Asymptotic estimates of the difference of products of Bessel functions by the integral of these functions

In the study of direct and inverse problems of finding the right-hand side of degenerate equations of mixed type with different boundary conditions, the problem arises of establishing asymptotic estimates for the differences of the products of cylindrical functions by the integral of these functions. Previously, on the basis of the established new formula for finding the finite binomial sum, the differences between the products of cylindrical functions and a definite integral of these functions are calculated through a generalized hypergeometric function. Using the asymptotic formula for large values of the argument for the generalized hypergeometric function, asymptotic estimates are established for large values of the parameter for the indicated differences of the Bessel functions of the first and second kind, as well as for modified Bessel functions.