A famous formula of Ramanujan states that for any \(\alpha ,\beta >0\) with \(\alpha \beta =\pi ^{2} \) and any nonzero integer \(m\) one has \[\frac{1}{\alpha ^{m} } \left\{\frac{1}{2} \zeta (2m+1)+\sum _{n=1}^{\infty }\frac{1}{n^{2m+1} (e^{2\alpha {\kern 1pt} n} -1)} \right\}=\frac{1}{(-\beta )^{m} } \left\{\frac{1}{2} \zeta (2m+1)+\sum _{n=1}^{\infty }\frac{1}{n^{2m+1} (e^{2\beta {\kern 1pt} n} -1)} \right\}\] \(-4^{m} \sum _{k=0}^{m+1}\frac{(-1)^{k} B_{2k} B_{2m+2-2k} }{(2k)!(2m+2-2k)!} {\kern 1pt} {\kern 1pt} \alpha ^{m+1-k} \beta ^{k} \).
Here, \(\zeta (s)\) is Riemann’s zeta function and \(B_{k} \) are the Bernoulli numbers. The authors obtain two-parameter extensions of this formula by developing new transformations for the Lambert series \(\sum _{n=1}^{\infty }\frac{n^{N-2m} \exp (-\alpha {\kern 1pt} n^{N} x)}{1-\exp (-n^{N} x)} \quad (0<\alpha \le 1,\; x>0,\; N\in \mathrm{N},\; m\in \mathrm{Z})\). An important tool in this work is an extension of Raabe’s cosine transform. The paper contains numerous interesting results including a relation between the values of \(\zeta (s)\) at certain odd integers.