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Explicit transformations of certain Lambert series. (English) Zbl 1492.33006

The purpose of this article is to find explicit transformations of certain Lambert series by using hypergeometric series. In other words, an exact transformation is obtained for the series \(\sum_{n=1 }^{\infty}\sigma_a(n)e^{-ny}\) for \( a\in\mathbb{C}\) and Re\((y) > 0 \) , where \(\sigma_a(n)\) is the generalized divisor function. Furthermore, formulas for a new generalization of the modified Bessel function of the second kind due to Koshliakov is obtained. Actually, a new two-variable generalization of the modified Bessel function, together with a novel modular-type transformation involving \(r k (n)\), the number of representations of \(n\) as a sum of \(k\) squares, is obtained. The proofs use Mellin transforms, Watson kernel, Parseval’s formula, Riemann zeta function, dominated convergence theorem, transformation of Guinand and L’Hopital’s rule.

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
11F11 Holomorphic modular forms of integral weight
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
44A20 Integral transforms of special functions

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