The generalized modified Bessel function and its connection with Voigt line profile and Humbert functions. (English) Zbl 1436.33020
Summary: Recently Dixit, Kesarwani, and Moll introduced a generalization \(K_{z, w}(x)\) of the modified Bessel function \(K_z(x)\) and showed that it satisfies an elegant theory similar to that of \(K_z(x)\). In this paper, we show that while \(K_{\frac{1}{2}}(x)\) is an elementary function, \(K_{\frac{1}{2}, w}(x)\) can be written in the form of an infinite series of Humbert functions. As an application of this result, we generalize the transformation formula for the logarithm of the Dedekind eta function \(\eta(z)\). We also establish a connection between \(K_{\frac{1}{2}, w}(x)\) and the cumulative distribution function corresponding to the Voigt line profile.
MSC:
| 33E20 | Other functions defined by series and integrals |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
Keywords:
generalized modified Bessel function; Humbert function; Voigt profile; distribution functionSoftware:
DLMFReferences:
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