Representation of the Modified Bessel Functions ∗

Some power series representations of the modified Bessel functions (McDonald functions Kα) are derived using the little known formalism of fractional derivatives. The resulting summation formulae are believed to be new. 1 Fractional derivatives There are several non-trivial examples in mathematics when some quantity, originally defined as integer, can radically extend its original range and assume fractional or even continuous values. The most common example is the gamma-function of Euler which is a natural, unique generalization of the ordinary factorial: n! ≡ n ∏ i=1 i = Γ (n+ 1) = ∞ ∫ 0 tedt (n > −1) The same thing may be performed with the order of derivatives which can also be made fractional. Although useful, fractional derivatives do not however create any essentially new calculus being rather some sort of particular, relatively simple, integral transforms. ∗Cracow Observatory preprint, no. 7/98 1 Following Oldham and Spanier (see e.g. [1]; cf. also [2], [3] and [4]) we define the fractional derivative ∂ x−a ≡ ( ∂ ∂x )s by an integral representation known as the Riemann-Liouville integral. Given a real number s < 0, define ∂ x−af (x) = 1 Γ (−s) x ∫ a (x− t) f (t) dt (1) where a < x is a fixed number, referred to as the boundary point. Since s < 0, the integral is convergent, provided that f behaves well. For s ≥ 0 we define ∂ x−a ≡ ∂ n x∂ s−n x−a where n is a positive integer chosen large enough so that s−n < 0 in order to assure convergence of the integral in definition (1). It is not difficult to show that, as expected, such a definition does not depend on n. One can further prove that the familiar Leibniz rule for product differentiation has the form ∂ x−a (fg) = ∞ ∑