In this paper, we study an asymptotic expansion of the heat kernel for a Laplace operator on a smooth Riemannian manifold without a boundary at enough small values of the proper time. The Seeley-DeWitt coefficients of this decomposition satisfy a set of recurrence relations, which we use to construct two function families of a special kind. Using these functions, we study the expansion of a local heat kernel for the inverse Laplace operator. We show that the new functions have some important properties. For example, we can consider the Laplace operator on the function set as a shift one. Also we describe various applications useful in theoretical physics and, in particular, we find a decomposition of Green's functions near the diagonal in terms of new functions. Comment: LaTeX, 22 pages, 3 figures; The second version contains more detailed description
