Series of Bessel and Kummer-Type Functions by Árpád Baricz Dragana Jankov Maširević & Tibor K. Pogány

Author:Árpád Baricz, Dragana Jankov Maširević & Tibor K. Pogány
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(3.34)

valid for all ℜ(b) > ℜ(a) > 0, we transform the Kapteyn–Kummer series into

(3.35)

Hence, for all β ≥ α ≥ 0 using (3.35) we get

(3.36)

Here we employ the Euler Beta function’s integral form and its connection to the Gamma function:

where . Indeed, specifying p = a + αn, q = b − a + (β − α)n (3.36) immediately follows. Finally, by virtue of e.g. Cauchy’s convergence test we get the convergence region of :

for any fixed real ζ.

The integral representation formula (3.34) of Kummer’s function enable us to re-formulate the series (3.35) into the following form

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