Explicit transformations for generalized Lambert series associated with the divisor function \(\sigma_a^{(N)}(n)\) and their applications. (English) Zbl 1540.11111
The main goal of this paper is to generalize (1.13) and (1.14) obtained in [A. Dixit et al., Res. Math. Sci. 9, No. 2, Paper No. 34, 54 p. (2022; Zbl 1492.33006)] in the setting of \(\sigma_a^{(N)}(n)e^{-ny}\), where \(\sigma_a^{(N)}(n)\) is the generalized sum-of-divisors function with \(N\)th powers of divisors. This has been studied by the authors in many places starting from [A. Dixit et al., Nagoya Math. J. 239, 232–293 (2020; Zbl 1462.11064)] based on the functional equation.
In this paper, the authors use the Voronoi summation formula [A. Dixit et al., “Voronoi summation formula for the generalized divisor function \(\sigma_{z}^{(k)}(n)\)”, Preprint, arXiv:2303.09937] to prove transformation formulas for Lambert series. It seems that the same results could be obtained by the use of the functional equation and the Hecke gamma transform and that the main interest lies in finding new special functions which express the results in a closed form.
In this paper, the authors use the Voronoi summation formula [A. Dixit et al., “Voronoi summation formula for the generalized divisor function \(\sigma_{z}^{(k)}(n)\)”, Preprint, arXiv:2303.09937] to prove transformation formulas for Lambert series. It seems that the same results could be obtained by the use of the functional equation and the Hecke gamma transform and that the main interest lies in finding new special functions which express the results in a closed form.
Reviewer: Shigeru Kanemitsu (Kitakyūshū)
MSC:
| 11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 11P82 | Analytic theory of partitions |
Keywords:
Ramanujan’s formula for \(\zeta(2m+1)\); asymptotic expansions; plane partitions; Dedekind eta transformation; Meijer \(G\)-functionSoftware:
DLMFReferences:
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