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 C k (z) = 1/k!{∂/∂v} k J v (z)∣ v=0 ( k = 0,1,2,3) of the Bessel function J 0 (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.