Applications of a theorem by A. B. Shidlovski

Source abstract

Abstract Shidlovski’s deep theorem on Siegel E-functions satisfying systems of linear differential equations is applied in this paper to the study of the arithmetic properties of the partial derivatives Ck(z) = 1/k!{∂/∂v}kJv(z)∣v=0 (k = 0,1,2,3) of the Bessel function J0(z). As a by-product, expressions involving Euler’s constant γ and the constant ζ(3) are obtained for which the transcendency can be established.