CoInvestigator(Kenkyūbuntansha) 
SAITOH Hitoshi Gunma Nat. College Tech., Prof., 教授 (10042607)
OWA Shigeyoshi Fukuoka Univ., Fac. Tech. and Sci., Prof., 理工学部, 教授 (50088506)
FUKUSHIMA Yukio Fukuoka Univ., Fac. Sci., Assoc. Prof., 理学部, 助教授 (40099007)
YOSHIDA Mamoru Fukuoka Univ., Fac. Sci., Prof., 理学部, 教授 (60078607)

Budget Amount *help 
¥2,300,000 (Direct Cost : ¥2,300,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1997 : ¥900,000 (Direct Cost : ¥900,000)

Research Abstract 
In this project, placing Fractional Calculus and its Applications to the center of the investigation, we studied jointly uniting individual storages of their research efforts. 1) Fractional Calculus : Fractional calculus operators involving Appell's function FィイD23ィエD2 as kernels were defined and analyzed, which are generalizations of that involving Gauss's hypergeometric function proposed by the head investigator in 1978. Further, we studied multidimensionability of fractional calculus operators. Two kinds of multidimensional extensions of fractional calculus operators of RiemannLiouville and Weyl were already proposed and discussed by us. Here we defined and investigated what involves Gauss's function. 2) Htransform and its application : As an extension of integral transforms involving various special functions) the transform with Hfunction as an integral kernel, Htransform, was studied. In the theory on the space LィイD2υ,2ィエD2, mapping properties, boundedness, injection and repre
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sentation property of the Htransform were first investigated, and, then, that the properties are extendable on the space LィイD2υ,γィエD2 for any ィイD2γィエD2 is shown. At the moment, precise properties of the Hfunction should be examined and used. The invertibility of the Htransform on LィイD2υ,γィエD2 was also studied. Such general results on the Htransform may produce mapping theorems of integral transforms involving various concrete special functions. Here, we investigated the Meijer transform, LommelMaitland transform, HardyTitchmarsh transform, and we obtained distinct results severally. 3) Fractional Integral and Differential Equations : Relating to fractional calculus, we studied the AbelVolterra type integral equations and investigated asymptotics near zero of the solutions. 4) Application of Fractional Calculus to Theory of Geometric Functions: Distortion theorems, starlikeness, convexity of analytic functions by using the fractional integral operator involving Gfunction as a kernel defined by V.S. Kiryakova are studied. The results deduce theorems for the Hohlov operator, RiemannLiouville fractional calculus operator, and fractional calculus operators involving the Gauss function and the Appell function FィイD23ィエD2 due to the present head investigator. For such investigations, the grantinaid was effectively used mainly for the trip expensises of the discussions of members of the project and of the presentations of them at meetings in foreign countries. Less
