Abstract: We present several inversion formulas and their applications in expansion and interpolation problems.
First, the concept of a generalized Stirling number pair can be characterized by a pair of inverse relations, in which the problem of expansion of A(t)f(g(t)) (a composition of any given formal power series) can be constructively solved with the aid of Sheffer-type differential operators. Using the generalized Stirling number pair, we establish an inversion formula that can be applied to find an expansion of an analytic function f in terms of a sequence of Sheffer-type polynomials. The result can be readily extended to the higher-dimensional setting.
Secondly, multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite-Fejér interpolation formulas of the Newton type are established using Carlita's inversion formulas. By letting q -> 1, the obtained formulas reduce to the corresponding multivariate polynomial interpolation formulas with combinatorial form.
Finally, we provide a wide class of Möbius inversion formulas in terms of the generalized Möbius functions and its application to the setting of the Selberg multiplicative functions.