Stirling functions of first kind in the setting of fractional calculus and generalized differences

Abstract

The purpose of this paper is to present a new approach to generalizations of Stirling numbers of the first kind by the application of differential and integration operators of fractional order and generalized, infinite differences. Such an approach allows us to extend the classical Stirling numbers of the first kind, s(n, k), to functions s(α, β), where both parameters n, k have been extended to complex α, β. Under such a construction the s(α, β) turn out to have the series representation—a major result of this paper for , with for any when β = 0. Various properties of the new Stirling functions are established, most generalize those known for the numbers s(n, k); some are new, i.e. a multiple sum formula for s(α, k), and an interesting connection between the s(α, β) and the Riemann zeta function for complex β with . Several connections between the s(α, β) and the Stirling functions of second kind, s(α, β), studied earlier by the authors, are deduced. Thus the s( − n, β) coincide with the Stirling functions S( − β, n) of second kind, apart from a multiplicative constant. Of fundamental importance is the orthogonality property of the s(α, k) and S(k, m). The basic tool here is the Shannon sampling theorem of signal analysis. The Riemann–Liouville fractional derivative is expressed in terms of Hadamard derivatives, which involve the powers of the operator δ = x(d/dx). The sampling representation of the Mittag–Leffler function as a function of α is one of the many new results. Finally, a new “infinite” or fractional order difference operator, Δα, is defined in terms of the s(α, k); it involves the powers of the operator Θ = xΔ. This calculus of “infinite” differences is applied to representative examples, including the factorial and exponential functions.