Abstract
Recently, Sun (Two q-analogues of Euler’s formula \(\zeta (2)=\pi ^2/6\). arXiv:1802.01473, 2018) obtained q-analogues of Euler’s formula for \(\zeta (2)\) and \(\zeta (4)\). 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 \(\zeta (6)=\pi ^6/945\). Our main results are stated in Theorems 2.1 and 2.2 below.
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1 Introduction
Recently, Sun [3] obtained a very nice q-analogue of Euler’s formula \(\zeta (2)=\pi ^2/6\).
Theorem 1.1
(Sun [3]) For a complex q with \(|q|<1\), we have:
Motivated by Theorem 1.1, the present author obtained the q-analogue of \(\zeta (4)=\pi ^4/90\) 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 q-analogues of Euler’s formula for \(\zeta (6)\) 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 \(\zeta (6)\). As we shall see shortly, the q-analogue formulation of \(\zeta (6)\) is more difficult as compared to \(\zeta (2)\) and \(\zeta (4)\) due to an extra term that shows up in the identity; however, in the limit as \(q\uparrow 1\) (where \(q\uparrow 1\) means q is approaching 1 from inside the unit disk), this term vanishes. We also state the q-analogue of \(\zeta (4)=\pi ^4/90\), 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 \(\zeta (6)=\pi ^6/945\) 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 \(\zeta (2k), k=4, 5,\ldots \) (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 \(q\uparrow 1\). The case \(k=3\) or the q-analogue of \(\zeta (6)\) 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).
2 Main Theorems
Theorem 2.1
For a complex q with \(|q|<1\), we have:
where \(P_2(x) = x^2+4x+1\). In other words, (2.1) gives a q-analogue of \(\zeta (4)=\pi ^4/90\).
Theorem 2.2
For a complex q with \(|q|<1\), we have:
where \(P_4(x) = x^4+236x^3+1446x^2+236x+1\) and \(\phi (q)=\displaystyle \prod \nolimits _{n=1}^\infty (1-q^n)\) is Euler’s function. In other words, (2.2) gives a q-analogue of \(\zeta (6)=\pi ^6/945\).
Remark 2.3
We note that \(\phi ^{12}(q)\) 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 \(\phi ^{12}(q)\) 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).
3 Some Useful Lemmas
Let \(q=e^{2\pi i\tau }\), \(\tau \in {\mathcal {H}}\) where \({\mathcal {H}}=\{\tau \in {\mathbb {C}} : \text{ Im }(\tau )>0\)}. Then, the Dedekind \(\eta \)-function defined by:
is a modular form of weight 1/2. Also, let us denote by \(\psi (q)\) the following sum:
where \(T_n=\dfrac{n(n+1)}{2}\) (for \(n=0,1,2,\ldots \)) are triangular numbers. Then, we have the following well-known result due to Gauss:
Lemma 3.1
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].
Theorem 3.2
Let \(\eta ^{12}(2\tau )=\sum \nolimits _{k=0}^\infty a(2k+1)q^{2k+1}\). Then, for a positive integer n, we have:
where
4 Proof of Theorem 2.2
Since \(\zeta (6)=\dfrac{\pi ^6}{945}\) has the following equivalent form:
it will be sufficient to get the q-analogue of (4.1). Now, from q-analogue of Euler’s Gamma function, we know that:
so that from (4.2), we have:
Next, we consider the following infinite series
where \(P_4(x)=x^4+236x^3+1446x^2+236x+1\).
By partial fractions, we have:
Lemma 4.1
With \(S_6(q)\) represented by (4.5), we have:
Proof
From (4.5), we have:
Also from (3.1), we have:
Thus, from above, we have:
where the last step follows from Theorem 3.2. This completes the proof of Theorem 2.2. \(\square \)
We also note that
where \(\lim _{q\uparrow 1}\;(1-q)^6\phi ^{12}(q)=0\) and \(q\uparrow 1\) indicates \(q\rightarrow 1\) from within the unit disk. Hence, combining Eqs. (4.1), (4.3), (4.7), and Lemma 4.1, Theorem 2.2 follows.
References
Goswami, A.: A \(q\)-analogue for Euler’s evaluations of the Riemann zeta function. Res. Number Theory 5(1), #Art. 3 (2019)
Ono, K., Robins, S., Wahl, P.T.: On the representation of integers as sums of triangular numbers. Aequationes Math. 50(1-2), 73–94 (1995)
Sun, Z.-W.: Two \(q\)-analogues of Euler’s formula \(\zeta (2)=\pi ^2/6\). arXiv:1802.01473 (2018)
Acknowledgements
Open access funding provided by Johannes Kepler University Linz. I am grateful to Prof. Krishnaswami Alladi for carrying out discussions pertaining to the function \(\phi (q)\) and for his encouragement. I also thank Prof. George Andrews for going through my proof and providing me a few useful references.
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In honour of Prof. George Andrews on his 80th birthday.
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Goswami, A. A \(\varvec{q}\)-Analogue for Euler’s \(\varvec{\zeta (6)=\pi ^6/945}\). Ann. Comb. 23, 801–806 (2019). https://doi.org/10.1007/s00026-019-00444-9
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DOI: https://doi.org/10.1007/s00026-019-00444-9