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 functio...

The aim of this paper is to outline the life and work of Chebyshev, creator in St. Petersburg of the largest prerevolutionary school of mathematics in Russia, who permitted himself to be equated only with Archimedes. Chebyshev, who was regularly in Paris, at the latest by 1852, if not already by 1842, a friend of Liouville and Hermite, was the author of ca 80 publications, covering approximation theory, probability theory, number theory, theory of mechanisms, as well as many problems of analysis and practical mathematics. He was also proud to be a constructor of various mechanisms, including an arithmomètre. Although the paper is intended for an approximation theorist readership, an attempt is made to give proportionate coverage of the broad spectrum of Chebyshev's achievements, emphasis being placed upon their background. The paper is based in part upon the authors' studies during 1985–1991

We characterize the function space whose elements have a Mellin transform with exponential decay at infinity. This result can be seen as a generalization of the Paley–Wiener theorem for Mellin transforms. As a byproduct in a similar spirit, we also characterize spaces of functions whose distances from Mellin–Paley–Wiener spaces have a prescribed asymptotic behavior. This leads to Mellin–Sobolev type spaces of fractional order.