On certain two-parameter deformations of multiple zeta values

Abstract

In a previous paper, we showed that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. The integrals appearing in this formula can be considered to be one-parameter deformations of q-double zeta values and thus two-parameter deformations of double zeta values. In this paper, we introduce certain integrals, regarded as two-parameter deformations of multiple zeta values, and investigate their properties. In particular, we consider two-parameter generalizations of the harmonic and shuffle product formulas, which are fundamental relations for multiple zeta values.

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Acknowledgements

The author would like to express my sincere gratitude to Professor Yoshihiro Takeyama for valuable comments and fruitful discussions. He also would like to thank the anonymous referees for their careful reading of the manuscript and their constructive comments.

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Appendix A: An alternative proof of the main theorem of [8]

Appendix A: An alternative proof of the main theorem of [8]

In this appendix, we give an alternative proof of the following theorem, established in [8], by using Theorem 5.2. We note that the conditions of the theorem are slightly weakened, compared to those in [8].

Theorem A.1

We fix \(p,q \in {{\mathbb {C}}}\) such that \(0<|p|,|q|<1\) and \(t \in {{\mathbb {C}}}\) satisfying \(1<|t|< \mathrm{min}\{|p|^{-1}, |q|^{-1} \}\). When \(|pq|<|x_1x_2t|<1\) and \(\mathrm{max}\{|p|, |q|\}<|x_2t|<1\), we have the following formula:

$$\begin{aligned}&\psi _{0,0}(x_1;p,q)\psi _{0,0}(x_2;p,q)+\psi _{0,0}(x_2;p,q)\psi _{0,0}(x_1x_2;p,q)-\psi _{0,0}(x_1;p,q)\psi _{0,0}(x_1x_2;p,q) \nonumber \\&\quad =\frac{1}{2\pi i}\int _{{\mathbb {T}}}\psi _{0,0}(x_1x_2tz;p,q)(\psi _0(x_2tz;p)\psi _0(t^{-1}z^{-1};q)+\psi _0(x_2tz;q)\psi _0(t^{-1}z^{-1};p))\frac{dz}{z}\nonumber \\&\qquad +\psi _{1,1}(pqx_1;p,q)-\psi _{1,1}(pqx_2;p,q)+\psi _{1,1}(x_1x_2;p,q). \end{aligned}$$
(A.1)

Proof

As described in Sect. 5, we can derive (5.2) from Theorem 5.2 (a). Furthermore the identity (5.2) can be written as follows:

$$\begin{aligned}&\psi _{0,0}(qa_1b_1^{-1};p,q)\psi _{0,0}\left( qa_2b_2^{-1};p,q\right) \nonumber \\&\quad =-\frac{1}{2\pi i}\int _{{\mathbb {T}}}\psi _{0,0}\left( qa_1b_1^{-1}b_2z;p,q\right) \psi _0(a_2z;p)\psi _0(b_2z;q)\frac{dz}{z}\nonumber \\&\qquad -\frac{1}{2\pi i}\int _{{\mathbb {T}}}\psi _{0,0}(a_2b_1b_2^{-1}z;p,q) \psi _0(a_1z;p)\psi _0(b_1z;q)\frac{dz}{z}+\psi _{1,1}\left( pqa_1a_2^{-1}b_1^{-1}b_2;p,q\right) . \end{aligned}$$
(A.2)

The identity (A.2) holds for \(|p|<|a_i|<1 \ (i=1,2), \ |q|<|b_1| \leqq |b_2| <1\). In (A.2). we put \(a_1=x_1,\, a_2=x_1x_2,\, b_1=b_2=qt\). Then, when \(|p|<|x_1t|<1, \ |p|<|x_1x_2t|<1\), we have

$$\begin{aligned}&\psi _{0,0}(x_1;p,q)\psi _{0,0}(x_1x_2;p,q)\nonumber \\&\quad =\frac{1}{2\pi i}\int _{{\mathbb {T}}}\psi _{0,0}(qx_1tz;p,q)\psi _0(x_1x_2tz;p)\psi _0(t^{-1}z^{-1};q)\frac{dz}{z} \nonumber \\&\qquad +\frac{1}{2\pi i}\int _{{\mathbb {T}}}\psi _{0,0}(x_1x_2tz;p,q)\psi _0(x_1tz;p)\psi _0(t^{-1}z^{-1};q)\frac{dz}{z}+\psi _{1,1}(pqx_2). \end{aligned}$$
(A.3)

Similarly, by (5.4) obtained from Theorem 5.2 (b), we have

$$\begin{aligned}&\psi _{0,0}(x_1;p,q)\psi _{0,0}(x_2;p,q) \nonumber \\&\quad =\frac{1}{2\pi i}\int _{{\mathbb {T}}} \psi _{0,0}(x_1x_2tz;p,q)\psi _0(x_2tz;p)\psi _0(t^{-1}z^{-1};q)\frac{dz}{z} \nonumber \\&\qquad +\frac{1}{2\pi i}\int _{{\mathbb {T}}}\psi _{0,0}(x_1x_2tz;p,q)\psi _0(x_1tz;p) \psi _0(t^{-1}z^{-1};q)\frac{dz}{z}+\psi _{1,1}(x_1x_2;p,q). \end{aligned}$$
(A.4)

The identity (A.4) is true for \(|p|<|x_i t|<1 \ (i=1,2), \ |p|<|x_1x_2t^2|<1\). Furthermore we set \(a_1=pt, \, a_2=x_1x_2t, \, b_1=x_2t, \, b_2=qt\) in Theorem 5.2 (b), expand both sides into Laurent series with respect to \(\beta _1-1,\, \beta _2-1,\, \alpha _1-1\) and \(\alpha _2-1\) repeatedly and compare the constant terms. Then, when \(|p|<|x_1x_2 t|<1, \ |p|<|x_1x_2t^2|<1, \ |q|<|x_2t|<1\), we obtain

$$\begin{aligned} \psi _{0,0}(x_2;p,q)\psi _{0,0}(x_1x_2;p,q) =&\frac{1}{2\pi i}\int _{{\mathbb {T}}}\psi _{0,0}(qx_1tz;p,q) \psi _{0}(x_1x_2tz;p)\psi _0(t^{-1}z;q)\frac{dz}{z}\nonumber \\&+\frac{1}{2\pi i}\int _{{\mathbb {T}}}\psi _{0,0}(x_1x_2tz;p.q) \psi _0(x_2tz;q)\psi _0(t^{-1}z^{-1};p)\frac{dz}{z}\nonumber \\&+\psi _{1,1}(pqx_1;p,q). \end{aligned}$$
(A.5)

By (A.3), (A.4), (A.5), we find that the identity (A.1) holds under the following conditions:

$$\begin{aligned} |p|<|x_it|<1 \ \ (i=1,2), \ |p|<|x_1x_2t|<|t|^{-1}, \ |q|<|x_2t|<1. \end{aligned}$$
(A.6)

We now regard both sides of (A.1) as functions of \(x_1x_2\) and \(x_2\). Then, by analytic continuation, it follows that the identity (A.1) holds for \(|pq|<|x_1x_2t|<1, \ \mathrm{max}\{|p|,|q|\}<|x_2t|<1\). Thus we finish the proof of the theorem. \(\square \)

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Kato, M. On certain two-parameter deformations of multiple zeta values. Res. number theory 6, 30 (2020). https://doi.org/10.1007/s40993-020-00205-1

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Keywords

  • Multiple zeta values
  • q-Analogues of multiple zeta values
  • Elliptic gamma function
  • Harmonic and Shuffle products

Mathematics Subject Classification

  • 33E30