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NIST
17
q
-Hypergeometric and Related Functions
Properties
17.4
Basic Hypergeometric Functions
17.6
ϕ
1
2
Function
§17.5
ϕ
0
0
,
ϕ
0
1
,
ϕ
1
1
Functions
ⓘ
Notes:
See
Andrews (
1976
, pp. 17, 19, 36)
and
Gasper and Rahman (
2004
, pp. 25–26)
.
Permalink:
http://dlmf.nist.gov/17.5
See also:
Annotations for
Ch.17
Euler’s Second Sum
ⓘ
Keywords:
Euler’s sums
,
Euler’s sums (first, second, third)
,
q
-hypergeometric function
See also:
Annotations for
§17.5
and
Ch.17
17.5.1
ϕ
0
0
(
-
;
-
;
q
,
z
)
=
∑
n
=
0
∞
(
-
1
)
n
q
(
n
2
)
z
n
(
q
;
q
)
n
=
(
z
;
q
)
∞
,
|
z
|
<
1
;
ⓘ
Symbols:
(
m
n
)
: binomial coefficient
,
(
a
;
q
)
n
:
q
-Pochhammer symbol (or
q
-shifted factorial)
,
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: basic hypergeometric (or
q
-hypergeometric) function
,
q
: complex base
,
n
: nonnegative integer
and
z
: complex variable
Referenced by:
§17.2(iii)
Permalink:
http://dlmf.nist.gov/17.5.E1
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.5
,
§17.5
and
Ch.17
compare (
17.3.2
).
q
-Binomial Series
ⓘ
Keywords:
q
-binomial series
See also:
Annotations for
§17.5
and
Ch.17
17.5.2
ϕ
0
1
(
a
;
-
;
q
,
z
)
=
(
a
z
;
q
)
∞
(
z
;
q
)
∞
,
|
z
|
<
1
;
ⓘ
Symbols:
(
a
;
q
)
n
:
q
-Pochhammer symbol (or
q
-shifted factorial)
,
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: basic hypergeometric (or
q
-hypergeometric) function
,
q
: complex base
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/17.5.E2
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.5
,
§17.5
and
Ch.17
compare (
17.2.37
).
q
-Binomial Theorem
ⓘ
Keywords:
q
-binomial theorem
See also:
Annotations for
§17.5
and
Ch.17
17.5.3
ϕ
0
1
(
q
-
n
;
-
;
q
,
z
)
=
(
z
q
-
n
;
q
)
n
.
ⓘ
Symbols:
(
a
;
q
)
n
:
q
-Pochhammer symbol (or
q
-shifted factorial)
,
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: basic hypergeometric (or
q
-hypergeometric) function
,
q
: complex base
,
n
: nonnegative integer
and
z
: complex variable
Permalink:
http://dlmf.nist.gov/17.5.E3
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.5
,
§17.5
and
Ch.17
This is (
17.2.35
) reformulated.
Euler’s First Sum
ⓘ
Keywords:
Euler’s sums
,
Euler’s sums (first, second, third)
,
q
-hypergeometric function
See also:
Annotations for
§17.5
and
Ch.17
17.5.4
ϕ
0
1
(
0
;
-
;
q
,
z
)
=
∑
n
=
0
∞
z
n
(
q
;
q
)
n
=
1
(
z
;
q
)
∞
,
|
z
|
<
1
;
ⓘ
Symbols:
(
a
;
q
)
n
:
q
-Pochhammer symbol (or
q
-shifted factorial)
,
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: basic hypergeometric (or
q
-hypergeometric) function
,
q
: complex base
,
n
: nonnegative integer
and
z
: complex variable
Referenced by:
§17.2(iii)
Permalink:
http://dlmf.nist.gov/17.5.E4
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.5
,
§17.5
and
Ch.17
compare (
17.3.1
).
Cauchy’s Sum
ⓘ
Keywords:
Cauchy’s sum
,
q
-hypergeometric function
See also:
Annotations for
§17.5
and
Ch.17
17.5.5
ϕ
1
1
(
a
c
;
q
,
c
/
a
)
=
(
c
/
a
;
q
)
∞
(
c
;
q
)
∞
,
|
c
|
<
|
a
|
.
ⓘ
Symbols:
(
a
;
q
)
n
:
q
-Pochhammer symbol (or
q
-shifted factorial)
,
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
;
b
1
,
…
,
b
s
;
q
,
z
)
or
ϕ
s
r
+
1
(
a
0
,
…
,
a
r
b
1
,
…
,
b
s
;
q
,
z
)
: basic hypergeometric (or
q
-hypergeometric) function
and
q
: complex base
Permalink:
http://dlmf.nist.gov/17.5.E5
Encodings:
TeX
,
pMML
,
png
See also:
Annotations for
§17.5
,
§17.5
and
Ch.17