VKiryakova_fcaa113

306 V. KiryakovaIn our subsequent works [9], [10], [15], etc, we have interpreted thePoisson- and Sonine-type transformations introduced by Dimovski, as operatorsof the generalized fractional calculus, giving alternative representationsfor them by means of single integrals involving special functions askernels.3. Notions of generalized fractional calculus (GFC),related generalized hypergeometric functionsHere we give only a brief account on some special functions used in thissurvey and on the operators of Generalized Fractional Calculus (GFC), introducedand studied by Kiryakova [15] and in a series of papers like [9],[10],[16]. Short survey on these operators, their origin and operational propertiescan be seen in the recent survey Kiryakova [18], that is also available onlineat http://www.math.bas.bg/˜fcaa/volume11/fcaa112/Kiryakova − fcaa112.pdfand http://www.diogenes.bg/fcaa/volume11/fcaa112/Kiryakova − fcaa112.pdf.Definition 6. Let m ≥ 1 be integer, β > 0, γ 1 , . . . , γ m and δ 1 ≥0, . . . , δ m ≥ 0 be real parameters. By a generalized (multiple, m-tuple)operator of integration of fractional multi-order δ = (δ 1 , . . . , δ m ) we meanan integral operator of the form∫ 1[I (γ k),(δ k )β,mf(x) = G m,0m,m σ∣ (γ k + δ k ) m ]10(γ k ) m f(xσ 1 β ) dσ. (21)1Also, each operator of the formRf(x) = x βδ 0I (γ k),(δ k )β,mf(x) with arbitrary δ 0 ≥ 0,is said to be a generalized (m-tuple) operator of fractional integration ofRiemann-Liouville type, or briefly: a generalized (R.-L.) fractional integral.Operator (21) is a typical representative of the so-called “generalizedoperators of fractional integration” having the general form (introduced byKalla, 1970-1979)If(x) =∫ 10Φ(σ) σ γ f(xσ)dσ,with a suitable special function Φ(σ) as a kernel-function. In our case, thekernel-function G m,0m,m is a specific case of the Meijer’s G-function, such toallow a general theory of GFC combined with useful applications.Definition 7. (see [12], Vol.1; [20]; [15], Appendix) By a Meijer’sG-function, we mean the generalized hypergeometric function defined bymeans of the contour integral in the complex plane