SOME FAMILIES OF INFINITE SERIES SUMMABLE BY MEANS OF FRACTIONAL CALCULUS

Abstract

In a Five-volume work published recently, K. Nishimoto [1] has presented a systematic account of the theory and applications of fractional calculus in a number of areas (such as ordinary and partial differential equations, special functions, and summation of series). In 2001, K. Nishimoto, D.-K. Chyan, S.-D. Lin and S.-T. Tu [11] derived the following interesting families of infinite series via fractional calculus, $$ \displaystyle\sum_{k=2}^{\infty} \frac{(-c)^k}{k(k-1)}\frac{(kz-c)}{(z-c)^{k-1}}=c^2 \biggr(\biggr|\displaystyle\frac{-c}{z-c}\biggr|<1\biggr).$$ The object of the present paper is to extend the above families of infinite series to more general closed form relations. Various numerical results are also provided.