A q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{q}$$\end{document}-Analogue for Euler’s ζ(6)=π6/945\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\zeta (6)=\pi ^6/945}$$\end{document}

Recently, Sun (Two q-analogues of Euler’s formula ζ(2)=π2/6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta (2)=\pi ^2/6$$\end{document}. arXiv:1802.01473, 2018) obtained q-analogues of Euler’s formula for ζ(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta (2)$$\end{document} and ζ(4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta (4)$$\end{document}. Sun’s formulas were based on identities satisfied by triangular numbers and properties of Euler’s q-Gamma function. In this paper, we obtain a q-analogue of ζ(6)=π6/945\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta (6)=\pi ^6/945$$\end{document}. Our main results are stated in Theorems 2.1 and 2.2 below.


Introduction
Recently, Sun [3] obtained a very nice q-analogue of Euler's formula ζ(2) = π 2 /6. Theorem 1.1. (Sun [3]) For a complex q with |q| < 1, we have: (1.1) Motivated by Theorem 1.1, the present author obtained the q-analogue of ζ(4) = π 4 /90 and noted that it was simultaneously and independently obtained by Sun in his subsequent revised paper. Theorem 1.2. (Sun [3]) For a complex q with |q| < 1, we have: Furthermore, Sun commented that one does not know how to find qanalogues of Euler's formula for ζ(6) and beyond, similar to Theorems 1.1 and 1.2. This further motivated the author to consider the problem, and indeed, we obtained the q-analogue of ζ(6). As we shall see shortly, the q-analogue formulation of ζ(6) is more difficult as compared to ζ (2) and ζ(4) due to an extra term that shows up in the identity; however, in the limit as q ↑ 1 (where q ↑ 1 means q is approaching 1 from inside the unit disk), this term vanishes. We also state the q-analogue of ζ(4) = π 4 /90, since we found it independently of Sun's result; however, we skip the proof of this, since it essentially uses the same idea as Sun.
We emphasize here that the q-analogue of ζ(6) = π 6 /945 is the first non-trivial case where we notice the occurrence of an interesting extra term which essentially is the twelfth power of a well-known function of Euler (see Theorem 2.2). After obtaining this result, we obtained q-analogues of Euler's general formula for ζ(2k), k = 4, 5, . . . (see [1]). Each of these q-analogues has an extra term that arises from the general theory of modular forms all of which approach zero in the limit q ↑ 1. The case k = 3 or the q-analogue of ζ(6) is special, since the extra term that we obtain in this case has a beautiful product representation, and has connections to well-known identities of Euler (see below).

Theorem 2.2.
For a complex q with |q| < 1, we have: Euler's function. In other words, (2.2) gives a q-analogue of ζ(6) = π 6 /945. Remark 2.3. We note that φ 12 (q) has a beautiful product representation and is uniquely determined by: In the general q-analogue formulation (see [1]), we do not have very elegant representations of these functions, although we obtain expressions for them similar to (2.3).

Remark 2.4.
Since the coefficients in the q-series expansion of φ 12 (q) are related to the pentagonal numbers by Euler's pentagonal number theorem, and the coefficients of the product in the right-hand side of (2.2) are related to the triangular numbers, it will be worthwhile to understand the relationships of these coefficients via identity (2.2).

Some Useful Lemmas
Let q = e 2πiτ , τ ∈ H where H = {τ ∈ C : Im(τ ) > 0}. Then, the Dedekind η-function defined by: is a modular form of weight 1/2. Also, let us denote by ψ(q) the following sum: where T n = n(n + 1) 2 (for n = 0, 1, 2, . . .) are triangular numbers. Then, we have the following well-known result due to Gauss: Thus, we have from Lemma 3.1 that: where t 12 (n) is the number of ways of representing a positive integer n as a sum of 12 triangular numbers. Next, we have the following well-known result of Ono, Robins and Wahl [2].
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