3.30–4.30 pm, Seminar Hall Applications of a theorem by A. B. Shidlovski Aritriya Mukhapadhyay Chennai Mathematical Institute. 081117 Abstract There is a theorem of Shidlovski on Siegel E functions which satisfy systems of linear differential equations over rational functions $\C(z)$, that is, the coefficients are rational functions of z. Suppose the coefficients of the system are such that they are analytic at an algebraic number say $\alpha$. Then the theorem of Shidlovski says that the maximum number of function values of these Siegel Efunctions evaluated at $\alpha$ that are algebraically independent over the field of rational numbers equals the maximum number of Siegel Efunctions that are algebraically independent over the field of rational functions of z. This theorem is applied to study the partial derivatives of the Bessel function of the first kind $J_0(z)$ where we get expressions involving Euler's constant $\gamma$ and $\zeta(3)$ whose transcendence can be established.
