An access to fractional differentiation via fractional difference quotients

Fractional calculus of periodic distributions

“…As described in [3], formula (1.10) can be used to motivate an alternative definition of a fractional integral, I α φ, expressed as a convolution integral, that will make sense for any α > 0 and φ ∈ L p 2π . The key to obtaining a formula for I α is to rewrite the right-hand side of (1.10) as…”

“…The Banach spaces L p 2π were also used by Butzer and Westphal [3] in their investigations into the difference quotient approach to fractional derivatives, where a (strong) Liouville-Grünwald fractional derivative D <α> of order α is defined by …”

“…Our main aim in this paper is to extend the work of Butzer and Westphal in [3] to the case of periodic distributions. Distributional versions of the Riemann-Liouville and Weyl fractional integrals (and the related Erdélyi-Kober operators) have been produced by Erdélyi [4] and McBride [7].…”

“…Sections 3 and 4 are concerned with the Fourier series and difference quotient approaches to fractional calculus on A 2π and A 2π . By modifying the arguments presented in [3], we establish that both approaches lead to equivalent definitions of the α th fractional derivative of a 2π-periodic distribution f . Finally, in Section 5, we consider fractional diffusion equations involving distributional initial conditions.…”

On generalizations of the series of Taylor, Lagrange, Laurent and Teixeira

“…The ordinary concept of n-th derivative (primitive) can be extended from positive v=+n (negative v=-n) integral order to rational, real or complex order v, by generalizing any of the definitions in the classical theory of functions: (i) the limit of the Leibnitz-Newton incremental ratio is generalized to a limit of finite differences, which yields an algebraic definition of derivative of complex order (Grunwald 1867; Butzer & Westphal 1974); (ii) the classical integral along the real axis, when extended to fractional order (Liouville 1832;Riemann 1847;Weyl 1971), leads to a concept of integration with complex order (Erdelyi 1940;Kober 1940); (iii) the Cauchy loop-integral can be extended to complex exponent, leading to the appearance of branch-cut(s), and requiring a suitable choice of paths of integration (Letnikov 1868;Nekrassov 1888; Lavoie & Tremblay & Osier 1974;Nishimoto 1984;Campos 1984). The theorems of expansion in series of ascending powers associated with Taylor (1715) and Lagrange (1770), occupy a central position in the theory of functions, and the series of Laurent (1843) and Teixeira (1900), which also involve descending powers, are useful to classify singularities in the complex plane.…”