We are interested in the special Heun equation, " special " means that all exponent differences are odd multiples of 1/2. Heun means 4 regular singularities.

The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) → −1 as y → ∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics.

Let H(B) denote the space of all holomorphic functions on the unit ball B Cn. We investigate the following integral operators: Tg( f )(z)= ∫ 1 0 f (tz) g(tz)(dt/t), Lg( f )(z)= ∫ 1 0 f (tz)g(tz)(dt/t), f H(B), z B, where g H(B), and h(z)= ∑n j=1 zj(∂h/∂zj)(z) is the radial derivative of h. The operator Tg can be considered as an extension of the Cesàro operator on the unit disk. The boundedness...

We use functional calculus methods to investigate qualitative properties of C0semigroups that are preserved by time-discretization methods. Preservation of positivity, concavity and other qualitative shape properties which can be described via positivity are treated in a Banach lattice framework. Preservation of contractivity (or norm-bound) of the semigroup is investigated in the Banach space ...