265B, doi:10.1098/rsta.1901.0006, ISSN 0264-3952, JSTOR 90809
Barnes, E. W. (1904), "On the theory of the multiple gamma function", Trans. Camb. 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