C. Angelantonj et al. / Nuclear Physics B 897 (2015) 781–820 787
and of the Hecke operator
4
F(s,κ,w)=H
(z)
κ
F(s, 1,w). (2.10)
For the special values s =
w
2
and s = 1 −
w
2
, within the domain of analyticity, F(s, κ, w)
becomes a harmonic Maaß form. For negative weight, the case of interest for most applica-
tions to string theory [6], the Niebur–Poincaré series is absolutely convergent at the latter value
s = 1 −
w
2
. Moreover, since the first cusp form for SL(2; Z) appears at weight twelve, for
w ∈{0, −2, −4, −6, −8, −12} the shadow of F(1 −
w
2
, κ, w) vanishes and the Niebur–Poincaré
series is actually holomorphic, hence valued in the polynomial ring generated by E
4
, E
6
and
Δ
−1
, where E
2n
are the holomorphic Eisenstein series and Δ is the modular discriminant. Other
Niebur–Poincaré series of interest are F(1 −
w
2
+n, κ, w), with n a positive integer, which can
be reached from the harmonic Maaß form F (1 −
w
2
+ n, κ, w − 2n) by acting with n modular
derivatives. In fact, one can prove [6–8] that any weak almost holomorphic modular form of
negative weight in the polynomial ring generated by
ˆ
E
2
, E
4
, E
6
and Δ
−1
, with
ˆ
E
2
being the
almost holomorphic Eisenstein series of weight two, can be uniquely decomposed into a linear
combination of Niebur–Poincaré series,
Φ(z) =
0<m≤κ
n
=0
d
(m) F
1 −
w
2
+, m, w
. (2.11)
The coefficients d
(m) are determined by the principal part of the q-Laurent expansion of Φ
itself [7]. The integer n counts the maximal power of
ˆ
E
2
in the ring decomposition of Φ, and is
known as the depth. An important fact is that for each value of , the shadows cancel in the linear
combination (2.11).
It will also be useful to introduce the non-holomorphic weight-w Eisenstein series
E(s,w;z) =
1
2
γ ∈Γ
∞
\Γ
y
s−
w
2
|
w
γ
=
1
2
(c,d)=1
y
s−
w
2
(c z + d)
w
|cz+d|
2s−w
(2.12)
which is absolutely convergent for Re (s) > 1, and admits the Fourier series expansion
E(s,w;z) = y
s−
w
2
+
4
1−s
i
−w
πΓ(2s −1)ζ (2s −1)
Γ(s+
w
2
)Γ(s −
w
2
)ζ(2s)
y
1−s−
w
2
+
m=0
i
−w
(4π|m|)
w
2
π
s
Γ(s+
w
2
sgn(m)) ζ(2s)
|m|
−s
σ
2s−1
(|m|) W
s,w
(my) e
2iπmx
. (2.13)
The Eisenstein series E(s, w) can be obtained formally as the limit κ → 0of |4πκ|
w
2
−s
F(s,
κ, w), and thus shares several of the properties of the Niebur–Poincaré series. Namely, it is an
eigenmode of the weight-w hyperbolic Laplacian
4
The Hecke operator H
(z)
n
acts on the Fourier modes of a weight-w modular form Φ =
m
Φ(m, y) e
2iπmx
as
H
(z)
n
Φ(m,y) = n
1−w
d|(m,n)
d
w−1
Φ(nm/d
2
,d
2
y/n).
It satisfies the commutative algebra H
(z)
κ
H
(z)
λ
=
d|(κ,λ)
d
1−w
H
(z)
κλ/d
2
.
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