Iseki's Formula

Let , , and

(1)

where

(2)

Then if either and , or and ,

(3)
where is a Bernoulli polynomial, and the second term on the right side can be written explicitly as

(4)

 

Dedekind Eta Function




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.