Some Families of Infinite Series Summable by Means of Fractional Calculus

Certain Classes of Infinite Series Deducible From Mellin-Barnes Type of Contour Integrals

“…Certain interesting single (or double) infinite series associated with hypergeometric functions (1.4) have recently been expressed in terms of Psi (or Digamma) function ψ(z) in (1.1), for example, see Nishimoto and Srivastava [8], Srivastava and Nishimoto [13], Saxena [10], Chen and Srivastava [5] and Srivastava and Choi [15], and so on. In this connection, with a view to unifying and extending those earlier results, we first establish two relations which some double infinite series involving hypergeometric functions are expressed in a single infinite series involving ψ(z).…”

“…The summation formula (1.2) and its obvious special cases were revived, in recent years, as illustrations emphasizing the usefulness of fractional calculus in evaluating infinite sums. For a detailed historical account of (1.2), and of its various consequences and generalizations have been presented by Nishimoto and Srivastava [8]. A systematic account of certain family of infinite series which can be expressed in terms of Digamma functions together with their relevant unification and generalization has been given by Srivastava [14], Al-Saqabi et al [1] and Aular de Duran et al [2].…”

“…From the aforementioned work of Nishimoto and Srivastava [8], we choose to recall here two interesting consequences of the summation formula (1.2), which are contained in Theorem 1 below.…”

Some Infinite Series and Functional Relations That Arose in the Context of Fractional Calculus

“…For a reasonably detailed historical account of the summation formula (1.2), and also of its numerous consequences and generalizations, one may refer to the work on the subject by Nishimoto and Srivastava [8], who also provided a number of relevant earlier references on summation of infinite series by means of fractional calculus. Many further developments on this subject are reported (among others) by Srivastava [14], Al-Saqabi et al [1], Aular de Durán et al [2], and Wu et al [18].…”

“…From the aforementioned work of Nishimoto and Srivastava [8], we choose to recall here two interesting consequences of the summation formula (1.2), which are contained in Theorem 1 (Nishimoto and Srivastava [8, p. 104] …”