Stirling Functions of the Second Kind in the Setting of Difference and Fractional Calculus

Abstract

The purpose of this article and companion ones is to present a new approach to generalizations of Stirling numbers of the first and the second kind in terms of fractional calculus analysis by using differences and differentiation operators of fractional order. Such an approach allows us to extend the classical Stirling numbers of the first and the second kind in a natural way not only to any positive, but also to any negative order. Moreover, an application of the fractional approach gives us the opportunity to extend the classical Stirling numbers to more general complex functions. In the present article we extend the classical Stirling numbers of the second kind, S(n, k), for the first parameter from a nonnegative integer number n to any complex α. Such constructions, S(α, k), will be defined for any complex α and by

when k >0, while S(0, 0) = 1 and when k = 0. We show that S(α, k) with positive α can be represented by the Liouville and Marchaud fractional derivatives of the exponential functions, while for negative α it can be interpreted in terms of Liouville fractional integrals. Many of the main properties of the above Stirling functions are established; they generalize those, well known, for the classical Stirling numbers S(n, k) (). Several new applications are presented. Thus, the sum , for any complex and , is represented as a finite sum involving the S(α + 1, k). Whereas the case α = n is a classical result, even the particular case α = −n gives a new application. Further, Hadamard-type fractional integrals and derivatives, basic in Mellin transform theory on (0, ∞), are represented in terms of infinite series involving the S(α, k) and the powers of the operator . Its corresponding classical discrete version, involving the S(n, k), plays an important role in computational mathematics, combinatorial analysis and discrete mathematics.

Keywords:

• Generalized Stirling numbers
• Stirling functions of the second kind
• Liouville fractional integrals and derivatives
• Marchaud and Hadamard-type fractional derivatives and integrals
• Fractional order differences
• Mellin transforms
• Mathematics Subject Classification: 33E99
• 11B73
• 26A33
• 05A10

Acknowledgment

The present investigation was supported in part by the Belarusian Fundamental Research Fund, by MCYT, by DGUI of G.A.CC and by ULL.