Atlas: Hadamard-type fractional calculus and generalized Stirling numbers by Anatoly A. Kilbas

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3rd International ISAAC Congress
August 20-25, 2001
Freie Universitaet Berlin
Berlin, Germany

Organizers
ISAAC Board

Hadamard-type fractional calculus and generalized Stirling numbers
by
Anatoly A. Kilbas
Belarusian State University, Minsk, Belarus

Hadamard-type fractional integration and differentiation operators J^\alpha_{0+, c}f and D^\alpha_{0+, c}f are defined for \alpha > 0, x > 0 and c in R=(-\infty, \infty) by

(J^\alpha_{0+, c}f)(x)= 1
\Gamma(\alpha)
ó
õ x

0
æ
è u
x
ö
ø c

æ
è log x
u
ö
ø \alpha-1

f(u) du
u

(1)
and

(D^\alpha_{0+, c}g)(x)=x^-c æ
è d
dx
ö
ø m

x^c(J^m-\alpha_{0+, c}g)(x), m=[\alpha]+1,
(2)
respectively. When c=0, these operators coincide with the Hadamard fractional integrals and derivatives of order \alpha > 0, see [1, Section 18.3].

We discuss a unified approach to the Hadamard-type fractional integrals (1) and derivatives (2) based on their representations in the forms

(J^\alpha_{0+, c}f)(x)= \infty
å
k=0
S_-c(\alpha, k)x^kf^(k)(x)

and

(D^\alpha_{0+, c}f)(x)= \infty
å
k=0
S_c(\alpha, k)x^kf^(k)(x),

where

S_c(\alpha, k) = 1
k!
k
å
j=0
k!
j!((k-j)!
(-1)^k-j(c+j)^\alpha.

When c=0, S(\alpha, k) \equiv S₀(\alpha, k) are known as Stirling numbers of fractional order \alpha > 0 [2].

References

1. Samko, S.G.; Kilbas, A.A. and Marichev, O.I. Fractional integrals and derivatives. Theory and applications. Gordon and Breach, Yverdon et alibi, 1993.

2. Butzer, P.L., Hauss, M. and Schmidt, M. Factorial functions and Stirling numbers of fractional order. Result. Math. 16 (1989), 16-48.

Date received: May 18, 2001