Noun
Appell-Lerch sum (plural Appell-Lerch sums)
(mathematics) A generalization of Lambert series with the form
μ
(
u
,
v
;
τ
)
=
a
1
2
θ
(
v
;
τ
)
∑
n
∈
Z
(
−
b
)
n
q
1
2
n
(
n
+
1
)
1
−
a
q
n
{\displaystyle \mu (u,v;\tau )={\frac {a^{\frac {1}{2}}}{\theta (v;\tau )}}\sum _{n\in Z}{\frac {(-b)^{n}q^{{\frac {1}{2}}n(n+1)}}{1-aq^{n}}}}
where
q
=
e
2
π
i
τ
,
a
=
e
2
π
i
u
,
b
=
e
2
π
i
v
{\displaystyle \displaystyle q=e^{2\pi i\tau },\quad a=e^{2\pi iu},\quad b=e^{2\pi iv}}
and
θ
(
v
,
τ
)
=
∑
n
∈
Z
(
−
1
)
n
b
n
+
1
2
q
1
2
(
n
+
1
2
)
2
.
{\displaystyle \theta (v,\tau )=\sum _{n\in Z}(-1)^{n}b^{n+{\frac {1}{2}}}q^{{\frac {1}{2}}\left(n+{\frac {1}{2}}\right)^{2}}.}