Harmonic-number summation identities, symmetric functions, and multiple zeta values

Abstract

We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently conjectured by J. Choi, and give several more families of identities of a similar nature.

A note on harmonic number identities, Stirling series and multiple zeta values

“…Before we turn to a proof of the main theorems we collect in this section several results for the truncated zeta values ζ n ({1} k ) and ζ n ({1} k ). Following Hoffman [15], we note two different explicit expressions for the polynomials P k and Q k due to MacDonald [23]. Lemma 1.…”

“…First, we discuss expressions for ζ n ({1} k ). These expressions are, for example in terms of the Stirling numbers of the first kind, well-known (see for example Adamchik [1], Prodinger [21], Hoffman [15], Hoffman et al [17]), but perhaps not all parts of it and not in this notation. Lemma 2 (Truncated multiple zeta values ζ n ({1} k )).…”

“…. , a q ), for some q ∈ N. Hence, an evaluation of the series (15) and thus of S can be reduced to evaluations of series of the form ∞ n=1 ζ n−1 (a 1 , . .…”

On harmonic numbers and nonlinear Euler sums

“…where h m = h m (x 1 , x 2 , · · · , x n ) and e m = e m (x 1 , x 2 , · · · , x n ) are respectively the complete and elementary symmetric functions of degree m, and p m = x m 1 + x m 2 + · · · + x m n is the mth power sum (for details introductions, see [28], the notation follows [31]).…”

“…(Theorem 2.7 can also be found in Entry 12(a) of Chapter 14 in Ramanujan's second notebook [5]). Hence, the elementary symmetric function e k and the complete symmetric function h k can be expressed in terms of the power sums, i.e., (see [28]) e k = P k (p 1 , p 2 , · · · , p k ) and h k = Q (p 1 , p 2 , · · · , p k ) .…”