modular relation
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2019 ◽  
Vol 196 ◽  
pp. 381-419 ◽  
Author(s):  
Eric D'Hoker ◽  
Michael B. Green ◽  
Pierre Vanhove

2018 ◽  
Vol 239 ◽  
pp. 232-293 ◽  
Author(s):  
ATUL DIXIT ◽  
RAJAT GUPTA ◽  
RAHUL KUMAR ◽  
BIBEKANANDA MAJI

A comprehensive study of the generalized Lambert series $\sum _{n=1}^{\infty }\frac{n^{N-2h}\text{exp}(-an^{N}x)}{1-\text{exp}(-n^{N}x)},0<a\leqslant 1,~x>0$, $N\in \mathbb{N}$ and $h\in \mathbb{Z}$, is undertaken. Several new transformations of this series are derived using a deep result on Raabe’s cosine transform that we obtain here. Three of these transformations lead to two-parameter generalizations of Ramanujan’s famous formula for $\unicode[STIX]{x1D701}(2m+1)$ for $m>0$, the transformation formula for the logarithm of the Dedekind eta function and Wigert’s formula for $\unicode[STIX]{x1D701}(1/N),N$ even. Numerous important special cases of our transformations are derived, for example, a result generalizing the modular relation between the Eisenstein series $E_{2}(z)$ and $E_{2}(-1/z)$. An identity relating $\unicode[STIX]{x1D701}(2N+1),\unicode[STIX]{x1D701}(4N+1),\ldots ,\unicode[STIX]{x1D701}(2Nm+1)$ is obtained for $N$ odd and $m\in \mathbb{N}$. In particular, this gives a beautiful relation between $\unicode[STIX]{x1D701}(3),\unicode[STIX]{x1D701}(5),\unicode[STIX]{x1D701}(7),\unicode[STIX]{x1D701}(9)$ and $\unicode[STIX]{x1D701}(11)$. New results involving infinite series of hyperbolic functions with $n^{2}$ in their arguments, which are analogous to those of Ramanujan and Klusch, are obtained.


2012 ◽  
Vol 23 (3) ◽  
pp. 183-190
Author(s):  
H.-L. Li ◽  
S. Kanemitsu ◽  
X.-H. Wang

2012 ◽  
Vol 66 (2) ◽  
pp. 411-427
Author(s):  
K. CHAKRABORTY ◽  
S. KANEMITSU ◽  
H. TSUKADA

2011 ◽  
Vol 65 (1) ◽  
pp. 39-53 ◽  
Author(s):  
K. CHAKRABORTY ◽  
S. KANEMITSU ◽  
X.-H. WANG

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