On the differential properties of Temlyakov-type and Temlyakov–Bavrin-type integrals

We establish a formula that expresses the value of the density of the Temlyakov-type integral with defining domain of type $A$ in terms of the integral itself by means of a differential operator. We extend this operator connection to the Temlyakov–Bavrin-type integral of order $k$. Our study bases on the method of linear differential operators with variable coefficients which is developed by A. V. Nelaev.