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Explicit transformations for generalized Lambert series associated with the divisor function \(\sigma _{a}^{(N)}(n)\) and their applications


Let \(\sigma _a^{(N)}(n)=\sum _{d^{N}|n}d^a\). An explicit transformation is obtained for the generalized Lambert series \(\sum _{n=1}^{\infty }\sigma _{a}^{(N)}(n)e^{-ny}\) for \(\text {Re}(a)>-1\) using the recently established Voronoï summation formula for \(\sigma _a^{(N)}(n)\) and is extended to a wider region by analytic continuation. For \(N=1\), this Lambert series plays an important role in string theory scattering amplitudes as can be seen in the recent work of Dorigoni and Kleinschmidt. These transformations exhibit several identities—a new generalization of Ramanujan’s formula for \(\zeta (2m+1)\), an identity associated with extended higher Herglotz functions, generalized Dedekind eta transformation, Wigert’s transformation, etc., all of which are derived in this paper, thus leading to their uniform proofs. A special case of one of these explicit transformations naturally leads us to consider generalized power partitions with “\(n^{2N-1}\) copies of \(n^{N}\).” Asymptotic expansion of their generating function as \(q\rightarrow 1^{-}\) is also derived which generalizes Wright’s result on the plane partition generating function. In order to obtain these transformations, several new intermediate results are required, for example, a new reduction formula for Meijer G-function and an almost closed-form evaluation of \(\left. \frac{\partial }{\partial \beta }E_{2N, \beta }(z^{2N})\right| _{\beta =1}\), where \(E_{\alpha , \beta }(z)\) is the two-variable Mittag–Leffler function.

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  1. Ramanujan gave equivalent definition of the generalized eta function for \(x>0\), but it can be easily extended to Re\((x)>0\).

  2. We emphasize here that the notation \(H_a^{(N)}(x)\) does not mean Nth derivative of some function \(H_a(x)\). This notation is used so as to be consistent with that used by Wigert [44] and Koshliakov [29] for the associated arithmetic as well as special functions. We retain it throughout the paper for other functions as well.

  3. Whenever the number of terms in the sequence \(\langle \cdot \rangle \) is M, \(M\ne N\), we explicitly write \(\langle \cdot \rangle |_{i=1}^{M}\).

  4. The second set of conditions, that is, \(a_j-a_t\notin {\mathbb {Z}}\), can be deleted provided we understand that if it does not hold, then a passage to the limit is required. See, for instance, [32, p. 178].


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We sincerely thank the referees for giving nice and important comments which enhanced the quality of the paper. Part of the work was done when the first author was an INSPIRE faculty at IISER Kolkata supported by the DST grant DST/INSPIRE/04/2021/002753. The major part of this work was done when the first author was a postdoctoral fellow at IIT Gandhinagar and was funded by the SERB NPDF grant PDF/2021/001224. The second author is supported by the Swarnajayanti Fellowship grant SB/SJF/2021-22/08 of SERB (Govt. of India). The third author was supported by CSIR SPM Fellowship under the grant number SPM-06/1031(0281)/2018-EMR-I. All of the authors sincerely thank the respective funding agencies for their support. The first author also thanks IIT Gandhinagar for its support.

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Banerjee, S., Dixit, A. & Gupta, S. Explicit transformations for generalized Lambert series associated with the divisor function \(\sigma _{a}^{(N)}(n)\) and their applications. Res Math Sci 10, 38 (2023).

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