Fractional Calculus and the Sums of Certain Families of Infinite Series

Although the concept of fractional calculus can be traced back to l′Hôpital′s naive (three centuries old) curiosity about the meaning of Leibniz′s nth derivative notation dny/dxn when n = 12, and many monumental works on the subject of fractional calculus have since appeared (and are still appearing) in the mathematical literature, the five-volume work published recently by K. Nishimoto ["Fractional Calculus," Vols. I-IV, 1984, 1987, 1989, 1991; "An Essence of Nishimoto′s Fractional Calculus (Calculus in the 21st Century): Integrations and Differentiations of Arbitrary Order," 1991] contains an interesting account of the theory and applications of fractional calculus in a number of areas of mathematical analysis (such as ordinary and partial differential equations, summation of series, et cetera). The main object of the present paper is to examine rather systematically some of the most recent contributions by Nishimoto et al. (cf., e.g., [J. College Engrg. Nihon Univ. Ser. B 32 (1991), 7-13, 15-21, 23-30; J. Fractional Calculus1 (1992), 7-16]; see also ["Fractional Calculus," Vol. IV, 1991; J. College Engrg. Nihon Univ. Ser. B33 (1992), 1-8]), L. Galué et al. [J. Fractional Calculus 1 (1992), 17-21], and others on the use of fractional calculus in finding the sums of numerous interesting families of infinite series. Various other classes of infinite sums found in the literature without using fractional calculus, and their hitherto unnoticed connections with certain known results, are also considered.