The Krätzel function and evaluation of integralsI

This paper is devoted to the study of the special function Z .z/ of z 2 C with two parameters 2 R and 2 C. In a special case such a function coincides with the McDonald function. Using the representations of Z .z/ in terms of the H-function, being different for > 0 and < 0, we deduce the asymptotic behavior of this function at zero and infinity and give applications to evaluation of integrals involving product of Z .x/ and the H- and Meijer G-function, the Bessel function of the first kind, the McDonald function and cosine and sine functions.