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Multiple gamma function

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In mathematics, the multiple gamma function ΓN 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).

Contents

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

Γ N ( w | a 1 , . . . , a N ) = exp ⁡ ( ∂ ∂ s ζ N ( s , w | a 1 , . . . , a N ) | s = 0 )

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

Properties

Considered as a meromorphic function of w, ΓN(w|a1,...) has no zeros, and has poles exactly at the values w=−(n1a1+...+nNaN) 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)= aw/a − 1/2Γ(w/a)/√(2π)
  • Γ N ( w | a 1 , . . . , a N ) = Γ N − 1 ( w | a 1 , . . . , a N − 1 ) Γ N ( w + a N | a 1 , . . . , a N )

    • References

    Multiple gamma function Wikipedia
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