In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number n. They were introduced by Frenkel & Turaev (1997) in their study of elliptic 6-j symbols.
For surveys of elliptic hypergeometric series see Gasper & Rahman (2004) or Spiridonov (2008).
The q-Pochhammer symbol is defined by
(
a
;
q
)
n
=
∏
k
=
0
n
−
1
(
1
−
a
q
k
)
=
(
1
−
a
)
(
1
−
a
q
)
(
1
−
a
q
2
)
⋯
(
1
−
a
q
n
−
1
)
.
(
a
1
,
a
2
,
…
,
a
m
;
q
)
n
=
(
a
1
;
q
)
n
(
a
2
;
q
)
n
…
(
a
m
;
q
)
n
.
The modified Jacobi theta function with argument x and nome p is defined by
θ
(
x
;
p
)
=
(
x
,
p
/
x
;
p
)
∞
θ
(
x
1
,
.
.
.
,
x
m
;
p
)
=
θ
(
x
1
;
p
)
.
.
.
θ
(
x
m
;
p
)
The elliptic shifted factorial is defined by
(
a
;
q
,
p
)
n
=
θ
(
a
;
p
)
θ
(
a
q
;
p
)
.
.
.
θ
(
a
q
n
−
1
;
p
)
(
a
1
,
.
.
.
,
a
m
;
q
,
p
)
n
=
(
a
1
;
q
,
p
)
n
⋯
(
a
m
;
q
,
p
)
n
The theta hypergeometric series r+1Er is defined by
r
+
1
E
r
(
a
1
,
.
.
.
a
r
+
1
;
b
1
,
.
.
.
,
b
r
;
q
,
p
;
z
)
=
∑
n
=
0
∞
(
a
1
,
.
.
.
,
a
r
+
1
;
q
;
p
)
n
(
q
,
b
1
,
.
.
.
,
b
r
;
q
,
p
)
n
z
n
The very well poised theta hypergeometric series r+1Vr is defined by
r
+
1
V
r
(
a
1
;
a
6
,
a
7
,
.
.
.
a
r
+
1
;
q
,
p
;
z
)
=
∑
n
=
0
∞
θ
(
a
1
q
2
n
;
p
)
θ
(
a
1
;
p
)
(
a
1
,
a
6
,
a
7
,
.
.
.
,
a
r
+
1
;
q
;
p
)
n
(
q
,
a
1
q
/
a
6
,
a
1
q
/
a
7
,
.
.
.
,
a
1
q
/
a
r
+
1
;
q
,
p
)
n
(
q
z
)
n
The bilateral theta hypergeometric series rGr is defined by
r
G
r
(
a
1
,
.
.
.
a
r
;
b
1
,
.
.
.
,
b
r
;
q
,
p
;
z
)
=
∑
n
=
−
∞
∞
(
a
1
,
.
.
.
,
a
r
;
q
;
p
)
n
(
b
1
,
.
.
.
,
b
r
;
q
,
p
)
n
z
n
The elliptic numbers are defined by
[
a
;
σ
,
τ
]
=
θ
1
(
π
σ
a
,
e
π
i
τ
)
θ
1
(
π
σ
,
e
π
i
τ
)
where the Jacobi theta function is defined by
θ
1
(
x
,
q
)
=
∑
n
=
−
∞
∞
(
−
1
)
n
q
(
n
+
1
/
2
)
2
e
(
2
n
+
1
)
i
x
The additive elliptic shifted factorials are defined by
[
a
;
σ
,
τ
]
n
=
[
a
;
σ
,
τ
]
[
a
+
1
;
σ
,
τ
]
.
.
.
[
a
+
n
−
1
;
σ
,
τ
]
[
a
1
,
.
.
.
,
a
m
;
σ
,
τ
]
=
[
a
1
;
σ
,
τ
]
.
.
.
[
a
m
;
σ
,
τ
]
The additive theta hypergeometric series r+1er is defined by
r
+
1
e
r
(
a
1
,
.
.
.
a
r
+
1
;
b
1
,
.
.
.
,
b
r
;
σ
,
τ
;
z
)
=
∑
n
=
0
∞
[
a
1
,
.
.
.
,
a
r
+
1
;
σ
;
τ
]
n
[
1
,
b
1
,
.
.
.
,
b
r
;
σ
,
τ
]
n
z
n
The additive very well poised theta hypergeometric series r+1vr is defined by
r
+
1
v
r
(
a
1
;
a
6
,
.
.
.
a
r
+
1
;
σ
,
τ
;
z
)
=
∑
n
=
0
∞
[
a
1
+
2
n
;
σ
,
τ
]
[
a
1
;
σ
,
τ
]
[
a
1
,
a
6
,
.
.
.
,
a
r
+
1
;
σ
,
τ
]
n
[
1
,
1
+
a
1
−
a
6
,
.
.
.
,
1
+
a
1
−
a
r
+
1
;
σ
,
τ
]
n
z
n
