We define and study the interpolated finite multiple harmonic q-series. A generating function of the sums of the interpolated finite multiple harmonic qseries with fixed weight, depth and i-height is computed. Some Ohno-Zagier type relation with corollaries and some evaluation formulas of the interpolated finite multiple harmonic q-series at roots of unity are given.respectively. Here for any m ∈ N, [m] is the q-integer [m] = 1−q m 1−q . It was proved in [2, Theorems 1.1, 1.2] that the values z n (k; ζ n ) and z ⋆ n (k; ζ n ) closely relate to the finite and the symmetric multiple zeta (star) values, where ζ n is a fixed primitive n-th root of unity. Motivated by these observations, the authors of [2] gave a re-interpretation of *

In the space of bounded real-valued functions on the interval (0, 1), we study the convergent sequences of q-analogues of multiple zeta values which do not converge to 0. And we obtain the derived sets of the set of some qanalogue of multiple zeta values.This q-analogue was first studied by Bradley [2] and independently by Zhao [4]. Here we introduce another q-analogue of multiple zeta values. Let r ∈ N, then we.( 1.2) Different from multiple zeta values, the multiple q-zeta values have a parameter q. Hence we work in the function space B(0, 1), which is the set of bounded realvalued functions on the open interval (0, 1). Since the multiple q-zeta values we consider here belong to B(0, 1) (see Remark 2.4), we just study the following two subspaces of B(0, 1):We define an order of B(0, 1) as follows. Let f, g ∈ B(0, 1). The function f is smaller than g, if f (q) < g(q) for any q ∈ (0, 1). We denote this by f < g. Then we can find the maximum element of QZ. Theorem 1.1. For any admissible multi-index k, we have ζ[k] ζ[2]. In other words, ζ[2] is the maximum element of QZ. While for the subspace QZZ, we only obtain an upper bound. Theorem 1.2. For any admissible multi-index k and any r ∈ N, we have ζ[k, r) < ζ[2]. In other words, ζ[2] is an upper bound of QZZ.

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In this paper, we compute the iterated derived sets of the set of multiple t-values under the usual topology of R. Our results imply that the set of multiple t-values, ordered by ≥, is a well-ordered set. We determine its type of order, which is ω 2 , where ω is the smallest infinite ordinal. There exists a unique bijection from the set of multiple t-values to N 2 , which reverses the orders. We provide some description of this bijection.

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