On Mordell–Tornheim double Eisenstein series

Abstract

Inspired by the Mordell–Tornheim double zeta function, in this paper, we introduce the Eisenstein series of Mordell–Tornheim type. Based on the theory of Cohen series and partial fraction decomposition, we explicitly compute these series. We end by discussing several identities about products of Eisenstein series and their Fourier expansions.

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Acknowledgements

The authors would like to thank F. Brunault and P. Charollois for their constructive suggestions on writing. Thanks also go to N. Diamantis for his helpful comments that greatly improved the manuscript.

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Correspondence to Hao Zhang.

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Appendix A: List of \(G(\tau ;k_1,k_2,k_3)\) with weight lower than 12

Appendix A: List of \(G(\tau ;k_1,k_2,k_3)\) with weight lower than 12

Mordell–Tornheim double Eisenstein series \(G(\tau ;k_1,k_2,k_3)\) with \(k=k_1+k_2+k_3\)

\(k=6\)
\(G(\tau ;1,2,3)=-\frac{\pi ^6}{945}E_6(\tau )\)
\(G(\tau ;2,2,2)=\frac{2\pi ^6}{945}E_6(\tau )\)
\(k=8\)
\(G(\tau ;1,2,5)=-\frac{\pi ^8}{14175}E_8(\tau )\)
\(G(\tau ;1,3,4)=-\frac{2\pi ^8}{14175}E_8(\tau )\)
\(G(\tau ;2,2,4)=\frac{2\pi ^8}{14175}E_8(\tau )\)
\(G(\tau ;2,3,3)=0\)
\(k=10\)
\(G(\tau ;1,2,7)=-\frac{\pi ^{10}}{155925}E_{10}(\tau )\)
\(G(\tau ;1,3,6)=-\frac{\pi ^{10}}{66825}E_{10}(\tau )\)
\(G(\tau ;1,4,5)=-\frac{\pi ^{10}}{93555}E_{10}(\tau )\)
\(G(\tau ;2,2,6)=\frac{2\pi ^{10}}{155925}E_{10}(\tau )\)
\(G(\tau ;2,3,5)=\frac{\pi ^{10}}{467775}E_{10}(\tau )\)
\(G(\tau ;2,4,4)=\frac{4\pi ^{10}}{467775}E_{10}(\tau )\)
\(G(\tau ;3,3,4)=-\frac{2\pi ^{10}}{467775}E_{10}(\tau )\)
\(k=12\)
\(G(\tau ;1,2,9)=-\frac{404\pi ^{12}}{638512875}\left( E_{12}(\tau )-\frac{51891840}{69791}\Delta (\tau )\right) \)
\(G(\tau ;1,3,8)=-\frac{326\pi ^{12}}{212837625}\left( E_{12}(\tau )+\frac{34594560}{112633}\Delta (\tau )\right) \)
\(G(\tau ;1,4,7)=-\frac{643\pi ^{12}}{638512875}\left( E_{12}(\tau )+\frac{1089728640}{444313}\Delta (\tau )\right) \)
\(G(\tau ;1,5,6)=-\frac{739\pi ^{12}}{638512875}\left( E_{12}(\tau )-\frac{1089728640}{510649}\Delta (\tau )\right) \)
\(G(\tau ;2,2,8)=\frac{808\pi ^{12}}{638512875}\left( E_{12}(\tau )-\frac{51891840}{69791}\Delta (\tau )\right) \)
\(G(\tau ;2,3,7)=\frac{34\pi ^{12}}{127702575}\left( E_{12}(\tau )+\frac{62270208}{11747}\Delta (\tau )\right) \)
\(G(\tau ;2,4,6)=\frac{43\pi ^{12}}{58046625}\left( E_{12}(\tau )+\frac{42456960}{29713}\Delta (\tau )\right) \)
\(G(\tau ;2,5,5)=\frac{38\pi ^{12}}{91216125}\left( E_{12}(\tau )-\frac{111196800}{13129}\Delta (\tau )\right) \)
\(G(\tau ;3,3,6)=-\frac{68\pi ^{12}}{127702575}\left( E_{12}(\tau )+\frac{62270208}{11747}\Delta (\tau )\right) \)
\(G(\tau ;3,4,5)=-\frac{19\pi ^{12}}{91216125}\left( E_{12}(\tau )-\frac{111196800}{13129}\Delta (\tau )\right) \)
\(G(\tau ;4,4,4)=\frac{38\pi ^{12}}{91216125}\left( E_{12}(\tau )-\frac{111196800}{13129}\Delta (\tau )\right) \)

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Wang, W., Zhang, H. On Mordell–Tornheim double Eisenstein series. Res. number theory 6, 33 (2020). https://doi.org/10.1007/s40993-020-00208-y

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