SUMMATION IDENTITIES AND SPECIAL VALUES OF
HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
Abstract. We prove hypergeometric type summation identities for a function
defined in terms of quotients of the p-adic gamma function by counting points
on certain families of hyperelliptic curves over Fq . We also find certain special
values of that function.
1. Introduction and statement of results
In [11], Greene introduced the notion of hypergeometric functions over finite
fields analogous to classical hypergeometric series. Since then many interesting
relations between special values of these hypergeometric functions and the number
of points on certain varieties over finite fields have been obtained. The arguments
of these functions are multiplicative characters of finite fields and, consequently,
results involving hypergeometric functions over finite fields are often restricted to
primes in certain congruence classes. For example, the expressions for the trace of
Frobenius map on families of elliptic curves given in [1, 2, 9, 15, 16] are restricted to
certain congruence classes to facilitate the existence of characters of specific orders.
To overcome these restrictions, in [18, 19], the third author defined a function
n Gn [· · · ] in terms of quotients of the p-adic gamma function which can best be
described as an analogue of hypergeometric series in the p-adic setting. He showed
[17, 18, 19] how results involving hypergeometric functions over finite fields can be
extended to almost all primes using the function n Gn [· · · ].
Let p be an odd prime, and let Fq denote the finite field with q elements. Let
Γp (·) denote the Morita’s p-adic gamma function, and let ω denote the Teichmüller
character of Fq with ω denoting its character inverse. For x ∈ Q we let bxc denote
the greatest integer less than or equal to x and hxi denote the fractional part of x.
Definition 1.1. [19, Definition 5.1] Let q = pr , for p an odd prime and r ∈ Z+ ,
and let t ∈ Fq . For n ∈ Z+ and 1 ≤ i ≤ n, let ai , bi ∈ Q ∩ Zp . Then we define
q−2
−1 X
a1 , a2 , . . . , an
(−1)jn ω j (t)
|t :=
n Gn
b1 , b2 , . . . , bn
q − 1 j=0
q
×
n r−1
Y
Y
k
k
jp
jp
−bhai pk i− q−1
c−bh−bi pk i+ q−1
c
(−p)
i=1 k=0
j
k
q−1 )p i)
Γp (hai pk i)
Γp (h(ai −
j
k
q−1 )p i)
.
Γp (h−bi pk i)
Γp (h(−bi +
Date: 20th August, 2014.
2010 Mathematics Subject Classification. Primary: 11G20, 33E50; Secondary: 33C99, 11S80,
11T24.
Key words and phrases. Character of finite fields, Gaussian hypergeometric series, Hyperelliptic curves, Teichmüller character, p-adic Gamma function.
We are grateful to Ken Ono for his comments on an initial draft of the article.
1
2
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
We note that the value of n Gn [· · · ] depends only on the fractional part of the
parameters ai and bi , and is invariant if we change the order of the parameters.
The aim of this paper is to prove summation identities for n Gn [· · · ]. In [19],
the third author showed that transformations for hypergeometric functions over
finite fields can be re-written in terms of n Gn [· · · ]. However, such transformations
will only hold for all p where the original characters existed over Fq , and hence
restricted to primes in certain congruence classes. It is a non-trivial exercise to
then extend these results to almost all primes. While numerous transformations
exist for the finite field hypergeometric functions, very few exist for n Gn [· · · ] in full
generality. The first and second authors [5, 6] provide transformations for 2 G2 [· · · ]q
by counting points on various families of elliptic curves over Fq . Recently, the third
author and Fuselier [10] provide two transformations for n Gn [· · · ]p when n = 3 and
n = 4, respectively. They also provide two transformations for n Gn [· · · ]p for any
n. In this paper we prove eight summation identities for the function n Gn [· · · ]q for
any n, which are listed below. Let φ be the quadratic character on F×
q extended to
all of Fq by setting φ(0) := 0.
Theorem 1.2. Let d ≥ 4 be even, and let p be an odd prime such that p d−1
2
d (b−y )d
and g(y) =
d(d − 1). Let a, b ∈ F×
q . For y ∈ Fq , let f (y) = a
a(d−1)
d−1
d(b−y 2 )
d
. Let l = gcd(d − 1, q − 1), and let χ be a multiplicative characa
a(d−1)
ter of order l. If b is not a square in Fq then
X
φ(y 2 − b)
y∈Fq
× d−1 Gd−1
3
2(d−1) ,
1
d,
1
2(d−1) ,
0,
1
d−1 ,
1
d,
2
d−1 ,
2
d,
3
2(d−1) ,
1
d,
...,
...,
= −1 − q · d−2 Gd−2
and
X
d−1
2(d−1) ,
d−2
2d ,
...,
...,
...,
...,
d+1
2(d−1) ,
d+2
2d ,
d−2
2(d−1) ,
d−2
2d ,
2d−3
2(d−1)
d−1
d
...,
...,
d
2(d−1) ,
d+2
2d ,
...,
...,
|f (y)
q
d−2
d−1
d−1
d
|f (0)
q
φ(y 2 − b)
y∈Fq
× d−1 Gd−1
1
2(d−1) ,
0,
= −1 + q · d−2 Gd−2
1
(d−1) ,
1
d,
d−1
2(d−1) ,
d−2
2d ,
2
(d−1) ,
2
d,
...,
...,
d+1
2(d−1) ,
d+2
2d ,
d−2
2(d−1) ,
d−2
2d ,
...,
...,
2d−3
2(d−1)
d−1
d
d
2(d−1) ,
d+2
2d ,
...,
...,
|g(y)
q
d−2
d−1
d−1
d
|g(0) .
q
If b is a square in Fq then
1+2
l−1
X
χj (−a) +
j=0
φ(y 2 − b)
y∈Fq
√
y6=± b
× d−1 Gd−1
X
1
2(d−1) ,
0,
3
2(d−1) ,
1
d,
...,
...,
d−1
2(d−1) ,
d−2
2d ,
d+1
2(d−1) ,
d+2
2d ,
...,
...,
2d−3
2(d−1)
d−1
d
|f (y)
q
IDENTITIES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
1
d−1 ,
1
d,
= −q · d−2 Gd−2
2
d−1 ,
2
d,
d−2
2(d−1) ,
d−2
2d ,
...,
...,
d
2(d−1) ,
d+2
2d ,
d−2
d−1
d−1
d
...,
...,
3
|f (0)
q
and
X
3+
φ(y 2 − b)
y∈Fq
√
y6=± b
1
2(d−1) ,
× d−1 Gd−1
0,
= −q · d−2 Gd−2
3
2(d−1) ,
1
d,
1
(d−1) ,
1
d,
d−1
2(d−1) ,
d−2
2d ,
...,
...,
2
(d−1) ,
2
d,
...,
...,
d+1
2(d−1) ,
d+2
2d ,
d−2
2(d−1) ,
d−2
2d ,
2d−3
2(d−1)
d−1
d
...,
...,
d
2(d−1) ,
d+2
2d ,
|g(y)
q
d−2
(d−1)
d−1
d
...,
...,
|g(0) .
q
Theorem 1.3. Let d ≥ 3 be odd, and let p be an odd prime such that p - d(d−1). Let
d−1
d−1
2
2
)
d (b−y )d
d
and g(y) = d(b−y
.
a, b ∈ F×
q . For y ∈ Fq , let f (y) = a
a(d−1)
a
a(d−1)
Let l = gcd(d − 1, q − 1), and let χ be a multiplicative character of order l.
If b is not a square in Fq then
X
0,
1
2d ,
d−1 Gd−1
y∈Fq
= q · d−1 Gd−1
1
d−1 ,
3
2d ,
1
2(d−1) ,
1
2d ,
...,
...,
d−3
2(d−1) ,
d−2
2d ,
3
2(d−1) ,
3
2d ,
d−1
2(d−1) ,
d+2
2d ,
d−2
2(d−1) ,
d−2
2d ,
...,
...,
...,
...,
d−2
d−1
2d−1
2d
d
2(d−1) ,
d+2
2d ,
| − f (y)
q
...,
...,
2d−3
2(d−1)
2d−1
2d
| − f (0)
q
and
φ(a) X
·
φ(y 2 − b)
q
y∈Fq
1
, ...,
0, d−1
× d−1 Gd−1
1
3
...,
2d ,
2d ,
1
2
, d−1
, ...,
= d−1 Gd−1 d−1
1
2
,
,
...,
d
d
d−3
2(d−1) ,
d−2
2d ,
d−1
2(d−1) ,
d−1
2d ,
d−1
2(d−1) ,
d+2
2d ,
d+1
2(d−1) ,
d+1
2d ,
d−2
d−1
2d−1
2d
...,
...,
...,
...,
| − g(y)
q
d−2
d−1 ,
d−2
d ,
1
2
d−1
d
| − g(0) .
q
If b is a square in Fq then
X
d−1 Gd−1
y∈Fq
√
y6=± b
= −2
l−1
X
1
d−1 ,
3
2d ,
0,
1
2d ,
...,
...,
d−3
2(d−1) ,
d−2
2d ,
d−1
2(d−1) ,
d+2
2d ,
...,
...,
d−2
d−1
2d−1
2d
| − f (y)
q
(χj φ)(−a) − q
j=0
× d−1 Gd−1
1
2(d−1) ,
1
2d ,
3
2(d−1) ,
3
2d ,
...,
...,
d−2
2(d−1) ,
d−2
2d ,
d
2(d−1) ,
d+2
2d ,
...,
...,
2d−3
2(d−1)
2d−1
2d
| − f (0)
q
4
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
and
−
2 φ(a) X
−
φ(y 2 − b)
q
q
y∈Fq
√
y6=± b
0,
1
2d ,
1
d−1 ,
1
d,
× d−1 Gd−1
= d−1 Gd−1
1
d−1 ,
3
2d ,
d−3
2(d−1) ,
d−2
2d ,
...,
...,
2
d−1 ,
2
d,
...,
...,
d−1
2(d−1) ,
d−1
2d ,
d−1
2(d−1) ,
d+2
2d ,
d−2
d−1
2d−1
2d
...,
...,
d+1
2(d−1) ,
d+1
2d ,
...,
...,
| − g(y)
q
d−2
d−1 ,
d−2
d ,
1
2
d−1
d
| − g(0) .
q
We now give one example to show how the above theorems are applied in specific
cases.
Example 1.4. Let a 6= 0 and p ≥ 5. Taking d = 3 and b = 1 in Theorem 1.3, we
deduce that
1
X
27
27
0, 12
, 34
2 2
4
(1 − y )
= −2 − 2φ(−a) − q · 2 G2 1
2 G2
1
5 |−
5 |−
4a3
4a3 q
6,
6
6,
6
q
y∈Fq
y6=±1
and
X
φ(y 2 − 1) 2 G2
y∈Fq
y6=±1
0,
1
6,
1
2
5
6
|−
27
2
(1
−
y
)
=
−2φ(a)
−
qφ(a)
·
G
2
2
4a3
q
1
2
2
3
1
2,
1
3,
|−
We also derive the following transformation for 2 G2 [· · · ].
−27b2
and
6= 1. Then
Theorem 1.5. Let q = pr , p > 3 be a prime. Let a, b ∈ F×
q
4a3
1
3 −27b2
3 −27b2
1
4,
4 |
4,
4 |
=
φ(−a)
·
G
.
2 G2
2
2
1
2
1
5
4a3 q
4a3 q
3,
3
6,
6
In particular, if
27b2
6= 1 then
4a6
1
3 27b2
4,
4 |
G
=
G
2 2
2 2
1
2
4a6 q
3,
3
1
4,
1
6,
3
4
5
6
|
27b2
4a6
.
q
Remark 1.6. In [5, 6], the first and second authors derived transformations for
2 G2 [· · · ]q with different parameters. Thus, we can derive many such transformations by combining the transformation of Theorem 1.5 with those in [5, 6].
Let d ≥ 2. In [4], the first and second authors expressed the number of distinct
zeros of the polynomials xd +ax+b and xd +axd−1 +b over Fq in terms d−1 Gd−1 [· · · ].
We now state four theorems from [4] which we will need to prove our main results.
Theorem 1.7. ([4, Theorem 1.2]) Let d ≥ 2 be even, and let p be an odd prime
d
such that p - d(d − 1). Let a, b ∈ F×
q . If N (x + ax + b = 0) denotes the number of
d
distinct zeros of the polynomial x + ax + b in Fq then
N (xd + ax + b = 0) = 1 + φ(−b)
"
1
3
, 2(d−1)
, ...,
× d−1 Gd−1 2(d−1)
1
0,
...,
d,
d−1
2(d−1) ,
d
2 −1
d ,
d+1
2(d−1) ,
d
2 +1
d ,
...,
...,
2d−3
2(d−1)
d−1
d
#
|α
,
q
27
4a3
.
q
IDENTITIES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
where α =
d
a
bd
a(d−1)
d−1
5
.
Theorem 1.8. ([4, Theorem 1.3]) Let d ≥ 3 be odd, and let p be an odd prime
d
such that p - d(d − 1). Let a, b ∈ F×
q . If N (x + ax + b = 0) denotes the number of
distinct zeros of the polynomial xd + ax + b in Fq then
N (xd + ax + b = 0) = 1 + φ(−a)
"
1
, . . . , (d−3)/2
0, d−1
d−1 ,
× d−1 Gd−1
3
d−2
1
...,
2d ,
2d ,
2d ,
where α =
d
a
bd
a(d−1)
d−1
(d−1)/2
d−1 ,
d+2
2d ,
...,
...,
d−2
d−1
2d−1
2d
#
|−α
,
q
.
Theorem 1.9. ([4, Theorem 1.4]) Let d ≥ 2 be even, and let p be an odd prime
d
d−1
+ b = 0) denotes the number
such that p - d(d − 1). Let a, b ∈ F×
q . If N (x + ax
d
d−1
of distinct zeros of the polynomial x + ax
+ b in Fq then
N (xd + axd−1 + b = 0) = 1 + φ(−b)
"
1
3
, 2(d−1)
, ...,
× d−1 Gd−1 2(d−1)
1
...,
0,
d,
where β =
bd
a
d
a(d−1)
d−1
d−1
2(d−1) ,
d
2 −1
d ,
d+1
2(d−1) ,
d
2 +1
d ,
...,
...,
2d−3
2(d−1)
d−1
d
#
|β
,
q
.
Theorem 1.10. ([4, Theorem 1.5]) Let d ≥ 3 be odd, and let p be an odd prime
d
d−1
+ b = 0) denotes the number
such that p - d(d − 1). Let a, b ∈ F×
q . If N (x + ax
d
d−1
of distinct zeros of the polynomial x + ax
+ b in Fq then
N (xd + axd−1 + b = 0) = 1 + φ(−ab)
"
1
0, d−1
, . . . , (d−3)/2
d−1 ,
× d−1 Gd−1
3
d−2
1
,
,
.
.
.
,
2d
2d
2d ,
where β =
bd
a
d
a(d−1)
d−1
(d−1)/2
d−1 ,
d+2
2d ,
...,
...,
d−2
d−1
2d−1
2d
#
|−β
,
q
.
2. Notations and Preliminaries
Throughout this paper p will denote an odd prime, Fq the finite field of q = pr
elements, Zp the ring of p-adic integers, Qp the field of p-adic numbers, Qp the
algebraic closure of Qp , and Cp the completion of Qp . Let Zq be the ring of integers
in the unique unramified extension of Qp with residue field Fq . Let µq−1 be the
group of (q − 1)-th roots of unity in C× .
×
2.1. Multiplicative characters and Gauss sums. Let Fc
q denote the group of
c
×
×
multiplicative characters χ on F with values in µ . F is a cyclic group of order
q
q−1
q
q − 1. Let ε and φ denote the trivial and quadratic characters, respectively. The
×
domain of each χ ∈ Fc
q is extended to Fq by setting χ(0) := 0. We now state the
orthogonality relations for multiplicative characters in the following lemma.
Lemma 2.1. ([13, Chapter 8]). We have
6
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
q − 1 if χ = ε;
0
if χ 6= ε.
x∈Fq
X
q − 1 if x = 1;
(2)
χ(x) =
0
if x 6= 1.
(1)
X
χ(x) =
c
χ∈F×
q
It is known that Z×
q contains all the (q − 1)-th roots of unity. Therefore, we can
×
×
consider multiplicative characters on F×
q to be maps χ : Fq → Zq .
We now introduce some properties of Gauss sums. For further details, see [7]
noting that we have adjusted results to take into account ε(0) = 0. Let ζp be a
fixed primitive p-th root of unity in Qp . The trace map tr : Fq → Fp is given by
2
tr(α) = α + αp + αp + · · · + αp
r−1
.
tr(α)
Then the additive character θ : Fq → Qp (ζp ) is defined by θ(α) := ζp
to see that θ(a + b) = θ(a)θ(b) and
X
(2.1)
θ(x) = 0.
. It is easy
x∈Fq
×
For χ ∈ Fc
q , the Gauss sum is defined by
X
G(χ) :=
χ(x)θ(x).
x∈Fq
If ζq−1 is a primitive (q − 1)-th root of unity in Qp , then G(χ) lies in Qp (ζp , ζq−1 ).
×
and denote by G the Gauss sum G(T m ).
We let T denote a fixed generator of Fc
q
m
Using (2.1) it is easy to show that G0 = −1. We will use the following results on
Gauss sums in the proof of our main results.
Lemma 2.2. ([7, Theorem 1.1.4 (a)]). If k ∈ Z and T k 6= ε, then
Gk G−k = qT k (−1).
Lemma 2.3. ([9, Lemma 2.2]). For all α ∈ F×
q ,
q−2
1 X
G−m T m (α).
θ(α) =
q − 1 m=0
Theorem 2.4. ([7, Davenport-Hasse Relation, Thm 11.3.5]). Let k be a positive
×
integer and let q = pr be a prime power such that q ≡ 1 (mod k). For χ, ψ ∈ Fc
q ,
Y
Y
k
−k
G(χψ) = −G(ψ )ψ(k )
G(χ).
χk =ε
χk =ε
×
2.2. p-adic gamma function and Gross-Koblitz formula. Let ω : F×
q → Zq
×
be the Teichmüller character. For a ∈ Fq , the value ω(a) is just the (q − 1)-th root
of unity in Zq such that ω(a) ≡ a (mod p). We note that ω|F×
is the Teichmüller
p
×
character on F× with values in Z× . Also, Fc
= {ω j : 0 ≤ j ≤ q − 2}.
p
p
q
We now recall the p-adic gamma function. For further details, see [14]. The
p-adic gamma function Γp is defined by setting Γp (0) = 1, and for n ∈ Z+ by
Y
Γp (n) := (−1)n
j.
0<j<n
p-j
IDENTITIES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
7
If x, y ∈ Z+ and x ≡ y (mod pk Z), then Γp (x) ≡ Γp (y) (mod pk Z). Therefore, the
function has a unique extension to a continuous function Γp : Zp → Z×
p . If x ∈ Zp
and x 6= 0, then Γp (x) is defined as
Γp (x) := lim Γp (xn ),
xn →x
where xn runs through any sequence of positive integers p-adically approaching x.
The following lemma will be used in the proofs of our main results.
Lemma 2.5. ([5, Lemma 3.1]). Let p be a prime and q = pr . For 0 ≤ j ≤ q − 2
and t ∈ Z+ with p - t, we have
r−1
i
r−1
Y tpi j t−1
Y
Y t−1
Y
pi j
hpi
ph
tj
ω(t )
i
i =
+
i
Γp h
Γp h
Γp h
q−1
t
t
q−1
i=0
i=0
h=1
h=0
and
ω(t−tj )
i
t−1
r−1
Y
Y
Y t−1
p (1 + h)
hpi
pi j
−tpi j
Γp h
Γp h
i
i =
−
i .
Γp h
q−1
t
t
q−1
i=0
i=0
r−1
Y
h=0
h=1
The Gross-Koblitz formula, which is given below, allows us to relate Gauss sums
and the p-adic gamma function. Let π ∈ Cp be the fixed root of xp−1 + p = 0 which
satisfies π ≡ ζp − 1 (mod (ζp − 1)2 ). Recall that ω denotes the character inverse of
the Teichmüller character ω.
Theorem 2.6. ([12, Gross-Koblitz]). For a ∈ Z and q = pr ,
r−1
P
Y api api
(p−1) r−1
h q−1
i
a
i=0
G(ω ) = −π
Γp h
i .
q−1
i=0
3. Proof of the results
Lemma 3.1. Let p be an odd prime and q = pr . Let d ≥ 4 be even and p - d(d − 1).
For 1 ≤ m ≤ q − 2 such that m 6= q−1
2 , 0 ≤ i ≤ r − 1 we have
−2mpi
mdpi
−m(d − 1)pi
−mpi
c+b
c+b
c−b
c+1
q−1
q−1
q−1
q−1
d−2
d−1
X
X
hpi
mpi
−hpi
mpi
=
bh
i−
c+
bh
i+
c.
d−1
q−1
d
q−1
b
(3.1)
h=1
h=1
h6= d
2
i
Proof. We express b md(d−1)p
c as d(d − 1)u + v for some u, v ∈ Z, where 0 ≤ v <
q−1
d(d − 1). By considering the cases v = 0, 1, . . . , d(d − 1) − 1 separately, we verify
i
mdpi
(3.1). For example, when v = 0 we deduce that b −2mp
q−1 c = −2u − 1, b q−1 c = du,
i
i
b −m(d−1)p
c = −(d − 1)u − 1 and b −mp
q−1
q−1 c = −u − 1. Substituting all these values
we find that the left hand side of (3.1) is equal to zero. Since p - d we have
(3.2)
d−1
X
h=1
h6= d
2
bh
d−1
X
−hpi
mpi
h
mpi
i+
c=
bh i +
c = (d − 2)u.
d
q−1
d
q−1
h=1
h6= d
2
8
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
Also, p - (d − 1) and hence
d−2
X
(3.3)
d−2
X
hpi
mpi
mpi
h
bh
i−
c=
i−
c = −(d − 2)u.
bh
d−1
q−1
d−1
q−1
h=1
h=1
Substituting for (3.2) and (3.3), we see that the right hand side of (3.1) is also
equal to zero, when v = 0. Similarly we check (3.1) for the other values of v. This
completes the proof of the lemma.
Lemma 3.2. Let p be an odd prime and q = pr . Let d ≥ 3 be odd and p - d(d − 1).
For 1 ≤ m ≤ q − 2 and 0 ≤ i ≤ r − 1 we have
−2mpi
mdpi
−m(d − 1)pi
−mpi
c+b
c+b
c−b
c+1
q−1
q−1
q−1
q−1
d−1
d−2
X
X
mpi
pi
mpi
−hpi
mpi
hpi
i−
c + bh i −
c+
bh
i+
c.
=
bh
d−1
q−1
2
q−1
d
q−1
b
h=1
h=1
Proof. The proof is similar to that of Lemma 3.1.
Lemma 3.3. Let p be an odd prime and q = pr . Let d ≥ 3 be odd and p - d(d − 1).
For 0 ≤ m ≤ q − 2 such that m 6= q−1
2 , 0 ≤ i ≤ r − 1 we have
−2mpi
2mdpi
−2m(d − 1)pi
−mpi
mdpi
−m(d − 1)pi
c+b
c+b
c−b
c−b
c−b
c
q−1
q−1
q−1
q−1
q−1
q−1
2d−1
2d−3
X −hpi
X
mpi
mpi
hpi
bh
i−
c+
i+
c.
=
bh
2(d − 1)
q−1
2d
q−1
b
h=1
h odd
h6=d
h=1
h odd
Proof. The proof is similar to that of Lemma 3.1.
Lemma 3.4. For 0 < m ≤ q − 2 we have
r−1
Y
(3.4)
Γp (h(1 −
i=0
m
mpi
)pi i)Γp (h
i) = (−1)r ω m (−1).
q−1
q−1
For 0 ≤ m ≤ q − 2 such that m 6=
r−1
Y
(3.5)
Γp (h( 12 −
i=0
q−1
2
we have
m
1
m
i
i
q−1 )p i)Γp (h( 2 + q−1 )p i)
i
i
Γp (h p2 i)Γp (h p2 i)
= ω m (−1).
Proof. Consider
Im =
=
r−1
Y
r−1
Y
m
mpi
−mpi
mpi
)pi i)Γp (h
i) =
Γp (h
i)Γp (h
i)
q−1
q−1
q−1
q−1
i=0
i=0
Pr−1 −mpi Q
P
mpi Q
i
r−1
r−1
mpi
(p−1) r−1
i=0 h q−1 i
Γ
π (p−1) i=0 h q−1 i i=0 Γp h −mp
i
π
h
i
i=0 p
q−1
q−1
Γp (h(1 −
.
Pr−1 −mpi
mpi
π (p−1) i=0 {h q−1 i+h q−1 i}
Now, applying Gross-Koblitz formula (Theorem 2.6), Lemma 2.2 and the fact that
i
mpi
h −mp
q−1 i + h q−1 i = 1, we obtain
Im =
G(ω
−m
)G(ω
(−p)r
m
)
=
q · ω m (−1)
= (−1)r ω m (−1).
(−1)r · q
IDENTITIES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
9
This completes the proof of (3.4). If m = 0 then clearly (3.5) is true. For m 6=
and using Lemma 2.5, we have
Jm =
r−1
Y
m
1
m
i
i
q−1 )p i)Γp (h( 2 + q−1 )p i)
i
i
Γp (h p2 i)Γp (h p2 i)
Γp (h( 21 −
i=0
=
i
i
2mp
Γp (h −2mp
q−1 i)Γp (h q−1 i)
r−1
Y
i=0
q−1
2 ,
Γp (h(1 −
mpi
m
i
q−1 )p i)Γp (h q−1 i)
.
Using (3.4), Gross-Koblitz formula (Theorem 2.6), Lemma 2.2, and the fact that
i
2mpi
h −2mp
q−1 i + h q−1 i = 1, we have
P
Pr−1 −2mpi Q
2mpi Q
i
r−1
r−1
2mpi
(p−1) r−1
i=0 h q−1 i
π (p−1) i=0 h q−1 i i=0 Γp h −2mp
i=0 Γp h q−1 i
q−1 i π
Jm =
Pr−1 −2mpi
2mpi
(−1)r ω m (−1)π (p−1) i=0 {h q−1 i+h q−1 i}
q ω 2m (−1)
G(ω −2m )G(ω 2m )
=
= ω m (−1).
=
m
qω (−1)
q ω m (−1)
Lemma 3.5. ([7, Lemma 10.4.1]) Let γ ∈ F×
q , and let k be a positive integer. Let
χ be a character on Fq of order d = gcd(k, q − 1). Then the number of solutions
x ∈ Fq of xk = γ is
N (xk = γ) =
d−1
X
χj (γ).
j=0
To prove Theorem 1.2 and Theorem 1.3, we will first express the number of
points on certain families of hyperelliptic curves over Fq in terms of the G-function.
For d ≥ 2 and a, b 6= 0, we consider the hyperelliptic curves Ed and Ed0 over Fq
given by
Ed : y 2 = xd + ax + b
and
Ed0 : y 2 = xd + axd−1 + b,
respectively. Let Nd and Nd0 denote the number of Fq -points on the curves Ed and
Ed0 , respectively. We now give explicit expressions for Nd and Nd0 in terms of the
G-function in the following theorems.
Theorem 3.6. Let d ≥ 4 be even, and let p be an odd prime such that p - d(d − 1).
Then
Nd = q − 1 − q
× d−2 Gd−2
1
d−1 ,
1
d,
2
d−1 ,
2
d,
...,
...,
d−2
2(d−1) ,
d−2
2d ,
d
2(d−1) ,
d+2
2d ,
...,
...,
where f is defined as in Theorem 1.2.
Proof. Let Ed (x, y) = xd + ax + b − y 2 . Using the identity
X
q, if Ed (x, y) = 0;
θ(zEd (x, y)) =
0, if Ed (x, y) 6= 0,
z∈Fq
d−2
d−1
d−1
d
|f (0) ,
q
10
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
we obtain
q · Nd =
X
θ(zEd (x, y))
x,y,z∈Fq
X
= q2 +
θ(zb) +
z∈F×
q
+
X
X
θ(bz)θ(−zy 2 ) +
y,z∈F×
q
X
θ(zxd )θ(zax)θ(zb)
x,z∈F×
q
θ(xd z)θ(axz)θ(bz)θ(−zy 2 )
x,y,z∈F×
q
= q 2 + A + B + C + D.
(3.6)
From (2.1), we find that A = −1. Applying Lemma 2.3 we have
X
B=
θ(bz)θ(−zy 2 )
y,z∈F×
q
q−2
X
X
X
1
m
l
2l
=
G
G
T
(b)T
(−1)
T
(y)
T l+m (z).
−m
−l
(q − 1)2
×
×
l,m=0
y∈Fq
z∈Fq
We now apply Lemma 2.1 to the inner sums on the right, which gives non zero sums
only if 2l ≡ 0 (mod q − 1) and l + m ≡ 0 (mod q − 1). Hence l = 0 or l = q−1
2 ; and
m = 0 or m = q−1
,
respectively.
Finally,
using
Lemma
2.2,
we
get
B
=
1
+
qφ(b).
2
Similarly,
X
D=
θ(xd z)θ(axz)θ(bz)θ(−zy 2 )
x,y,z∈F×
q
q−2
X
1
=
G−m G−l G−n G−k T l (a)T n (b)T k (−1)
(q − 1)4
l,m,n,k=0
X
X
X
l+md
T l+m+n+k (z).
T 2k (y)
T
(x)
×
(3.7)
x∈F×
q
y∈F×
q
z∈F×
q
The inner sums are non zero only if l + md ≡ 0 (mod q − 1), 2k ≡ 0 (mod q − 1),
and l + m + n + k ≡ 0 (mod q − 1). This implies that l ≡ −md (mod q − 1), k = 0
q−1
or k = q−1
(mod q − 1),
2 ; and n ≡ m(d − 1) (mod q − 1) or n ≡ m(d − 1) + 2
q−1
respectively. Accounting for these values in (3.7), noting that T
= ε, we obtain
D=
q−2
1 X
G−m Gmd G−m(d−1) G0 T −md (a)T m(d−1) (b)
q − 1 m=0
+
=
q−2
q−1
q−1
1 X
G−m Gmd G−m(d−1)+ q−1 G q−1 T −md (a)T m(d−1)+ 2 (b)T 2 (−1)
2
2
q − 1 m=0
q−2
−1 X
G−m Gmd G−m(d−1) T −md (a)T m(d−1) (b)
q − 1 m=0
q−2
φ(−b) X
+
G−m Gmd G−m(d−1)+ q−1 G q−1 T −md (a)T m(d−1) (b).
2
2
q − 1 m=0
Expanding C in a similar fashion, using Lemma 2.3, it follows that the first term
of the last expression for D will be equal to −C. Now substituting the expressions
IDENTITIES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
11
for A, B, C, and D in (3.6) we have
d−1 q−2
φ(−b) X
b
m
q · Nd = q + qφ(b) +
.
G−m Gmd G−m(d−1)+ q−1 G q−1 T
2
2
q − 1 m=0
ad
d−1 q−2
φ(−1) X
b
2
m
q−1
q−1
= q + qφ(b) +
(3.8)
G
Gmd G−m(d−1) G
T
,
2
q − 1 m=0 −m+ 2
ad
2
where we have replaced m by m − q−1
2 in the last sum. Using Davenport-Hasse
relation (Theorem 2.4) for k = 2 and ψ = T −m we deduce that
G−m+ q−1 =
(3.9)
G q−1 G−2m T m (4)
2
G−m
2
.
Substituting (3.9) into (3.8) and using Lemma 2.2 yields
q
qφ(b)
+
q−1
q−1
d−1 q−2
X G−2m Gmd G−m(d−1)
q
4b
+
Tm
.
q − 1 m=1
G−m
ad
q · Nd = q 2 + qφ(b) +
m6= q−1
2
Now we take T to be the inverse of the Teichmüller character, that is, T = ω, and
then using Gross-Koblitz formula (Theorem 2.6) we deduce that
d−1 q−2
X
4b
q
qφ(b)
q
(p−1)s m
ω
q · Nd = q + qφ(b) +
+
+
π
q−1
q−1
q − 1 m=1
ad
2
m6= q−1
2
×
i
i
i
r−1
Y
−m(d−1)p
mdp
i)
Γp (h −2mp
q−1 i)Γp (h q−1 i)Γp (h
q−1
i=0
Γp (h −mp
q−1 i)
i
,
o
Pr−1 n
i
−m(d−1)pi
mdpi
−mpi
where s = i=0 h −2mp
i
+
h
i
+
h
i
−
h
i
. Using Lemma 2.5,
q−1
q−1
q−1
q−1
canceling q from both the sides, and rearranging the terms, we have
Nd = q + φ(b) +
q−2
X
1
φ(b)
1
bd−1 dd
+
+
π (p−1)s ω m
q − 1 q − 1 q − 1 m=1
ad (d − 1)d−1
m6= q−1
2
m
Γp (h( 21 − q−1
)pi i)Γp (h( 21 +
mpi
m
i
×
Γp (h
i)Γp (h(1 −
)p i)
i
i
q−1
q−1
Γp (h p2 i)Γp (h p2 i)
i=0
Q
1+h
m
m
r−1
d−1
i
Y d−3
Y Γp (h( hd + q−1
)pi i)
h=0 Γp (h( d−1 − q−1 )p i)
×
.
Qd−2
i
hpi
Γp (h hpd i)
h=1 Γp (h d−1 i)
i=0
h=1
r−1
Y
h6= d
2
m
i
q−1 )p i)
12
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
Applying Lemma 3.4 we then get
q−2
X
1
φ(b)
1
bd−1 dd
r (p−1)s m
Nd = q + φ(b)+
(−1) π
+
+
ω
q − 1 q − 1 q − 1 m=1
ad (d − 1)d−1
m6= q−1
2
×
(3.10)
h
−
Γp (h( d−1
r−1
Y d−2
Y
m
d−1
i
Y
q−1 )p i)
m
i
q−1 )p i)
.
i
Γp (h hpd i)
Γp (h( hd +
i
hp
Γp (h d−1
i)
i=0 h=1
h=1
h6= d
2
o
Pr−1 n −mpi
−m(d−1)pi
−2mpi
mdpi
c
−
b
c
−
b
c
−
b
c
,
b
i=0
q−1
q−1
q−1
q−1
Simplifying the term s we obtain s =
which is an integer. Plugging this expression in (3.10) we have
q−2
X
1
φ(b)
q
bd−1 dd
m
Nd = q + φ(b) +
+
+
ω
q − 1 q − 1 q − 1 m=1
ad (d − 1)d−1
m6= q−1
2
Pr−1
−m(d−1)pi
−mpi
−2mpi
mdpi
b q−1 c−b q−1 c−b q−1 c−b
c−1
i=0
q−1
× (−p)
×
r−1
Y d−2
Y
h
Γp (h( d−1
−
Γp (h( hd +
m
d−1
i
Y
q−1 )p i)
m
i
q−1 )p i)
.
i
Γp (h hpd i)
i
hp
Γp (h d−1
i)
i=0 h=1
h=1
h6= d
2
Now using Lemma 3.1 we obtain
Nd = q + φ(b) +
q−2
X
φ(b)
q
1
bd−1 dd
+
+
ωm
q − 1 q − 1 q − 1 m=1
ad (d − 1)d−1
m6= q−1
2
×
r−1
Y
−
Pd−2
(−p)
hpi
mpi
h=1 bh d−1 i− q−1
P
c+ d−1
i
h=1, h6= d
2
i
bh −hp
i+ mp
d
q−1 c
i=0
×
(3.11)
r−1
Y
Y d−2
h
Γp (h( d−1
−
m
d−1
i
Y
q−1 )p i)
m
i
q−1 )p i)
.
i
Γp (h hpd i)
Γp (h( hd +
i
hp
Γp (h d−1
i)
i=0 h=1
h=1
h6= d
2
Now for m = 0 we have the following identities:
d−2
X
bh
h=1
hpi
mpi
i−
c+
d−1
q−1
d−1
X
bh
h=1, h6= d
2
−hpi
mpi
i+
c=0
d
q−1
and
ωm
bd−1 dd
ad (d − 1)d−1
Also for m =
q−1
2
d−2
X
h
−
Γp (h( d−1
m
d−1
i
Y
q−1 )p i)
i
i=0 h=1
hp
Γp (h d−1
i)
h=1
h6= d
2
m
i
q−1 )p i)
i
Γp (h hpd i)
Γp (h( hd +
we have
bh
h=1
r−1
Y d−2
Y
hpi
mpi
i−
c+
d−1
q−1
d−1
X
h=1, h6= d
2
bh
−hpi
mpi
i+
c = 0,
d
q−1
= 1.
IDENTITIES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
13
and by Lemma 2.5 we have
r−1
m
m
h
Y Γp (h( hd + q−1
Y d−2
Y Γp (h( d−1
− q−1
)pi i) d−1
)pi i)
bd−1 dd
m
ω
= φ(b).
i
i
ad (d − 1)d−1 i=0
Γp (h hp i)
Γp (h hp i)
h=1
h=1
h6= d
2
d−1
d
Using all these four identities in (3.11) we deduce that
q−2
q X m
bd−1 dd
Nd = q − 1 +
ω
q − 1 m=0
ad (d − 1)d−1
×
r−1
Y
(−p)
Pd−2 hpi
Pd−1
mpi
−
h=1 bh d−1 i− q−1 c+
i
h=1, h6= d
2
i
bh −hp
i+ mp
d
q−1 c
i=0
×
r−1
Y d−2
Y
h
−
Γp (h( d−1
m
d−1
i
Y
q−1 )p i)
Γp (h( hd +
hpi
Γp (h d−1 i)
i=0 h=1
=q−1−q
× d−2 Gd−2
1
d−1 ,
1
d,
h=1
h6= d
2
2
d−1 ,
2
d,
i
h=1
h6= d
2
d−2
2(d−1) ,
d−2
2d ,
...,
...,
where we have used the fact that
m
d−1
Y Γp (h( hd + q−1
)pi i)
Γp (h hpd i)
m
i
q−1 )p i)
i
Γp (h hpd i)
=
d−1
Y
h=1
h6= d
2
d
2(d−1) ,
d+2
2d ,
...,
...,
d−2
d−1
d−1
d
|f (0) ,
q
Γp (h( −h
d +
m
i
q−1 )p i)
.
i
Γp (h −hp
d i)
Theorem 3.7. Let d ≥ 3 be odd, and let p be an odd prime such that p - d(d − 1).
Then
Nd = q − qφ(−ab)
1
,
× d−1 Gd−1 2(d−1)
1
,
2d
3
2(d−1) ,
3
2d ,
d−2
2(d−1) ,
d−2
2d ,
...,
...,
d
2(d−1) ,
d+2
2d ,
...,
...,
2d−3
2(d−1)
2d−1
2d
| − f (0) ,
q
where f is defined as in Theorem 1.2.
Proof. Following the proof of Theorem 3.6 we have
q−2
φ(−ab) X
q−1 G
q−1 G
q−1
q · Nd = q + qφ(b) +
G
q − 1 m=0 −m+ 2 md+ 2 −m(d−1)+ 2
d−1 b
(3.12)
× G q−1 T m
.
2
ad
2
Using Davenport-Hasse relation (Theorem 2.4) for k = 2 and ψ = T −m , T md , and
T −m(d−1) , and Lemma 2.2, in (3.12) we obtain
q−2
φ(b)
qφ(−ab) X G−2m G2md G−2m(d−1) m bd−1
Nd = q + φ(b) +
+
T
.
q−1
q−1
G−m Gmd G−m(d−1)
ad
m=0
m6= q−1
2
14
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
Letting T = ω, and using the Gross-Koblitz formula (Theorem 2.6) we get that
q−2
φ(b)
qφ(−ab) X (p−1)s m bd−1
Nd = q + φ(b) +
ω
+
π
q−1
q−1
ad
m=0
m6= q−1
2
×
i
i
i
r−1
Y
−2m(d−1)p
2mdp
i)
Γp (h −2mp
q−1 i)Γp (h q−1 i)Γp (h
q−1
i=0
−m(d−1)p
mdp
Γp (h −mp
i)
q−1 i)Γp (h q−1 i)Γp (h
q−1
i
i
i
,
where
r−1 X
mdpi
−m(d − 1)pi
−mpi
c+b
c+b
c
b
q−1
q−1
q−1
i=0
r−1 X
−2mpi
2mdpi
−2m(d − 1)pi
−
b
c+b
c+b
c ,
q−1
q−1
q−1
i=0
s=
which is an integer. Using Lemma 2.5 and Lemma 3.4 we deduce that
q−2
qφ(−ab) X (p−1)s m
−bd−1 dd
φ(b)
+
π
ω
Nd = q + φ(b) +
q−1
q−1
ad (d − 1)d−1
m=0
m6= q−1
2
×
r−1
Y 2d−3
Y
h
−
Γp (h( 2(d−1)
m
i
q−1 )p i)
i
hp
Γp (h 2(d−1)
i)
i=0 h=1
h odd
×
2d−1
Y
Γp (h( −h
2d +
m
i
q−1 )p i
.
i
Γp (h −hp
2d i)
h=1
h odd
h6=d
We now calculate the term under summation for m = q−1
2 separately using Lemma
2.5 and Lemma 3.3, and deduce the required expression.
Theorem 3.8. Let d ≥ 4 be even, and let p be an odd prime such that p - d(d − 1).
Then
Nd0 = q − 1 − qφ(b)
1
,
× d−2 Gd−2 d−1
1
d,
2
d−1 ,
2
d,
...,
...,
d−2
2(d−1) ,
d−2
2d ,
d
2(d−1) ,
d+2
2d ,
...,
...,
d−2
(d−1)
d−1
d
|g(0) ,
q
where g is defined as in Theorem 1.2.
Proof. The proof proceeds along similar lines to the proofs of Theorems 3.6 and 3.7
so we omit the details for reasons of brevity.
Theorem 3.9. Let d ≥ 3 be odd, and let p be an odd prime such that p - d(d − 1).
Then
Nd0 = q − qφ(b)
1
,
× d−1 Gd−1 d−1
1
,
d
2
d−1 ,
2
d,
...,
...,
d−1
2(d−1) ,
d−1
2d ,
where g is defined as in Theorem 1.2.
d+1
2(d−1) ,
d+1
2d ,
...,
...,
d−2
d−1 ,
d−2
d ,
1
2
d−1
d
| − g(0) ,
q
IDENTITIES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
15
Proof. The proof proceeds along similar lines to the proofs of Theorems 3.6 and
3.7 so we omit the details for reasons of brevity. However, we apply Lemma 3.2 to
deduce the final expression.
Remark 3.10. Putting d = 3 in Theorem 3.9 we can derive [5, Theorem 3.4].
Proof of Theorem 1.2: Consider the hyperelliptic curves Ed : y 2 = xd + ax + b
and Ed0 : y 2 = xd + axd−1 + b, where a, b 6= 0. We have
(3.13)
Nd = #{(x, y) ∈ F2q : xd + ax + b − y 2 = 0} =
X
N (xd + ax + b − y 2 = 0),
y∈Fq
d
2
where, for a given y, N (x + ax + b − y = 0) denotes the number of distinct zeros of
the polynomial xd + ax + b − y 2 . If b is not a square in Fq then the term b − y 2 6= 0
for all y ∈ Fq . Applying Theorem 1.7 we have
N (xd + ax + b − y 2 = 0) = 1 + φ(y 2 − b)
#
"
3
d−1
d+1
2d−3
1
2(d−1) ,
2(d−1) , . . . ,
2(d−1) ,
2(d−1) , . . . ,
2(d−1)
|f (y) ,
× d−1 Gd−1
d
d
1
d−1
2 −1
2 +1
0,
...,
...,
d,
d ,
d ,
d
q
d−1
2
)d
where f (y) = ad (b−y
. Now putting the value of N (xd + ax + b − y 2 = 0) in
a(d−1)
(3.13), and then applying Theorem 3.6 we easily derive the first summation identity.
To derive the second summation identity we consider the hyperelliptic curve Ed0 and
the proof is similar to that of the first summation identity. If b is not a square in
Fq , using Theorem 1.9 and Theorem 3.8√we derive the second summation identity.
If b is a square in Fq , then for y = ± b the term b − y 2 = 0. Hence
Nd = #{(x, y) ∈ F2q : xd + ax + b − y 2 = 0}
X
=
N (xd + ax + b − y 2 = 0) + 2 · N (x(xd−1 + a) = 0).
y∈Fq
√
y6=± b
Using Lemma 3.5 we have N (x(xd−1 + a) = 0) = 1 +
gcd(d − 1, q − 1) and χ is a character of order l. Thus,
(3.14)
Nd = 2 + 2 ·
l−1
X
j=0
χj (−a) +
X
Pl−1
j=0
χj (−a), where l =
N (xd + ax + b − y 2 = 0).
y∈Fq
√
y6=± b
Now applying Theorem 1.7 and Theorem 3.6 in (3.14), we deduce the third summation identity. Again, if b is a square then
Nd0 = #{(x, y) ∈ F2q : xd + axd−1 + b − y 2 = 0}
X
=
N (xd + axd−1 + b − y 2 = 0) + 2 · N (xd−1 (x + a) = 0)
y∈Fq
√
y6=± b
(3.15)
=
X
y∈Fq
√
y6=± b
N (xd + axd−1 + b − y 2 = 0) + 4.
16
RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
Now applying Theorem 1.9 and Theorem 3.8 in (3.15) we derive the fourth summation identity of the theorem. This completes the proof of the theorem.
Proof of Theorem 1.3: Here d is odd. Following the proof of Theorem 1.2
and applying Theorem 3.7, Theorem 3.9, Theorem 1.8, and Theorem 1.10, we can
derive all the four summation identities. This completes the proof of the theorem.
Proof of Theorem 1.5: In [19, Theorem 1.2], the third author gave a formula
for the number of points on the elliptic curve y 2 = x3 + ax + b as a special value of
2 G2 [· · · ]p when the j-invariant of the curve is different from 0 and 1728. We have
verified that the result is also true for Fq . Thus, from [19, Theorem 1.2] we have
1
, 34 −27b2
2
2
3
4
,
#{(x, y) ∈ Fq : y = x + ax + b} = q − φ(b) · q · 2 G2 1
2 |
4a3 q
3,
3
−27b2
6= 1. Now taking d = 3 in Theorem 3.7, we have
4a3
1
, 34 −27b2
#{(x, y) ∈ F2q : y 2 = x3 + ax + b} = q − q · φ(−ab) · 2 G2 41
|
,
5
4a3 q
6,
6
where a, b 6= 0 and
where a, b 6= 0. Comparing both the identities completes the proof.
4. special values of n Gn [· · · ] for n = 2, 3, 4
Finding special values of hypergeometric function is an important and interesting
problem. Many special values of hypergeometric functions over finite fields are
obtained (see for example [1, 3, 8, 20]). These results can be re-written in terms
of 2 G2 [· · · ] and 3 G3 [· · · ]. However, no special value of n Gn [· · · ] is obtained in full
generality to date. In [4], the first and second author expressed the number of
distinct zeros of the polynomials xd + ax + b and xd + axd−1 + b over Fq in terms of
values of the function d−1 Gd−1 [· · · ]. We now look at those expressions more closely
and derive certain special values of the function n Gn [· · · ] when n = 2, 3, 4.
Theorem 4.1. Let a, b, c ∈ F×
q be such that a + b + c = 0 and ab + bc + ca 6= 0.
Then, for p ≥ 5, we have
27a2 b2 c2
0, 21
G
(4.1)
= A · φ(−(ab + bc + ca)),
|
−
2 2
1
5
4(ab + bc + ca)3 q
6,
6,
where A = 2 if all of a, b, c are distinct and A = 1 if exactly two of a, b, c are equal.
If a, b, c ∈ F×
q are such that ab + bc + ca = 0 and a + b + c 6= 0. Then, for p ≥ 5,
we have
27abc
0, 12
G
(4.2)
|
−
= A · φ(−abc(a + b + c)).
2 2
1
5
4(a + b + c)3 q
6,
6,
Proof. We have (x − a)(x − b)(x − c) = x3 − (a + b + c)x2 + (ab + bc + ca)x − abc.
Now if a + b + c = 0 and ab + bc + ca 6= 0, then putting d = 3 in Theorem 1.8, we
can easily deduce (4.1). Similarly, (4.2) follows from Theorem 1.10.
Example 4.2. Put a = b = 1 and c = −2 in (4.1), then for all p ≥ 5, we have
0, 21
|1
= φ(3).
2 G2
1
5
6,
6,
q
IDENTITIES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING
17
Theorem 4.3. If p ≥ 5 then we have
1
, 12 , 56
6
|1 = φ(−3) + φ(6).
3 G3
0, 14 , 34
q
Proof. We have
x4 −
3a4
a
3a2
a3
x+
= (x − )2 (x2 + ax +
).
2
16
2
4
Hence
a3
3a4
3a2
x+
= 0) = 1 + N (x2 + ax +
= 0) = 2 + φ(−2).
2
16
4
Now applying Theorem 1.7 on the left side of (4.3) we obtain the result.
(4.3)
N (x4 −
Theorem 4.4. If p > 7 and p 6= 23, then
1
3
1
55
,
,
0,
0,
4
2
4
= φ(−1) + φ(3) + φ(−1) · 2 G2 1
4 G4
1
3
7
9 |− 4
,
,
,
4
10
10
10
10
6,
q
1
2
5
6,
27
|
4
.
q
Proof. We have x5 +ax4 +a5 = (x3 −a2 x+a3 )(x2 +ax+a2 ). Let f (x) = x5 +ax4 +a5 .
Then f 0 (x) = 5x4 + 4ax3 . If f (x) has a repeated zero, say c, then f (c) = 0 and
f 0 (c) = 0. Solving these two equations, we have 3381 = 3.72 .23 = 0 in Fq . Hence,
if p 6= 3, 7, 23, we have
N (x5 + ax4 + a5 = 0) = N (x3 − a2 x + a3 = 0) + N (x2 + ax + a2 = 0).
Now the result easily follows from Theorem 1.10 and Theorem 1.8.
5. concluding remarks
The technique used to derive the summation identities for n Gn [· · · ]q in this paper
and for 2 G2 [· · · ]q in [5, 6] is based on counting points on families of certain algebraic
varieties and counting zeros on certain families of polynomials. This technique is
quite involved. Also, in finding the special values of the function n Gn [· · · ] when
n = 2, 3, 4, we factored the polynomials xd + ax + b and xd + axd−1 + b into
polynomials of the same form of lower degree when d = 5, 4, 3. However, such
factorizations do not exist when d > 5. Hence, our method can’t be applied to
deduce special values of n Gn [· · · ] when n ≥ 5. Therefore, it would be very beneficial
if we could derive these summation identities for n Gn [· · · ] and special values more
directly using properties of p-adic gamma function.
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RUPAM BARMAN, NEELAM SAIKIA, AND DERMOT McCARTHY
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Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi110016, INDIA
E-mail address: rupam@maths.iitd.ac.in
Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi110016, INDIA
E-mail address: nlmsaikia1@gmail.com
Department of Mathematics & Statistics, Texas Tech University, Lubbock, TX 794091042, USA
E-mail address: dermot.mccarthy@ttu.edu