Certain relationships among polygamma functions, Riemann zeta function and generalized zeta function

Abstract

Many useful and interesting properties, identities, and relations for the Riemann zeta function ζ (s) and the Hurwitz zeta function ζ (s, a) have been developed. Here, we aim at giving certain (presumably) new and (potentially) useful relationships among polygamma functions, Riemann zeta function, and generalized zeta function by modifying Chen's method. We also present a double inequality approximating ζ (2r + 1) by a more rapidly convergent series.

Inequalities for the inverses of the polygamma functions

“…Functions ψ ′ (x), ψ ′′ (x), ψ ′′′ (x), ... are called polygamma functions in the literature. Polygamma functions are also very important functions and appear in the evaluations of many series and integrals [2,9,12,13,16,18]. They are also related with many special functions such as the Riemann zeta function, Hurwitz zeta function, Clausen's function and generalized harmonic numbers.…”

Rapidly Converging Series forζ(2n+1)from Fourier Series

“…On the other hand, various rapidly converging series for (2 + 1) ( ∈ N) have been developed by many authors (see, e.g., [25,26]; see also [1,Chapter 4] and the references cited in the chapter). Very recently, Choi and Chen [27] gave a double inequality approximating (2 + 1) ( ∈ N) by a more rapidly convergent series. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for (2 + 1) ( ∈ N).…”