Iseki,s Formula
المؤلف:
Iseki, S
المصدر:
"The Transformation Formula for the Dedekind Modular Function and Related Functional Equations." Duke Math. J. 24
الجزء والصفحة:
...
23-12-2019
1177
Iseki's Formula
Let ![R[z]>0](http://mathworld.wolfram.com/images/equations/IsekisFormula/Inline1.gif)
, 
, and
![Lambda(alpha,beta,z)=sum_(r=0)^infty[lambda((r+alpha)z-ibeta)+lambda((r+1-alpha)z+ibeta)],](http://mathworld.wolfram.com/images/equations/IsekisFormula/NumberedEquation1.gif) |
(1)
|
where
Then if either 
and 
, or 
and 
,
 |
(4)
|
where 
is a Bernoulli polynomial, and the second term on the right side can be written explicitly as
 |
(5)
|
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.
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