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A construction of q-analogue of Dedekind sums

Published online by Cambridge University Press:  22 January 2016

Junya Satoh
Junya Satoh*
Affiliation:
Konan Women’s Junior College, Takaya-cho 172, Konan 483, Japan
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If one looks back the classical proof (cf. Carlitz [4]) of the reciprocity law for Dedekind sums in order to construct q-analogue of Dedekind sums which also have the reciprocity law, one can soon see that the following elementary equation is essential in the proof:

for any distinct complex numbers u and v, where means the generating function of Euler numbers associated to u. So we must extend the above equation to the generating function of q-Euler numbers for our purpose.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 127 , September 1992 , pp. 129 - 143
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

References

[ 1 ] Apostol, T. M., Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J., 17 (1950), 147–157.Google Scholar
[ 2 ] Carlitz, L., q-Bernoulli numbers and polynomials, Duke Math. J., 15 (1948), 987–1000.Google Scholar
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[ 6 ] Satoh, J., q-Analogue of Riemann’s ζ-function and q-Euler numbers, J. Nunmber Theory, 31 (1989), 346–362.CrossRefGoogle Scholar