On some class of exponential type operators

Starting from a differential equation \(\frac{\partial}{\partial t} W (\lambda, t, u)=\frac{\lambda(u-t)}{p(t)}W(\lambda,t,u)-\beta W(\lambda,t,u)\) for the kernel of an operator \(S_\lambda(f,t) = \int_{A}^{B}W(\lambda,t,u)f(u)du\) with the normalization condition \(\int_A^B W(\lambda, t, u)du = 1\) we prove some properties which are similar to properties proved by Ismail and May for the exponential operators. In particular, we show that all these operators are approximation operators. Moreover, a method of determining \(S_\lambda\) for a given function \(p\) is introduced