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Iseki's Formula

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Iseki's formula gives a transformation law for the function Lambda(alpha,beta,z) defined below. Let Re[z]>0, 0<=alpha,beta<=1, and

Lambda(alpha,beta,z)=sum_(r=0)^infty[lambda((r+alpha)z-ibeta)+lambda((r+1-alpha)z+ibeta)],
(1)

where

lambda(x)=-ln(1-e^(-2pix))
(2)
=sum_(m=1)^(infty)(e^(-2pimx))/m.
(3)

Then if either 0<=alpha<=1 and 0<beta<1, or 0<alpha<1 and 0<=beta<=1,

Lambda(alpha,beta,z)=Lambda(1-beta,alpha,z^(-1))-pizsum_(n=0)^2(2; n)(iz)^(-n)B_(2-n)(alpha)B_n(beta),
(4)

where B_k(x) is a Bernoulli polynomial, and the second term on the right side can be written explicitly as

-piz(alpha^2alpha+1/6)+pi/z(beta^2-beta+1/6)+2pii(alpha-1/2)(beta-1/2).
(5)

See also

Dedekind Eta Function

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References

Apostol, T. M. "Iseki's Transformation Formula" and "Deduction of Dedekind's Functional Equation from Iseki's Formula." §3.5-3.6 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 53-61, 1997.Iseki, S. "The Transformation Formula for the Dedekind Modular Function and Related Functional Equations." Duke Math. J. 24, 653-662, 1957.

Referenced on Wolfram|Alpha

Iseki's Formula

Cite this as:

Weisstein, Eric W. "Iseki's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IsekisFormula.html

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