A direct proof of Kim's identities

Details A direct proof of Kim's identities A direct proof of Kim's identities

1998-01-15

arxiv.org/abs/cond-mat/9801148v1

J.Phys.A31:1105-1108,1998

Physics

Condensed Matter

Statistical Mechanics

Latex, 6 pages, accepted by J. Physics A

Scientific paper

10.1088/0305-4470/31/3/020

As a by-product of a finite-size Bethe Ansatz calculation in statistical mechanics, Doochul Kim has established, by an indirect route, three mathematical identities rather similar to the conjugate modulus relations satisfied by the elliptic theta constants. However, they contain factors like $1 - q^{\sqrt{n}}$ and $1 - q^{n^2}$, instead of $1 - q^n$. We show here that there is a fourth relation that naturally completes the set, in much the same way that there are four relations for the four elliptic theta functions. We derive all of them directly by proving and using a specialization of Weierstrass' factorization theorem in complex variable theory.

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