157

数理解析研究所講究録
第2013巻 2016年 157-169

TRANSFORMATION FORMULAE AND ASYMPTOTIC EXPANSIONS FOR
DOUBLE HOLOMORPHIC EISENSTEIN SERIES OF TWO COMPLEXVARIABLES (SUMMARIZED VERSION)

MASANORI KATSURADA, DEPT. MATH., FAC. ECON., KEIO UNIV.

(慶磨義塾大学経済学部桂田昌紀)
AND

TAKUMI NODA, DEPT. MATH., FAC. ENG., NIHON UNIV

(日本大学工学部野田工)

ABSTRACT. This is a summarized version of the forthcoming paper [10].

The main object of study in [10] is the double holomorphic Eisenstein series \overline{$\zeta$_{\mathrm{Z}^{2} (s;z) definedby (1.10) with (1.9) below, having two complex variables s=(s_{1},s_{2}) and two parameters z=(z_{1},z_{2}) satisfying either z\in(\mathfrak{H}^{+})^{2} or z\in(\mathfrak{H}^{-})^{2} for which its transformation properties andasymptotic aspects are studied when the distance |Z2-Z1| becomes small and large under certain
natural settings on the movement of z\in(\mathfrak{H}^{\pm})^{2} Let$\epsilon$\underline{(w} ) be the signature defined by (1.2). We,,.

establish in [10] complete asymptotic expansions of $\zeta$_{\mathbb{Z}^{2} (s;z) when z moves within either thepoly‐sector (\mathfrak{H}^{+})^{2} or (\mathfrak{H}^{-})^{2} so as that the pivotal parameter $\eta$ defined by (1.7) with (1.2) and(1.5), tends to 0 through |\arg $\eta$|< $\pi$/2 in the ascending order of $\eta$ (Theorem 1); this further,,

counterpart expansions exist for $\zeta$_{\mathrm{Z}^{2} (s;z) in the descending order of $\eta$ as$\eta$\rightarrow\infty through |\arg $\eta$|< $\pi$/2 (Theorem 2). Our second main formula in Theorem 2 naturally

leads us to show that

reduces to various expressions ofs\in \mathbb{Z}^{2} (Corollaries 2. 1-2.14 ).

\overline{$\zeta$_{\mathb {Z}^{2} (s;z) at s\in \mathbb{Z}^{2}(j=1,2),are\overline{$\zeta$_{\mathrm{Z}^{2} (s;z)(in finitely closed forms) at any integer lattice points
A major portion of these results reveals that specific values ofclosely linked to WeierstraB elliptic functions $\beta$(w|2 $\pi$(1, $\epsilon$(z_{j})z_{j}))the classical Eisenstein senes \mathscr{S}_{r}(q_{j}) in (5.9), reformulated by Ramanujan [18], as

well as the Jordan‐Kronecker type functions $\eta$_{\mathrm{J}_{1} (w;q_{j}) and ($\rho$_{2}(w;q_{j}) in (5.23), each defined withthe distinct bases q_{j}=e( $\epsilon$(z_{j})z_{J})(j=1,2) the latter two of which were extensively utilized by,

Ramanujan in the course of developing his theory of Eisenstein series, elliptic functions and theta
functions (cf. [1][2][21]). As for the methods used in [10], crucial rôles are played by a class of
Mellin‐Uarnes type integrals, manipulated with several properties of confluent hypergeometricfunctions.

1. INTRODUCTION

Throughout the present article, s and s=(s_{1},s_{2}) are complex variables with s= $\sigma$+it ands_{j}=$\sigma$_{j}+it_{j}(j=1,2) z and z=(Z1,Z2) complex parameters with z=x+iy and z_{j}=x_{\dot{j}}+iy_{j}(j=1,2) and \mathb {C}^{\times} denotes the universal covering of \mathbb{C}^{\times}=\mathbb{C}\backslash \{0\} ; note that \arg w\in(-\infty, +\infty)
is uniquely determined for any w\in\overline{\mathb {C}^{\times} The principal leaves of the upper and lower half‐planes,,.

are denoted

respectively by\mathfrak{H}^{+}=\{z\in\overline{\mathbb{C}^{\times}}|0<\arg z< $\pi$\}and\mathfrak{H}^{-}=\{z\in\overline{\mathbb{C}^{\times}}|- $\pi$<\arg z<0\}.

We suppose throughout the article that z=(z_{1},z_{2}) belongs to either(1.1)(\mathfrak{H}^{+})^{2} or (\mathfrak{H}^{-})^{2} and that,{\rm Im} z1\neq{\rm Im} z2.

It is the principal aim of the paper [10] (announced partially in [8][17]) to study transfor‐
mation properties and asymptotic aspects of the double holomorphic Eisenstein series, de‐s=(s_{1},s_{2})\in \mathbb{C}^{2} and two parameters
z=(z_{1},z_{2})\in(\mathfrak{H}^{\pm})^{2} ; this further leads us to establish its complete asymptotic expansions both

fined by (1.10) with (1.9) below, having two variables

20|0 Mathematics Subject Classification. Pnmary 11\mathrm{M}36 ; Secondary 11\mathrm{E}45, 11\mathrm{M}06.
Key words andphrases. Eisenstein series, Mellin‐Bames integral, asymptotic expansion.
Research supported in part by Grant‐in‐Aid for Scientific Research (No. 26400021) from JSPS.

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KATSURADA AND NODA

in the ascending (Theorem 1) and descending (Theorem 2) orders when z moves within thepoly‐sectors (\mathfrak{H}^{\pm})^{2} so as that the distance |z_{2}-z_{1}| becomes small and large respectively. Our\mathrm{m}\mathrm{a}\cdot \mathrm{n} formulae in Theorem 2 naturally reduce to various expressions (in finitely closed forms)\overline{$\zeta$_{\mathb {Z}^{2} (s;z)

for evaluating specific values ofat any integer lattice pints s\in \mathbb{Z}^{2} (Corollanes 2.1−
2.14), as well as its certain central values (Corollary 2.15).
We prepare here several notations necessary for describing our results. The symbol $\epsilon$(w) isdefined for any w\in\overline{\mathb {C}^{\times} (except when \arg w=0 ) by$\epsilon$(w)=\mathrm{s}\mathrm{g}\mathrm{n}(\arg w)=\{(1.2)

ifargw >0,
ifargw <0,+1-1

and the (vector‐like) notations

andZ12=z2-Z1
Z21=z1-z2
(1.3)
their
to
be
are
restricted as
frequently used;arguments
and(1.4)
0<|\arg_{Z12}|< $\pi$
0<|\arg z_{21}|< $\pi$,where the first equalities come from (1.1). It is readily seen under (1.4) thatare

Z21=e^{- $\epsilon$(z) $\pi$ i}z_{12}12

(1.5)

and

Z12=e^{- $\epsilon$(z) $\pi$ i}z_{21}21,

and also that$\epsilon$(z_{21})=- $\epsilon$(z_{12})

(1.6)

.

We next introduce a new parameter $\eta$ defined by$\eta$=\displaystyle \frac{1}{2}e^{- $\epsilon$(z_{12}) $\pi$ i/2_{Z12} =\frac{1}{2}e^{- $\epsilon$(z_{21}) $\pi$ i/2_{\mathrm{Z}21} ,

(1.7)
or

equivalently byz_{12}=2e^{ $\epsilon$(Z12) $\pi$ i/2} $\eta$andZ21=2e^{ $\epsilon$(z_{21}) $\pi$ i/2} $\eta$,

which plays pivotal rôles in describing our results; its argument satisfies under (1.4) that|\displaystyle \arg $\eta$|=|\arg z12-\frac{1}{2} $\epsilon$(z12) $\pi$|=|\arg z21-\frac{1}{2} $\epsilon$(z21) $\pi$|<\frac{ $\pi$}{2}.

(1.8)

We remark that the introduction of $\eta$ above is made with the intention to the forthcoming

study [11] (partially announced in [9]), where the non‐holomorphic case z=(z,\overline{z})\in \mathfrak{H}^{+x}\mathfrak{H}^{-}or z=(\overline{z},z)\in \mathfrak{H}^{-}\times \mathfrak{H}^{+} is to be treated, where Z21=z-\overline{z}=2e^{ $\pi$ i/2}y or Z21=\overline{z}-z=2e^{- $\pi$ i/2}yholds; this in comparison with (1.7) suggests that $\eta$ is a complexification of the real parametery.

Throughout the following, the notation \{s\rangle=s_{1}+s_{2} for any s=(s_{1},s_{2})\in \mathbb{C}^{2} is used, and\overline{$\zeta$_{\mathb {Z}^{2^{\text{ま} } (s;z) denotes the double holomorphic Eisenstein series of
two parameters(1.9)

tWO variables S\in \mathbb{C}^{2} and withz\in(\mathfrak{H}^{\pm})^{2} defined by,\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} ^{\pm}(s;z)=\sum_{m,n=-\infty}^{\infty}(m+nz1)^{-s_{1} (m+nZ2)^{-s_{2} ({\rm Re}\langle s\}>1),(m,n)\neq(0,0)

where the branch of each summand is to be chosen such that \arg(m+nZj) falls within the range(- $\pi$, $\pi$] in \overline{$\zeta$_{\mathb {Z}^{2} ^{+}(s;z),

and within [- $\pi,\ \pi$ ) in\overline{$\zeta$_{\mathb {Z}^{2} }^{-}(s;z) for j=1,2 We
,

.

now

main object by taking (from a viewpoint of symmetry) the arithmetical mean of(1.10)

introduce the\overline{$\zeta$_{\mathb {Z}^{2^{\text{ま} } (s;z):\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (s;z)=\frac{1}{2}\{\overline{$\zeta$_{\mathb {Z}^{2} ^{+}(s;z)+\overline{$\zeta$_{\mathb {Z}^{2} ^{-}(s;z)\},for which we establish at first in [10] complete asymptotic expansions when z=(z_{1},z_{2}) moves
within either the poly‐sector
to 0 through the sector(\mathfrak{H}^{+})^{2} or (\mathfrak{H}^{-})^{2} so as that the pivotal parameter $\eta$ in (1.7) tends,|\arg $\eta$|< $\pi$/2 (Theorem 1). The case N=0 of our first main result(2.5) with (2.6) (resp. (2.7)) and (2.10) reduces to a transformation formulae for \overline{$\zeta$_{\mathb {Z}^{2} (s;z) in{}_{1}F_{1}(_{v}^{ $\lambda$};Z) (Corollary 1.1); this further leads us toterms of Kummers hypergeometric function

159
DOUBLE HOLOMORPHIC EISENSTEIN SERIES OF \mathrm{T}\mathrm{V}\mathrm{O} COMPLEX VARIABLES

show that counterpart$\pi$/2 (Theorem 2).

expansions exist for \overline{$\zeta$_{\mathb {Z}^{2} (s;z) when $\eta$ tends to\inftythrough |\arg $\eta$|<

The use of Mellin‐Barnes type integrals in [10], manipulated with several properties of hy‐
pergeometric functions, is crucial throughout the proofs; the transference from Theorem 1 toTheorem 2 is for instance achieved by a classical connection formula relating Kummers con‐fluent hypergeometric functions of the first and second kind.
As for asymptotic aspects of relevant Eisenstein series (of one complex variable), Matsumoto[15] obtained complete asymptotic expansions (with respect z) of holomorphic Eisenstein se‐\mathrm{n}\mathrm{e}\mathrm{s}while the second author [16] studied an asymptotic formula (as t\rightarrow+\infty ) for the non‐holomorphic Eisenstein series E_{0}(s;z) (of weight O). Complete asymptotic expansions for theclassical Epstein zeta‐function $\zeta$_{\mathbb{Z}^{2} (s;z) as y={\rm Im} z\rightarrow+\infty have been established by the firstauthor [5], in which similar expansions were also derived for the Laplace‐Mellin transform of$\zeta$_{\mathbb{Z}^{2} (s;z) The main formula in [5] for $\zeta$_{\mathbb{Z}^{2} (s;z) is readily switched to that for E_{0}(s;z) by the,.

relation E_{0}(s;z)=y^{s}$\zeta$_{\mathbb{Z}^{2}}(s;z)/2 $\zeta$(s) where $\zeta$(s) denotes the Riemann zeta‐function; this could,

further be transferred to complete asymptotic expansions as y\rightarrow+\infty for E_{k}(s;z) (of any evenweight k) by the authors [7] upon using Maal] weight change operators. Furthermore, complete
asymptotic expansions for a more general Epstein zeta‐function $\psi$_{\mathbb{Z}^{2}}(s;a,b; $\mu$, v;z) as y\rightarrow+\infty
have recently been established by the first author [6], together with those for the Riemann‐

Liouville transform of $\zeta$_{\mathbb{Z}^{2} (s;z) We further mention that Eisenstein type series (of two complex
variables) relevant to (1.10) have recently been treated in [12][13][14].
The authors would like to thank Professor Kohji Matsumoto for valuable comments on thepresent work.
.

2. MAIN RESULTS
Let\mathscr{D}_{j}^{\pm}(j=1,2) be the domains in \mathfrak{H}^{\pm} defined by,\mathscr{D}_{1}^{\pm}=\{z\in(\mathfrak{H}^{\pm})^{2}||{\rm Im} Z1|<|{\rm Im} z2|\},\mathscr{D}_{2}^{\pm}=\{z\in(\mathfrak{H}^{\pm})^{2}| {\rm Im} z1|>|{\rm Im} z_{2}We shall split the statements of our results into the following two cases (with the equivalentdescription in terms of (1.2)):(2.1)

Case i)Case ii)z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-}\Leftrightarrow $\epsilon$(z1)= $\epsilon$(z2)= $\epsilon$(z_{12}) ;z\in \mathscr{D}_{2}^{+}\cup \mathscr{D}_{2}^{-}\Leftrightarrow $\epsilon$(z1)= $\epsilon$(z_{2})= $\epsilon$(z_{21}).

\overline{$\zeta$_{\mathb {Z}^{2} (s;z) that the assertions in Case i)

It is seen from the definition (1.10) with (1.9) of
readily be switched to those in Case ii) by the replacements

(2.2)

can

\left\{ begin{ar y}{l \mathb {C}^{2}\nis=(s_{1},s_{2})(s_{2},s_{1})=\hat{s}\in\mathb {C}^{2},\ \mathscr{D}_{1}^{+}\cup\mathscr{D}_{1}^{-}\niz=(Z1,z2) (z2,Z1)=\hat{z}\in\mathscr{D}_{2}^{+}\cup\mathscr{D}_{2}^{-} \end{ar y}\right.

and vice versa; however, the statements are to be given in both the cases for clarifying symmetryof our results.
We frequently use the notatione(w)=e^{2 $\pi$ iw} for any w\in\overline{\mathb {C}^{\times}.

Let $\Gamma$(s) be the gamma func‐tion, (s)_{n}= $\Gamma$(s+n)/ $\Gamma$(s) for any n\in \mathbb{Z} the shifted factorial of s Further let {}_{1}F_{1}(_{v}^{ $\lambda$};Z ) and.

U( $\lambda$;v;Z) denote respectively the first and second solutions of Kummers hypergeometric dif‐ferential equation, defined byfor |Z|<+\infty (cf. [20]), and(2.3){}_{1}F_{1}(_{v}^{$\lambda$};Z)=\displaystyle\sum_{k=0}^{\infty}\frac{($\lambda$)_{k}{(v)_{k} !}Z^{k}

U( $\lambda$;v Z)=\displaystyle \frac{1}{ $\Gamma$( $\lambda$)}\int_{0}^{\infty e^{i $\varphi$} e^{-wZ}w^{ $\lambda$-1}(1+w)^{v- $\lambda$-1}dw

for {\rm Re} $\lambda$>0 and | $\varphi$+\arg Z|< $\pi$/2 with any fixed angle $\varphi$\in(- $\pi$, $\pi$) where the path of inte‐,

gration is the ray from the origin to \infty e^{i $\varphi$} (cf. [20]). Next let$\sigma$_{w}(l)=\displaystyle \sum_{0<h|l}h^{w} and $\Phi$_{r,s}(e(z)),

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KATSURADA AND NODA

the function defined for any z\in \mathfrak{H}^{+} by$\Phi$_{r,s}(e(z) =\displaystyle \sum_{h,k=1}^{\infty}e(hkz)h^{r} =\displaystyle\sum_{l=1}^{\infty}$\sigma$_{r-s}(l)e(lz)
が

(2.4)

,

which was first introduced and studied by Ramanujan [18] for the purpose of giving various
evaluations of the holomorphic Eisenstein series E_{k}(z) with k=2,4 , 6.
We now state our first main result.
Theorem 1. Let $\eta$ be given by (1.7). Then the double holomorphic Eisenstein series\overline{$\zeta$_{\mathb {Z}^{2} (s;z)

in (1.10) with (1.9) can be continued to an entirefunction to the s ‐space \mathb {C}^{2} and theformula,\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (s;z)=2\cos^{2}(\frac{ $\pi$}{2}\{s\rangle) $\zeta$(\{s\rangle)+\overline{$\zeta$_{\mathb {Z}^{2^{*} }(s;z)(2.5)
holds, where\overline{$\zeta$_{\mathb {Z}^{2^{*} }(s;z) is representedfor any integer N\geq 0as:i) if z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-},\displaystyle \overline{$\zeta$_{\mathb {Z}^{2^{*} }(s;z)=\frac{2(2 $\pi$)^{\langle s\rangle} { $\Gam a$(\{s\rangle)}\cos(\frac{ $\pi$}{2}\{s\rangle)\{S_{N}(s;z)+R_{N}(s;z)\}(2.6)

in the region $\sigma$_{2}>-N ;ii) if z\in \mathscr{D}_{2}^{+}\cup \mathscr{D}_{2}^{-},\displaystyle\overline{$\zeta$_{\mathb {Z}^{2^{*} (s;z)=\frac{2( $\pi$)^{\ s\} {$\Gam a$(\langles\}) \cos(\frac{$\pi$}{2}\{s\rangle)\{S_{N}(\hat{s};\hat{z})+R_{N}(\hat{s};\hat{z})\}(2.7)

in the region $\sigma$_{1}>-N.

Here in both the cases i) and ii), S_{N}(s;z) isfurther expanded as

S_{N}(s;z)=\displaystyle \sum_{n=0}^{N-1}\frac{(-1)^{n}(s_{2})_{n} {(\{s\})_{n}n!}$\Phi$_{\langle s)-1+n,n}(e( $\epsilon$(z1)_{Z1}) (4 $\pi \eta$)^{n},(2.8)

giving the asymptotic series in the ascending order of $\eta$ as $\eta$\rightarrow 0 through the sector |\arg $\eta$|<$\pi$/2 ; the remainder RN (s;z) is expressed by a certain Mellin‐Barnes type integral and satisfies
the estimate

R_{N}(s;z)=O(e^{-2 $\pi$|{\rm Im} z|}1| $\eta$|^{N})(2.9)$\eta$\rightarrow 0 through |\arg $\eta$|\leq $\pi$/2- $\delta$ with any small $\delta$>0 while z\in(\mathfrak{H}^{\pm})^{2} moves within|{\rm Im} z_{j}|\geq y_{0}>0(j=1,2) Here the constant implied in the O ‐symbol depends at most on s,y_{0}, N and $\delta$ Furthermore, the explicit expression
as

,

.

.

(2.10)

R_{N}(s;z)=\displaystyle \frac{(-1)^{N}(s_{2})_{N} {(N-1)!(\{s\})_{N} (4 $\pi \eta$)^{N}\sum_{h,k=1}^{\infty}e(hk $\epsilon$(Z1)z1)h^{\langle s\}-1+N}k^{N}\displaystyle \times\int_{0}^{1}(1- $\xi$)^{N-1}{ _{1}F_{1}(_{\{s\rangle+N}^{s_{2}+N};-4 $\pi$ hk $\eta \xi$)d $\xi$

holds in the same region of s above, where the case N=0 should read without the factor (-1) !

and the $\xi$ ‐integration.

Remark The n‐th indexed tern on the right side of (2.8) is of order(2.11)=e^{-2 $\pi$|{\rm Im} z_{1}|}| $\eta$|^{n} since,$\Phi$_{r,s}(e( $\epsilon$(z)z))=e( $\epsilon$(z)z)+O(e^{-4 $\pi$|\mathrm{k}\mathrm{n}z|})=e^{-2 $\pi$|{\rm Im} z|}by (2.4); this shows that the presence of the bound in (2.9) is reasonable.
The case N=0 of Theorem 1 yields the following result.

as {\rm Im} z\rightarrow\pm\infty ,Corollary 1.1. Thefunction \overline{$\zeta$_{\mathb {Z}^{2^{*} }(s;z) defined by (2.5) can be continued to an entirefunctionof s to the s ‐space \mathb {C}^{2} and the following transformation formulae holdfor all s\in \mathbb{C}^{2} :,161
DOUBLE HOLOMORPHIC EISENSTEIN SERIES OF TWO COMPLEX VARIABLESi) if z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-},(2.12)\displaystyle \overline{$\zeta$_{\mathb {Z}^{2^{*} }(s;z)=\frac{2(2 $\pi$)^{\langle s\rangle} { $\Gamma$(\{s\rangle)}\cos(\frac{ $\pi$}{2}\langle S\})1(_{\{s\rangle}^{s_{2} ;-4 $\pi$ hk $\eta$);ii) if z\in \mathscr{D}_{2}^{+}\cup \mathscr{D}_{2}^{-},(2.13)\displaystyle\overline{$\zeta$_{\mathb {Z}^{2^{*} (s;z)=\frac{2( $\pi$)^{\s\rangle}{$\Gam a$(\{s\rangle)}\cos(\frac{$\pi$}{2}\langles\rangle)\sum_{h,k=1}^{\infty}e(hk$\epsilon$(z_{2})z_{2})h^{\langles\rangle-1}{_1}F_{1}(_{\s\rangle}^{s_{1};-4$\pi$hk$\eta$).

We now state our second main result.

Theorems\in \mathbb{C}^{2}:2.

\overline{$\zeta$_{\mathb {Z}^{2^{*} }(s;z) be defined by (2.5). Then the following formulae hold for anyLeti) if z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-},(2.14)\displaystyle\overline{$\zeta$_{\mathb {Z}^{2^{*} (s;z)=2( $\pi$)^{\langles\} \cos(\frac{$\pi$}{2}\{s\})\{ frac{e^{$\epsilon$(z_{12})$\pi$is_{2} {$\Gam a$(s1)}T_{1}(s;z)+\frac{e^{$\epsilon$(z_{21})$\pi$is_{1} {$\Gam a$(s_{2}) T_{2}(s;z)\};ii) if z\in \mathscr{D}_{2}^{+}\cup \mathscr{D}_{2}^{-},(2.15)\displaystyle\overline{$\zeta$_{\mathb {Z}^{2^{*} (s;z)=2( $\pi$)^{\s\rangle}\cos(\frac{$\pi$}{2}\{s\})\{ frac{e^{$\epsilon$(z_{21})$\pi$is_{1} {$\Gam a$(s_{2})T_{1}(\hat{s};\hat{z})+\frac{e^{$\epsilon$(z)$\pi$is_{2}12}{$\Gam a$(s_{1})T_{2}(\hat{s}\cdot\hat{z})\}.

T_{j}(s;z)(j=1, 2) are representedfor any integerN \geq 0 as
(2.16)
T_{j}(s;z)=S_{j,N}(s;z)+R_{j,N}(s;z)
in the region -N<$\sigma$_{j}<N+1(j=1,2) where

Here in both the cases i) and ii),,

(2.17)(2.18)

S_{1,N}(s;z)=\displaystyle \sum_{n=0}^{N-1}\frac{(-1)^{n}(s_{2})_{n}(1-s_{1})_{n} {n!}$\Phi$_{s_{1}-n-1,-s_{2}-n}(e( $\epsilon$(z $\iota$)_{Z1}) \{4 $\pi$ e^{ $\epsilon$(z) $\pi$ i}12 $\eta$\}^{-s_{2}-n},S_{2,N}(s;z)=\displaystyle \sum_{n=0}^{N-1}\frac{(-1)^{n}(s_{1})_{n}(1-s_{2})_{n} {n!}$\Phi$_{s_{2}-n-1,-s_{1}-n}(e( $\epsilon$(z_{2})_{Z2}) (4 $\pi \eta$)^{-s_{1}-n},both giving the asymptotic series in the descending order of $\eta$ as $\eta$\rightarrow\infty through the sector|\arg $\eta$|< $\pi$/2 ; the remainders R_{j,N}(s;z)(j=1,2) are expressed by certain Mellin‐Barnes
type integrals, and satisfy the estimates

R_{1,N}(s;z)=O(e^{-2 $\pi$|{\rm Im} z1}|| $\eta$|^{-$\sigma$_{2}-N})
R_{2,N}(s;z)=O(e^{-2 $\pi$|{\rm Im} z2}|| $\eta$|^{-$\sigma$_{1}-N})(2.19)(2.20),

respectively as $\eta$\rightarrow\infty through |\arg $\eta$|\leq $\pi$/2- $\delta$ with any small $\delta$>0 while z\in(\mathfrak{H}^{\pm})^{2} moves
within |{\rm Im} z_{j}|\geq y_{0}>0(j=1,2) Here the constants implied in the O ‐symbols depend at most
,

.

on

s, y_{0}, N and $\delta$(2.21).

Furthermore, the explicit expressions

R_{1,N}(s;z)=\displaystyle \frac{(-1)^{n}(s_{2})_{N}(1-s_{1})_{N} {(N-1)!}\sum_{h,k=1}^{\infty}Z1\displaystyle \times\int_{0}^{1}$\xi$^{-s_{2}-N}(1- $\xi$)^{N-1}U(S_{2}+N;\langle s\};4 $\pi$ hke^{ $\epsilon$(z) $\pi$ i}12 $\eta$/ $\xi$)d $\xi$,(2.22)

R_{2,N}(s;z)=\displaystyle \frac{(-1)^{N}(s_{1})_{N}(1-s_{2})_{N} {(N-1)!}\sum_{h,k=1}^{\infty}e(hk $\epsilon$(z_{2})z_{2})h^{\{s\}-1}\displaystyle \mathrm{x}\int_{0}^{1}$\xi$^{-s_{1}-N}(1- $\xi$)^{N-1}U(s_{1}+N;\{s\rangle;4 $\pi$ hk $\eta$/ $\xi$)d $\xi$

hold in the same region of s above, where the case N=0 should read without the factor (-1) !

and the $\xi$ ‐integration.
Remark

By virtue of (1.2), it is ensured that |e( $\epsilon$(z_{j})z_{j})|<1(j=1,2) in (2.5)-(2.22).

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KATSURADA AND NODA

3. SPECIFIC VALUES AT S\in \mathbb{N}^{2} AND

S\in(-\mathbb{N}_{0})^{2}

The following Sections 3‐6 are devoted to showing that our second main formula (2.5) withand (2.22) reduces to various expressions (in closed(2.14) (resp. (2.15)), (2.17),(2.18\underline{),(}2.21)

foml) of the particular values of $\zeta$_{\mathbb{Z}^{2} (s;z) and its partial derivatives when s is at any integer
lattice points, as well as its central values at s=(s,s)(s\in \mathbb{C}) For this, let B_{n}(n\in \mathrm{N}_{0}) denotethe n‐th Bemoulli number (cf. [3]), and wnte 0=(0,0) e_{1}=(1,0) e_{2}=(0,1) and 1=(1,1)
for brevity. We hereafter use the customary notation (q;q)_{\infty}=\displaystyle \prod_{k=1}^{\infty}(1-q^{k}) for any complex q
with |q|<1 and set q_{j}=e( $\epsilon$(z_{j})z_{j})(j=1,2) The assertions only when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} are to
be stated in the sequel, since those when z\in \mathscr{D}_{2}^{+}\cup \mathscr{D}_{2}^{-} are readily obtained from the former by(2.2).
.

,

,

,

.

,

Corollary 2.1. For specific values of \overline{$\zeta$_{\mathb {Z}^{2} (s;z) and its first derivatives at any lattice points
s=m\in \mathrm{N}^{2} with m=(m_{1},m_{2}) the following formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :,i) if \langle m\rangle=m_{1}+m_{2} is even,\displaystyle\overline{$\zeta$_{\mathb {Z}^{2} (m;z)=-\frac{(2$\pi$i)^{\langle$\tau$n)}B_{\ 7r$\iota$\rangle} {\ m\}! +2( $\pi$i)^{\ 7n)}(3.1)

\displaystyle \times\{\frac{(-1)^{m_{2} }{(m_{1}-1)!}s_{1,m_{1} (m;z)+\frac{(-1)^{m_{1} }{(m_{2}-1)!}S_{2,m_{2} (m;z)\},and in particular for m=1,

\displaystyle\overline{$\zeta$_{\mathb {Z}^{2} (1;z)=\frac{$\pi$^{2}{3}+\frac{2$\pi$}{$\eta$}\log\frac{(q_{1};q_{1})_{\infty}{(q_{2},q_{2})_{\infty}

(3.2)

;

ii) if \langle m)=m1+m_{2} is odd,

\overline{$\zeta$_{\mathbb{Z}^{2} }(m;z)=0,(3.3)

andfu rther for j=1,2,(3.4)\displaystyle \frac{\partial$\zeta$_{\mathb {Z}^{2} {\partial s_{j} (m;z)=\frac{1}{2}(2 $\pi$ i)^{\langle $\iota$\rangle+1}7 $\gamma$\{\frac{(-1)^{m_{1} {(m_{2}-1)!}S_{1,m_{1} (m;z)+\frac{(-1)^{m_{2} {(m_{1}-1)!}S_{2,m_{2} (m;z)\}.

Here in both the cases above, S_{j,m}(m;z)(j=1,2) are given (in finite closedforms) as

(3.5)

(3.6)

S_{1,m_{1} (m;z)=\displaystyle \sum_{n=0}^{m_{1}-1}\left(\begin{ar ay}{l} m_{1}-1\ n \end{ar ay}\right)(m_{2})_{n}$\Phi$_{m_{1}-n-1,-m_{2}-n}(q_{1})(-4 $\pi \eta$)^{-m_{2}-n},S_{2,m_{2} (m;z)=\displaystyle \sum_{n=0}^{m_{2}-1}\left(\begin{ar ay}{l} m_{2}-1\ n \end{ar ay}\right)(m_{1})_{n}$\Phi$_{m_{2}-n-1,-m_{1}-n}(q_{2})(4 $\pi \eta$)^{-m_{1}-n}.

Remark Formula (3.2) gives a two variable analogue of the classical Kronecker limit formulafor the (one variable) Epstein zeta‐function as s\rightarrow 1 (cf. [19]), while (3.1), (3.3) and (3.4) may

be regarded as its variants at s=m\in \mathbb{N}^{2} ; similar reductions from asymptotic expansions arealso observed in [5][6].

Corollary 2.2. For specific values of \overline{$\zeta$_{\mathb {Z}^{2} (s;z) at any lattice points s=-m\in(-\mathrm{N}_{0})^{2} withm=(m_{1},m_{2}) the following formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :,\overline{$\zeta$_{\mathbb{Z}^{2} }(-m;z)=\{_{0}^{-1}

(3.7)

=0,.
otherwiseifm

Corollary 2.3. For specific values of (\partial\overline{$\zeta$_{\mathbb{Z}^{2} }/\partial_{S_{j} )(s;z)(j=1,2) at any lattice points s=-m\in(-\mathbb{N}_{0})^{2} with m=(m_{1},m_{2}) the following formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :i) if m=0 for j=1,2,
,

,

(3.8)

\displaystyle \frac{\partial$\zeta$_{\mathb {Z}^{2} }{\partial s_{j} (0;z)=-2\log\sqrt{2 $\pi$}+2S_{j,1}(0;z)=-2\log\{\sqrt{2 $\pi$}(q_{j};q_{j})_{\infty}\},163
DOUBLE HOLOMORPHIC EISENSTEIN SERIES OF TWO COMPLEX VARIAHLESii) if \langle m\rangle=m1+m_{2} is even with \{m\rangle\geq 2,(3.10)(3.9)\displayte\frac{}(-m;z)=\frac{}$\zeta({m\rangle+1)\frac{}S_2,m{1}+(-m;z)\frac{ptil\overn{$\zeta_{mthb{Z}^2 \partil^{\facs_1}{$\zeta_{mthb{Z}^2 \partils_{2}\partil(-m;z)=\frac{m\}!(2$pi)^{\lange7$\am iota$\rngle}(2$\pi)^{$\gam $n\}{m!$\zeta({m\}+1)frac{2m_1}!(2$\pi)^{7n}(2$\pi)lange7$\am iota$)2m_{}!S1,m_{2}+(-m;z),iii) if{m\rangle=m_{1}+m_{2} is odd with \langle m} \geq 1,(3.11)
and further(3.12)\displaystyle \frac{\partial$\zeta$_{\mathb {Z}^{2} {\partial s_{k} (-m;z)=0 (k=1,2),\displayst le\frac{\partial^{2}\overline{$\zeta$_{\mathb {Z}^{2} {\partials_{1}\partials_{2}(-m;z)=-\frac{B_{\langle$\gam a$rt\angle+1}{\m\rangle+1}$\pi$^{2}-\frac{1}2( $\pi$ )\{7$\gam a$l)-1}\times\{m_{1}!S_{1,m_{2}+1}(-m;z)+m_{2}!S_{2,m_{1}+1}(-m;z)\}.

Here in the cases i)‐iii) above, S_{j,m+1}(-m;z)(j=1, 2) are given (in finitely closedforms) as(3.13)(3.14)

S_{1,m_{2}+1}(-m;z)=\displaystyle \sum_{n=0}^{m_{2} \left(\begin{ar ay}{l} m_{2}\ n \end{ar ay}\right)(1+m_{1})_{n}$\Phi$_{-m_{1}-n-1,m_{2}-n}(q_{1})(-4 $\pi \eta$)^{m_{2}-n},S_{2,m_{1}+1}(-m;z)=\displaystyle \sum_{n=0}^{1}m\left(\begin{ar ay}{l} m_{1}\ n \end{ar ay}\right)(1+m_{2})_{n}$\Phi$_{-m_{2}-n-1,m_{1}-n}(q_{2})(4 $\pi \eta$)^{m_{1}-n}.

Remark Formula (3.8) gives a two variable analogue of the classical Kronecker limit formulafor (the derivative of) Epstein zeta‐function at s=0 (cf. [19]), while (3.9)-(3.12) may be re‐garded as its variants at s=-m\in(-\mathbb{N}_{0})^{2} ; similar reductions from asymptotic expansions arealso observed in [5][6].

4. SPECIFIC VALUES IN CONNECTION WITH GENERALIZED LAMBERT SERIES
We evaluate in this section several specific values ofLambert series defined for any s,w\in \mathbb{C} with

(4.1)

\overline{$\zeta$_{\mathb {Z}^{2} (s;z) in terms of the generalized

w\neq q^{-k}(k\in \mathbb{N}) by\displaystyle\mathscr{L}_{s}(w;q)=\sum_{k=1}^{\infty}\frac{k^{s}wq^{k} {1-wq^{k} .

Let $\delta$(m)(m\in \mathbb{Z}) be the symbol which equals 1 or 0 according to m=0 or otherwise, and(n,k\in \mathbb{N}_{0}) denote the Stirling numbers of the first kind defined by(4.2)\mathfrak{S}_{k}^{n}\displaystyle\frac{(e^{w}-1)^{k}{k!}=\sum_{n=0}^{\infty}\frac{\mathfrak{S}_{k}^{n}{n!}w^{n}.

Proposition 1. For any r\in \mathbb{N}_{0} and s\in \mathbb{C} we have the relation
(4.3)
where

(4.4)

$\Phi$_{r,s}(q)=\displaystyle\sum_{j=0}^{r}\mathfrak{S}_{j}^{r}\mathscr{L}_{s}^{(j)}(q),\displaystyle\mathscr{L}_{s}^{(j)}(q)=(\frac{\partial}{\partialw})^{j}\mathscr{L}_{s}(w;q)_{w=1}=j!\sum_{k=1}^{\infty}\frac{k^{s}q^{k\{j+$\delta$(j)\} {(1-q^{k})^{j+1}(j\in \mathbb{N}_{0}).

We remark that the Lambert series of the type in (4.4) play underlying rôles in Ramanujanstheories of Eisenstein series, theta functions and elliptic functions (cf. [1][2][21]), where, for
e.g.,

(4.5)

P(q)=1-24\mathscr{L}_{1}^{(0)}(q) , Q(q)=1+240\mathscr{L}_{3}^{(0)}(q) , R(q)=1-504\mathscr{L}_{5}^{(0)}(q).

164
KATSURADA AND NODA

arethe classical Eisenstein series reformulated by Ramanujan [18]. It is possible to transfer,through (4.3), from the expressions in (3.5), (3.6), (3.13) and (3.14) to those in terms of \mathscr{L}_{s}^{(j)}(q)(j\in \mathbb{N}_{0}) We can for instance show the following Corollaries 2.4−2.6.
.

Corollary 2.4. For specific values of \overline{$\zeta$_{\mathb {Z}^{2} (s;z) and its first derivatives at s=1 (1,2), (2, 1)
and 2 1, the foUowing formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :
,

\cdot

\displaystyle\overline{$\zeta$_{\mathb {Z}^{2} (1;z)=\frac{$\pi$^{2}{3}+\frac{2$\pi$}{$\eta$}\mathscr{L}_{-1}^{(0)}(q_{1})-\frac{2$\pi$}{$\eta$}\mathscr{L}_{-1}^{(0)}(q_{2})

(4.6)
(4.7)
(4.8)

,

\displaystyle\frac{\partial\overline{$\zeta$_{\mathb {Z}^{2} {\partials_{j}(1,2;z)=-\frac{$\pi$^{2}{2$\eta$^{2}\{ mathscr{L}_{-2}^{(0)}(q_{1})-\mathscr{L}_{-2}^{(0)}(q_{2})\}+\frac{2$\pi$^{3}{$\eta$}\mathscr{L}_{-1}^{(1)}(q_{2})\displaystyle\frac{\partial$\zeta$_{\mathb {Z}^{2} {\partials_{j} (2,1;z)=\frac{$\pi$^{2} {2$\eta$^{2} \{ mathscr{L}_{-2}^{(0)}(q_{1})-\mathscr{L}_{-2}^{(0)}(q_{2})\}-\frac{2$\pi$^{3} {$\eta$}\mathscr{L}_{-1}^{(1)}(q_{1}),,both for j=1,2 and
,

(4.9)

\displaystyle\overline{$\zeta$_{\mathb {Z}^{2} (2\cdot1;z)=\frac{$\pi$^{4} {45}+\frac{2$\pi$^{2} {$\eta$^{2} \{ mathscr{L}_{-2}^{(1)}(q_{1})+\mathscr{L}_{-2}^{(1)}(q_{2})\}-\frac{$\pi$}{$\eta$^{3} \{ mathscr{L}_{-3}^{(0)}(q_{1})-\mathscr{L}_{-3}^{(0)}(q_{2})\}.

Corollary 2.5. For specific values of (\partial\overline{$\zeta$_{\mathbb{Z}^{2} }/\partial sj)(s;z) at s=0, -1, -2\cdot 1, -m_{j}e_{j} with evenm_{j}(j=1,2) the following formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :,(4.10)(4.11)(4.12)(4.13)(4.14)\displaystyle \frac{\partial\overline{$\zeta$_{\mathb {Z}^{2} }{\partial s_{j} (0;z)=-\log\sqrt{2 $\pi$}+2\mathscr{L}_{-1}^{(0)}(q_{j}) (j=1,2)\displaystyle \frac{\partial$\zeta$_{\mathb {Z}^{2} {\partials1}(-1;z)=-\frac{1}{2$\pi$^{2} $\zeta$(3)+\frac{2 $\eta$}{$\pi$}\mathscr{L}_{-2}^{(1)}(q_{1})-\frac{1}{$\pi$^{2} \mathscr{L}_{-3}^{(0)}(q_{1})\displaystyle\frac{\partial$\zeta$_{\mathb {Z}^{2} {\partials_{2} (-1;z)=-\frac{1}{2$\pi$^{2} $\zeta$(3)-\frac{2$\eta$}{$\pi$}\mathscr{L}_{-2}^{(1)}(q_{2})-\frac{1}{$\pi$^{2} \mathscr{L}_{-3}^{(0)}(q_{2}),,\displaystyle\frac{\partial\overline{$\zeta$_{\mathb {Z}^{2} {\partials_{1}(-2\cdot1;z)=\frac{3}{2$\pi$^{2} $\zeta$(5)+\frac{4$\eta$^{2}{$\pi$^{2}\{ mathscr{L}_{-3}^{(1)}(q_{1})+\mathscr{L}_{-3}^{(2)}(q_{1})\},-\displaystyle \frac{6 $\eta$}{$\pi$^{3} \mathscr{L}_{-4}^{(1)}(q_{1})+\frac{3}{$\pi$^{4} \mathscr{L}_{-5}^{(0)}(q_{1}),+\displaystyle \frac{6 $\eta$}{$\pi$^{3} \mathscr{L}_{-4}^{(1)}(q_{2})+\frac{3}{$\pi$^{4} \mathscr{L}_{-5}^{(0)}(q_{2}),\displaystyle\frac{\partial\overline{$\zeta$_{\mathb {Z}^{2} {\partials_{2}(-2\cdot1;z)=\frac{3}{2$\pi$^{2} $\zeta$(5)+\frac{4$\eta$^{2}{$\pi$^{2}\{ mathscr{L}_{-3}^{(1)}(q_{2})+\mathscr{L}_{-3}^{(2)}(q_{2})\}and for any even m_{j}\in \mathbb{N}_{0}(j=1,2)(4.15),,\displaystyle\frac{\partial\overline{$\zeta$_{\mathb {Z}^{2} {\partials_{j} (-m_{j}e_{j};z)=\frac{m_{j}! {(2$\pi$i)^{m_{J} $\zeta$(m_{j}+1)+\frac{2m_{j}! {(2$\pi$i)^{m_{j} \mathscr{L}_{-m_{j}-1}^{(0)}(q_{j}).

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DOUBLE HOLOMORPHIC EISENSTEIN SERIES OF TWO COMPLEX VARIABLES

Corollary 2.6. For spec fic values (\partial^{2}\overline{$\zeta$_{\mathb {Z}^{2} }/\partial s_{1}\partial s_{2})(s;z) at s=-e_{j} for j=1,2, (-2, -1)(-1, -2) the following formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :

and

,

,

(4.16)(4.17)(4.18)\displaystyle\frac{\partial^{2}\overline{$\zeta$_{\mathb {Z}^{2} {\partials1\partials_{2} (-e_{1};z)=-\frac{$\pi$^{2} {12}-2$\pi\eta$\mathscr{L}_{-1}^{(1)}(q_{2})-\frac{1}{2}\{ mathscr{L}_{-2}^{(0)}(q_{1})+\mathscr{L}_{-2}^{(0)}(q2)\},\displaystyle\frac{\partial^{2}\overline{$\zeta$_{\mathb {Z}^{2} {\partials_{1}\partials_{2} (-e_{2};z)=-\frac{$\pi$^{2} {12}+2$\pi\eta$\mathscr{L}_{-1}^{(1)}(q_{1})-\frac{1}{2}\{ mathscr{L}_{-2}^{(0)}(q_{1})+\mathscr{L}_{-2}^{(0)}(q_{2})\},\displaystyle\frac{\partial^{2}\overline{$\zeta$_{\mathb {Z}^{2} {\partial_{S1}\partials_{2} (-2,-1;z)=\frac{$\pi$^{2} {120}+2$\eta$^{2}\{ mathscr{L}_{-2}^{(1)}(q_{2})+\mathscr{L}_{-2}^{(2)}(q_{2})\}-\frac{$\eta$}{$\pi$}\{ mathscr{L}_{-3}^{(1)}(q_{1})-2\displaystyle \mathscr{L}_{-3}^{(1)}(q_{2})\}+\frac{3}{4$\pi$^{2} \{\mathscr{L}_{-4}^{(0)}(q_{1})+\mathscr{L}_{-4}^{(0)}(q_{2})\},(4.19)\displaystyle\frac{\partial^{2}\overline{$\zeta$_{\mathb {Z}^{2} {\partials1\partials_{2} (-1,-2;z)=\frac{$\pi$^{2} {120}+2$\eta$^{2}\{ mathscr{L}_{-2}^{(1)}(q_{1})+\mathscr{L}_{-2}^{(2)}(q_{1})\}-\frac{$\eta$}{$\pi$}\{2\mathscr{L}_{-3}^{(1)}(q_{1})-\displaystyle \mathscr{L}_{-3}^{(1)}(q_{2})\}+\frac{3}{4$\pi$^{2} \{\mathscr{L}_{-4}^{(0)}(q_{1})+\mathscr{L}_{-4}^{(0)}(q2)\}.

Using the relation (4.3), we can further show that S_{j,m}(m;z) in (3.5) and (3.6), as well as

S_{j,m+1}(-m;z) in (3.13) and (3.14) are (as functions of z) the elements from the \mathbb{Z}[(4 $\pi \eta$)^{\pm}]modules spanned by certain sets of the generalized Lambert series\mathscr{L}_{s}^{(j)}(q).

Corollary 2.7. For thefunctions in (3.5), (3.6), (3.13) and (3.14), the following algebraic prop‐
erties hold whenz\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :i) at any lattice points s=m\in \mathbb{N}^{2} with m=(m_{1},m_{2})(4.20)(4.21),

S_{1,m_{1} (m;z)\in\langle\{\mathscr{L}_{l-\{7 $\gamma \iota$)+1}^{(j)}(q_{1})|0\leq j\leq l\leq m_{1}-1\}\rangle_{\mathbb{Z}[1/4 $\pi \eta$]},S_{2,m_{2} (m;z)\in\langle\{\mathscr{L}_{l-\{7 $\gamma$ l\rangle+1}^{(j)}(q_{2})|0\leq j\leq l\leq m_{2}-1\}\rangle_{\mathbb{Z}[1/4 $\pi \eta$]};ii) at any lattice points s=-m\in(-\mathbb{N}_{0})^{2} with m=(m_{1},m_{2})(4.22)(4.23),

S_{1,m_{2}+1}(-m;z)\in\langle\{\mathscr{L}_{l-\langle rn\rangle-1}^{(j)}(q_{1})|0\leq j\leq l\leq m_{2}\}\rangle_{\mathbb{Z}[4 $\pi \eta$]},S_{2,m_{1}+1}(-m;z)\in\langle\{\mathscr{L}_{l-\{7r $\iota$\rangle-1}^{(\dot{j})}(q_{2})|0\leq j\leq l\leq m_{1}\}\rangle_{\mathbb{Z}[4 $\pi \eta$]}.
5. SPECIFIC VALUES AT s\in \mathbb{N}\times(-\mathbb{N}_{0})\mathrm{O}\mathrm{R}s\in(-\mathbb{N}_{0})\times \mathbb{N}Our second main formula in Theorem 2 further yields various evaluations (in finitely closed

forms) for specific values of \overline{$\zeta$_{\mathb {Z}^{2} (s;z) and its partial derivatives at any lattice points s=m ineither \mathbb{N}\times(-\mathbb{N}_{0}) or (-\mathbb{N}_{0})\times \mathbb{N}.

Corollary 2.8. For specific values of \overline{$\zeta$_{\mathb {Z}^{2} (s;z) at any lattice points s=m\in \mathbb{N}\mathrm{x}(-\mathbb{N}_{0}) with(m_{1}, - m2) the following formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-}upon N_{1}=\displaystyle \min(m_{1},m_{2}+1) :i) if \{m\}=m1-m_{2} is even,

m=

(5.1)

,

\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (m;z)=2 $\zeta$(m_{1}-m_{2})+\frac{2(2 $\pi$ i)^{m_{1}-m_{2} (-1)^{m_{2} {(m_{1}-1)!}S_{1,N_{1} (m;z)ii) if \{m\rangle=m_{1}-m_{2} is odd,

(5.2)

andfurtherfor k=1 2,\overline{$\zeta$_{\mathbb{Z}^{2} }(m;z)=0,

,

(5.3)

\displaystyle \frac{\partial$\zeta$_{\mathb {Z}^{2} {\partial s_{k} (m;z)=\frac{$\pi$^{2} {2} $\delta$(\{m\}-1)+\frac{(2 $\pi$ i)^{m_{1}-m_{2}+1} {2(m_{1}-1)!}S_{1,N_{1} (m;z)

.

;

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KATSURADA AND NODA

Here in both the cases i) and ii), S_{1,N_{1}}(m;z) is given by

S_{1,N_{1} (m;z)=\displaystyle \sum_{n=0}^{N_{1}-1}\left(\begin{ar ay}{l} m_{2}\\ n \end{ar ay}\right)1\cdot(5.4)

Corollary 2.9. For specific values of \overline{$\zeta$_{\mathb {Z}^{2} (s;z) at any lattice points s=m\in(-\mathbb{N}_{0})\mathrm{x}\mathbb{N} withm=(-m,m) the following formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} upon N_{2}=\displaystyle \min(m1+1,m_{2}) :i) if \langle m\rangle=m_{2}-m_{1} is even,,\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (m;z)=2 $\zeta$(m_{2}-m1)+\frac{2(2 $\pi$ i)^{m_{2}-m_{1} (-1)^{m_{1} {(m_{2}-1)!}S_{2,N_{2} (m;z)

(5.5)

;

ii) if \langle m\}=m_{2}-m_{1} is odd,

\overline{$\zeta$_{\mathbb{Z}^{2} }(m;z)=0,(5.6)

andfurtherfor k=1 2,,\displaystyle \frac{\partial$\zeta$_{\mathb {Z}^{2} {\partial s_{k} (m;z)=\frac{$\pi$^{2} {2} $\delta$(\{m\}-1)+\frac{(2 $\pi$ i)^{m_{2}-m_{1}+1} {2(m_{2}-1)!}S_{2,N_{2} (m;z)

(5.7)

.

Here in both the cases i) and ii), S_{2,N_{2}}(m;z) is given by

S_{2,N_{2} (m;z)=\displaystyle \sum_{n=0}^{N_{2}-1}\left(\begin{ar ay}{l} m_{1}\ n \end{ar ay}\right)(1-m_{2})_{n}$\Phi$_{m_{2}-n-1,m_{1}-n}(q_{2})(4 $\pi \eta$)^{m_{1}-n}.

(5.8)

We next define for any r\in \mathbb{N}_{0} the functions \mathscr{S}_{r}(q) by\displaystle\mathscr{S}_r(q)=\frac{1}2 $\zeta$(-r)+$\Phi$_{0,r}(q)=\left\{ begin{ar y}{l \frac{B_1}{2+$\Phi$_{0, }(q)&\mathrm{i}\mathrm{f}r=0,\ -\frac{B_r+1}{2(r+1)} $\Phi$_{0,r}(q)&\mathrm{i}\mathrm{f}r\geq1, \end{ar y}\right.

(5.9)

which was first introduced and studied by Ramanujan [18] in the course of developing his theoryofEisenstein series and elliptic functions (see, for e.g., [1][2][21]); the Eisenstein ser es in (4.5),
due to Ramanujan [18], are for instance connected with \mathscr{S}_{r}(q) as

P(q)=-24\mathscr{S}_{1}(q) , Q(q)=240\mathscr{S}_{3}(q) , R(q)=-504\mathscr{S}_{5}(q),while Weierstra13 elliptic function $\beta$(w|2 $\pi$(1,z)) associated with the basis 2 $\pi$(1,z) for z\in \mathfrak{H}^{+},
is expanded into the Laurent series involving \mathscr{S}_{r}(q) with q=e(z) in its coefficients as,(5.10)\displaystyle \wp(w|2 $\pi$(1,z) =\frac{1}{w^{2} +2\sum_{n=1}^{\infty}\frac{(-1)^{n+1} {(2n)!}\mathscr{S}_{2n+1}(q)w^{2n}

for 0<|w|<2 $\pi$\displaystyle \min(1, |z|) The case m=m_{j}e_{j}(j=1,2) of Corollanes 2.8 and 2.9 in partic‐.

ular implies the following relations:

Corollary 2.10. Thefollowing formulae hold for any m_{j}\in \mathbb{N}(j=1,2) when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :i) if m_{j}\in \mathbb{N}(j=1,2) is even,(5.11)\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (m_{j}e_{j};z)=2 $\zeta$(m_{j})+\frac{2(2 $\pi$ i)^{m_{j} {(m_{j}-1)!}$\Phi$_{m_{j}-1,0}(q_{j})=\frac{2(2 $\pi$ i)^{m_{j} {(m_{j}-1)!}\mathscr{S}_{m_{j}-1}(q_{j}),and in particular when m_{j}\geq 4,(5.12)\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (m_{j}e_{j};z)=\frac{(2 $\pi$)^{m_{j} {(m_{j}-1)!}(\frac{\partial}{\partial w})^{m_{j}-2}\{ $\beta$(w|2 $\pi$(1, $\epsilon$(z_{j})z_{j}) -\frac{1}{w^{2} \}_{w=0};ii) if m_{j}\in \mathrm{N}(j=1, 2) is odd,

(5.13)
andfurtherfor k=1 2,\overline{$\zeta$_{\mathbb{Z}^{2} }(m_{j}e_{j};z)=0,,(5.14)\displaystyle \frac{\partial$\zeta$_{\mathb {Z}^{2} {\partial s_{k} (m_{j}e_{j};z)=\frac{$\pi$^{2} {2} $\delta$(m_{j}-1)+\frac{(2 $\pi$ i)^{m_{j}+1} {2(m_{j}-1)!}$\Phi$_{m_{j}-1,0}(q_{j})=\frac{(2 $\pi$.i)^{m_{j}+1} {2(m_{i}-1)!}\mathscr{S}_{m_{j}-1} (qj).

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DOUBLE HOLOMORPHIC EISENSTEIN SERIES OF \mathfrak{R}\mathrm{V}\mathrm{O} COMPLEX VARIABLES

Remark It is interesting to note that the even index cases of \mathscr{S}_{r}(q) appear in the formulae for\partial\overline{$\zeta$_{\mathb {Z}^{2} /\partial_{S_{k} above, whereas theser es,cases

cannot be observed in Ramanujans theories of Eisenstein

elliptic functions and theta functions.

Corollary 2.11. For specific values of \overline{$\zeta$_{\mathb {Z}^{2} (s;z) and its derivatives at any lattice points s=(m_{1}, -1)\in \mathbb{N}\times(-\mathbb{N}_{0}) the following formulae hold when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :,i) if m_{1} is odd,
(5.15)\displaystyle \overline{$\zeta$_{\mathb {Z}+(1-m_{1})$\Phi$_{m_{1}-2,0}(q_{1})\},^{2} (m_{1}, -1;z)=2 $\zeta$(m_{1}-1)-\frac{2(2 $\pi$ i)^{m_{1}-1} {(m_{1}-1)!}\{-4 $\pi \eta \Phi$_{m_{1}-1,1}(q_{1})and in particular if m_{1}\geq 3,(5.16)\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (m_{1}, -1;z)=\frac{2(2 $\pi$ i)^{m_{1} {(m_{1}-2)!}\{\mathscr{S}_{m_{1}-2}(q_{1})+\frac{4 $\pi \eta$}{m_{1}-1}$\Phi$_{m_{1}-1, }(q_{1}ii) if m_{1} is even,\overline{$\zeta$_{\mathbb{Z}^{2} }(m1, -1;z)=0,(5.17)

andfurtherfor k=1 2,,(5.18)\displaystyle\frac{\partial\overline{$\zeta$_{\mathb {Z}^{2} }{\partials_{k} (m_{1},-1;z)=\frac{$\pi$^{2} {2}$\delta$(m_{1}-2)-\frac{(2$\pi$i)^{m_{1} {2(m_{1}-2)!}\{-4$\pi\eta\Phi$_{m_{1}-1, }(q_{1})+(1=\displaystyle \frac{(2 $\pi$ i)^{m_{1} {2(m_{1}-2)!}\{\mathscr{S}_{m_{1}-2}(q_{1})+\frac{4 $\pi \eta$}{m_{1}-1}$\Phi$_{m_{1}-1,1}(q_{1})\}.
Corollary 2.12. For specific values of(-1,m_{2})\in(-\mathbb{N}_{0})\times \mathbb{N},\overline{$\zeta$_{\mathb {Z}^{2} (s;z) and its derivatives at any lattice points

thefollowing formulae hold whenz\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-}:

i) if m_{2} is odd,
(5.19)\displaystyle \overline{$\zeta$_{\mathb {Z}^+(1-m_{2})$\Phi$_{m_{2}-2,0}(q_{2})\},{2} (-1,m_{2};z)=2 $\zeta$(m_{2}-1)-\frac{2(2 $\pi$ i)^{m_{1-1} {(m_{2}-1)!}\{4 $\pi \eta \Phi$_{m_{2}-1,1}(q_{2})and in particular if m_{2}\geq 3,(5.20)\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (-1,m_{2};z)=\frac{2(2 $\pi$ i)^{m_{2}-1} {(m_{2}-2)!}\{\mathscr{S}_{m_{2}-2}(q_{2})-\frac{4 $\pi \eta$}{m_{2}-1}$\Phi$_{m_{2}-1,1}(q_{2})\};ii) if m_{2} is even,(5.21)

andfurther for k=1 2,\overline{$\zeta$_{\mathbb{Z}^{2} }(-1,m_{2};z)=0,,(5.22)\displaystyle\frac{\partial\overline{$\zeta$_{\mathb {Z}^{2} }{\partials_{k} (-1,m_{2};z)=\frac{$\pi$^{2} {2}$\delta$(m_{2}-2)-\frac{(2$\pi$i)^{m_{2} {2(m_{2}-1)!}\{4$\pi\eta\Phi$_{m_{2}-1, }(q_{2})+(1-m_{2})$\Phi$_{m_{2}-2,0}(q_{2})=\displaystyle \frac{(2 $\pi$ i)^{m_{2} {2(m_{2}-2)!}\{\mathscr{S}_{m_{2}-2}(q_{2})-\frac{4 $\pi \eta$}{m_{2}-1}$\Phi$_{m_{2}-1,1}(q_{2})\}.

We next define the functions $\psi$_{j}(w;q)(j=1,2) by the Fourier series expansions(5.23)$\psi$_{1}(w;q)=\displaystyle \frac{1}{4}\cot\frac{w}{2}+\sum_{n=1}^{\infty}\frac{q^{n} {1-q^{n} \sin(w;q)=-\displaystyle \frac{1}{24}+\sum_{n=1}^{\infty}\frac{q^{n} {(1-q^{n})^{2} \cos

Wh

nw ,

nw

s=

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KATSURADA AND NODA

for |{\rm Im} w|<2 $\pi$|{\rm Im} z| with q =e(z) and their analytic continuations. These were first introduced
by Ramanujan, who made extensive use of these functions for developing his theories of elliptic
functions, theta functions and Eisenstein series (see, for e.g., [1][2][21]); WeierstraB elliptic
function \wp(w|2 $\pi$(1,z)) for z\in \mathfrak{H}^{+} is in fact connected with these functions as,$\beta$(w|2 $\pi$(1,z) =2\displaystyle \{$\psi$_{2}(0;q)-\frac{\partial}{\partial w}$\psi$_{1}(w;q)\}.

In view of the Laurent series expansions$\psi$_{1}(w;q)=\displaystyle \frac{1}{2w}+\sum_{n=1}^{\infty}\frac{(-1)^{n-1} {(2n-1)!}\mathscr{S}_{2n-1}(q)w^{2n-1},(5.24)$\phi$_{2}(w;q)=-\displaystyle \frac{1}{24}+\sum_{n=0}^{\infty}\frac{(-1)^{n} {(2n)!}$\Phi$_{1,2n}(q)w^{2n}

for 0<|w|<2 $\pi$ min (1, |z|)Corollary 2.13.

,

we can show thei) for any m_{j}\in \mathbb{N}(j=1, 2)(5.25)

following relations.

Thefollowing evaluations are valid ifz\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-}:,\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (m_{j}e_{j};z)=-\frac{(2 $\pi$)^{m_{J} {(m_{j}-1)!}(\frac{\partial}{\partial w})^{m_{J}-1}\{$\phi$_{1}(w;q_{j}) \mathrm{w}=0



and in particular for any m_{j}\geq 3,(5.26)\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (m_{j}e_{j};z)=\frac{(2 $\pi$)^{m_{j} {(m_{j}-1)!}(\frac{\partial}{\partial w})^{m_{1}-2}\{\wp(w|2 $\pi$(1 , $\epsilon$(z_{j})z_{j}) -\frac{1}{w^{2} \}_{w=0},

which unifies (5.11) and (5.12) for m_{j}\geq 3 ;ii) for any m_{1}\in \mathbb{N} with m_{1}\geq 2,(5.27)\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (m_{1}, -1;z)=\frac{2(2 $\pi$)^{m-1}1}{(m_{1}-2)!}[-(\frac{\partial}{\partial w})^{m_{1}-2}\{$\psi$_{1}(w;q_{1})-\frac{1}{2w}\}_{w=0}-\displaystyle\frac{4$\pi\eta$}{m_{1}-1}(\frac{\partial}{\partialw})^{m_{1}-1}$\psi$_{2}(w;q_{1})_{w=0}];iii) for any m_{2}\in \mathbb{N} with m_{2}\geq 2,(5.28)\displaystyle \overline{$\zeta$_{\mathb {Z}^{2} (-1,m_{2};z)=\frac{2(2 $\pi$)^{m-1}2}{(m_{2}-2)!}[-(\frac{\partial}{\partial w})^{m_{2}-2}\{$\psi$_{1}(w;q_{2})-\frac{1}{2w}\}_{w=0}+\displaystyle\frac{4$\pi\eta$}{m_{2}-1}(\frac{\partial}{\partialw})^{m_{2}-1}$\phi$_{2}(w;q_{2})_{w=0}];

Using again the relation (4.3), we can show the following results.
Corollary 2.14. For the functions in (5.4) and (5.8), the following algebraic properties holdwhenz\in \mathscr{D}_{1}^{+}\cup \text{多_{}1}^{-}:

i) at any s=m=(m_{1}, -m_{2})\in \mathbb{N}\times(-\mathbb{N}_{0})(5.29),

S_{1,N_{1} (m;z)\in\{\{\mathscr{L}_{\{7n\rangle-1+l}^{(j)0\leq j\leq l;}(q_{1})_{\max(0,1-\langle m\})\leq l\leq m_{2} \}\}_{\mathb {Z}[4 $\pi \eta$]}ii) at any s=m=(-m_{1},m_{2})\in(-\mathbb{N}_{0})\times \mathrm{N},(5.30)

S_{2,N_{2} (m;z)\in\{\{\mathscr{L}_{\langle 7r $\iota$\}-1+l}( q_{2})_{\max(0,1-\langle m\rangle)\leq l\leq m_{1} \}\}_{\mathb {Z}[4 $\pi \eta$]};

169
DOUBLE HOLOMORPHIC EISENSTEIN SERIES OF TWO COMPLEX VARIABLES

6. CENTRAL VALUES
Next let K_{v}(Z) denote the Bessel function of the third kind (cf. [4]), and write d(l)=$\sigma$_{0}(l)for l\in \mathbb{N} Then the case N=0 of our main fomlula (2.5) with (2.14), (2.21) and (2.22) yields.

the following results on the central values of\overline{$\zeta$_{\mathb {Z}^{2} (s;z) along the complex line s=s1(s\in \mathbb{C}),and further its extremal central value at s=(1/2)1.

Corollary 2.15. The foUowing formula holds for any s\in \mathbb{C} when z\in \mathscr{D}_{1}^{+}\cup \mathscr{D}_{1}^{-} :(6.1)\displaystyle \frac{\overline{$\zeta$_{\mathb {Z}^{2} (s1;z)}{\cos( $\pi$ s)}=2\cos( $\pi$ s) $\zeta$(2s)+\frac{4i$\pi$^{s}$\eta$^{1/2-s} { $\Gam a$(s)}[\sum_{l=1}^{\infty}l^{1/2-s}$\sigma$_{2s-1}(l)e(l $\epsilon$(Z1)(Z1+z_{2})/2)\times\{ $\epsilon$(Z12)K_{S-1/21}12,s\rightarrow 1/2 asserts thatwhose limitingcase as(6.2)\displaystyle\frac{\partial\overline{$\zeta$_{\mathb {Z}^{2} {\partials_{j} (1/2;z)=\frac{$\pi$^{2} {2}-2$\pi$i\sum_{l=1}^{\infty}d(l)e(l$\epsilon$(z_{1})(z1+z_{2})/2)\times\{ $\epsilon$(Z12)K_{0}(2 $\pi$ le^{ $\epsilon$(z_{12}) $\pi$ i} $\eta$)+ $\epsilon$(z21)K_{0}(2 $\pi$ l $\eta$)\}for j=1,2.

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