3.30–4.30 pm, Seminar Hall
Applications of a theorem by A. B. Shidlovski
Chennai Mathematical Institute.
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 E-functions evaluated at $\alpha$ that are algebraically independent over the field of rational numbers equals the maximum number of Siegel E-functions 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.