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.

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 $k=k_1+k_2+k_3$

$k=6$

$G(\tau ;1,2,3)=-\frac{\pi ^6}{945}E_6(\tau )$

$G(\tau ;2,2,2)=\frac{2\pi ^6}{945}E_6(\tau )$

$k=8$

$G(\tau ;1,2,5)=-\frac{\pi ^8}{14175}E_8(\tau )$

$G(\tau ;1,3,4)=-\frac{2\pi ^8}{14175}E_8(\tau )$

$G(\tau ;2,2,4)=\frac{2\pi ^8}{14175}E_8(\tau )$

$G(\tau ;2,3,3)=0$

$k=10$

$G(\tau ;1,2,7)=-\frac{\pi ^{10}}{155925}E_{10}(\tau )$

$G(\tau ;1,3,6)=-\frac{\pi ^{10}}{66825}E_{10}(\tau )$

$G(\tau ;1,4,5)=-\frac{\pi ^{10}}{93555}E_{10}(\tau )$

$G(\tau ;2,2,6)=\frac{2\pi ^{10}}{155925}E_{10}(\tau )$

$G(\tau ;2,3,5)=\frac{\pi ^{10}}{467775}E_{10}(\tau )$

$G(\tau ;2,4,4)=\frac{4\pi ^{10}}{467775}E_{10}(\tau )$

$G(\tau ;3,3,4)=-\frac{2\pi ^{10}}{467775}E_{10}(\tau )$

$k=12$

$G(\tau ;1,2,9)=-\frac{404\pi ^{12}}{638512875}\left( E_{12}(\tau )-\frac{51891840}{69791}\Delta (\tau )\right)$

$G(\tau ;1,3,8)=-\frac{326\pi ^{12}}{212837625}\left( E_{12}(\tau )+\frac{34594560}{112633}\Delta (\tau )\right)$

$G(\tau ;1,4,7)=-\frac{643\pi ^{12}}{638512875}\left( E_{12}(\tau )+\frac{1089728640}{444313}\Delta (\tau )\right)$

$G(\tau ;1,5,6)=-\frac{739\pi ^{12}}{638512875}\left( E_{12}(\tau )-\frac{1089728640}{510649}\Delta (\tau )\right)$

$G(\tau ;2,2,8)=\frac{808\pi ^{12}}{638512875}\left( E_{12}(\tau )-\frac{51891840}{69791}\Delta (\tau )\right)$

$G(\tau ;2,3,7)=\frac{34\pi ^{12}}{127702575}\left( E_{12}(\tau )+\frac{62270208}{11747}\Delta (\tau )\right)$

$G(\tau ;2,4,6)=\frac{43\pi ^{12}}{58046625}\left( E_{12}(\tau )+\frac{42456960}{29713}\Delta (\tau )\right)$

$G(\tau ;2,5,5)=\frac{38\pi ^{12}}{91216125}\left( E_{12}(\tau )-\frac{111196800}{13129}\Delta (\tau )\right)$

$G(\tau ;3,3,6)=-\frac{68\pi ^{12}}{127702575}\left( E_{12}(\tau )+\frac{62270208}{11747}\Delta (\tau )\right)$

$G(\tau ;3,4,5)=-\frac{19\pi ^{12}}{91216125}\left( E_{12}(\tau )-\frac{111196800}{13129}\Delta (\tau )\right)$

$G(\tau ;4,4,4)=\frac{38\pi ^{12}}{91216125}\left( E_{12}(\tau )-\frac{111196800}{13129}\Delta (\tau )\right)$