Buescu, JorgePaixão, António2021-01-142021-01-142020-08BUESCU, J.; PAIXÃO, António Carlos – Positive-definiteness and integral representations for special functions. Positivity. ISSN 1385-1292. Vol. 25, N.º 2 (2020), pp. 731-7501385-12921572-9281http://hdl.handle.net/10400.21/12608It is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier-Laplace transform of a unique exponentially finite measure on R. In this paper we apply this complex integral representation to specific families of special functions, including the Gamma, zeta and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions. In the case of the zeta function this process leads to a new proof of an integral representation on the critical strip.engPositive definite functionsFourier-Laplace transformCharacteristic functionsHolomorphyGamma functionZeta functionExponentially convex functionsjournal article10.1007/s11117-020-00784-4