The Krätzel function and evaluation of integrals

This paper is devoted to the study of the special function Zρν(z) of z∈Cz∈C with two parameters ρ∈Rρ∈R and ν∈Cν∈C. In a special case such a function coincides with the McDonald function. Using the representations of Zρν(z) in terms of the HH-function, being different for ρ>0ρ>0 and ρ<0ρ<0, we deduce the asymptotic behavior of this function at zero and infinity and give applications to evaluation of integrals involving product of Zρν(x) and the HH- and Meijer GG-function, the Bessel function of the first kind, the McDonald function and cosine and sine functions.