BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES

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BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES
ASSOCIATED TO SUPERCONGRUENCES
DERMOT McCARTHY
Abstract. We establish two binomial coefficient–generalized harmonic sum identities using
the partial fraction decomposition method. These identities are a key ingredient in the proofs
of numerous supercongruences. In particular, in other works of the author, they are used to
establish modulo pk (k > 1) congruences between truncated generalized hypergeometric series,
and a function which extends Greene’s hypergeometric function over finite fields to the p-adic
setting. A specialization of one of these congruences is used to prove an outstanding conjecture
of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the p-th
Fourier coefficient of a particular modular form.
1. Introduction and Statement of Results
(i)
For non-negative integers i and n, we define the generalized harmonic sum, Hn , by
n
X
1
(i)
Hn :=
ji
j=1
(i)
and H0 := 0. In [3] Chu proves the following binomial coefficient-generalized harmonic sum
identity using the partial fraction decomposition method. If n is a positive integer, then
n X
n+k 2 n 2
(1)
(1)
(1)
1 + 2kHn+k + 2kHn−k − 4kHk
= 0.
(1.1)
k
k
k=1
This identity had previously been established using the WZ method [1] and was used by Ahlgren
and Ono in proving the Apéry number supercongruence [2].
In [4], [5] the author establishes various supercongruences between truncated generalized
hypergeometric series, and a function which extends Greene’s hypergeometric function over
finite fields to the p-adic setting. Specifically, let p be an odd prime and let n ∈ Z+ . For
mi
i
1 ≤ i ≤ n + 1, let m
di ∈ Q ∩ Zp such that 0 < di < 1. Let Γp (·) denote Morita’s p-adic gamma
function, bxc denote the greatest integer less than or equal to x and hxi denote the fractional
part of x, i.e. x − bxc. Then define
mn+1
m1 m2
n+1 G d1 , d2 , . . . , dn+1
p
j p−2
i
n+1 n+1
m
Y Γp h m
−1 X −b i − j c
di − p−1 i
j j
:=
(−1) Γp p−1
(−p) di p−1 .
mi
p−1
Γp di
j=0
i=1
2010 Mathematics Subject Classification. Primary: 05A19; Secondary: 11B65.
This work was supported by the UCD Ad Astra Research Scholarship program.
1
2
DERMOT McCARTHY
Note that when p ≡ 1 (mod di ) this function recovers Greene’s hypergeometric function over
finite fields. For a complex number a and a non-negative integer n let (a)n denote the rising
factorial defined by
(a)0 := 1
and
(a)n := a(a + 1)(a + 2) · · · (a + n − 1) for n > 0.
Then, for complex numbers ai , bj and z, with none of the bj being negative integers or zero, we
define the truncated generalized hypergeometric series
"
r Fs
a1 , a2 , a3 , . . . , ar z
b1 , b2 , . . . , bs
#
:=
m
X
(a1 )n (a2 )n (a3 )n · · · (ar )n z n
.
(b1 )n (b2 )n · · · (bs )n
n!
n=0
m
An example of one the supercongruence results from [5] is the following theorem.
Theorem 1.1 ([5] Thm. 2.7). Let r, d ∈ Z such that 2 ≤ r ≤ d − 2 and gcd(r, d) = 1. Let
2
p be an odd primesuch that
≡ ±1
p d−1
(mod d) or p ≡ ±r (mod d) with r ≡ ±1 (mod d). If
1
r
d−r
s(p) := Γp d Γp d Γp d Γp d , then
4G
1 r
d, d, 1
−
r
d, 1
−
1
d p
"
≡ 4 F3
r
d,
1
d,
1,
1 − dr , 1 −
1,
1
1
d
#
1
+ s(p)p
(mod p3 ).
p−1
A specialization of this congruence is used to prove an outstanding supercongruence conjecture
of Rodriguez-Villegas, which relates a truncated generalized hypergeometric series to the p-th
Fourier coefficient of a particular modular form [4],[6]. Similar results to Theorem 1.1 exist for
4 G with other parameters, and also 2 G and 3 G.
The main results of the current paper, Theorems 1.2 and 1.3 below, are two binomial coefficient–
generalized harmonic sum identities which factor heavily into the proofs of all the 4 G congruences. Taking particular values for n, m, l, c1 and c2 in these identities allows the vanishing of
certain terms in the proofs. Note that letting m = n in Theorem 1.2 recovers (1.1).
Theorem 1.2. Let m, n be positive integers with m ≥ n. Then
n X
m+k
m
n+k
n
(1)
(1)
(1)
(1)
(1)
1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk
k
k
k
k
k=0
+
m
X
k−n
(−1)
k=n+1
m+k
k
m
k
n+k . k−1
= (−1)m+n .
k
n
BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES
3
Theorem 1.3. Let l, m, n be positive integers with l > m ≥ n ≥ 2l and c1 , c2 ∈ Q some
constants. Then
(
n X
m+k
m
n+k
n
(1)
(1)
(1)
(1)
(1)
1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk
k
k
k
k
k=0
· c1
+ c2
(1)
Hk+n
(2)
Hk+m
−
(1)
Hk+l−n−1
−
(2)
Hk+l−m−1
+
(1)
c2 Hk+m
)
+
m
X
−
(1)
Hk+l−m−1
k−n
(−1)
k=n+1
(2)
(2)
− k c1 Hk+n − Hk+l−n−1
m+k
k
m
k
n+k . k−1
k
n
(1)
(1)
(1)
(1)
· c1 Hk+n − Hk+l−n−1 + c2 Hk+m − Hk+l−m−1 = 0.
The remainder of this paper is spent proving Theorems 1.2 and 1.3.
2. Proofs
We first develop two algebraic identities of which the binomial coefficient–harmonic sum
identities are limiting cases.
Theorem 2.1. Let x be an indeterminate and let m, n positive integers with m ≥ n. Then
n X
m+k
m
n+k
n
k=0
k
k
(
·
+
k
−k
+
(x + k)2
k
(1)
(1)
(1)
(1)
(1) )
1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk
x+k
m
X
x(1 − x)n (1 − x)m
(−1)k−n m + k
m
n+k . k−1
=
.
x+k
k
k
k
n
(x)n+1 (x)m+1
(2.1)
k=n+1
Proof. Using partial fraction decomposition we can write
n m
X
x(1 − x)n (1 − x)m
A X
Bk
Ck
Dk
f (x) :=
= +
+
+
(x)n+1 (x)m+1
x
(x + k)2 x + k
x+k
k=1
k=n+1
for some A, Bk , Ck and Dk ∈ Q. We now isolate these coefficients by taking various limits of
f (x) as follows.
A = lim xf (x) = lim
x→0
x→0
(1 − x)n (1 − x)m
= 1.
(1 + x)n (1 + x)m
4
DERMOT McCARTHY
For 1 ≤ k ≤ n,
x(1 − x)n (1 − x)m
Bk = lim (x + k)2 f (x) = lim
x→−k (x)2 (x
k
x→−k
=
=
+ k + 1)n−k (x + k + 1)m−k
−k(k + 1)n (k + 1)m
(−k)2k (1)n−k (1)m−k
−k(k + 1)n (k + 1)m
(−1)2k k!2 (n − k)!(m − k)!
m+k
= −k
k
m
k
n+k
k
n
,
k
and, using L’Hôspital’s rule,
(x + k)2 f (x) − Bk
x→−k
x+k
Ck = lim
"
#
d
= lim
(x + k)2 f (x)
x→−k dx
"
#
x(1 − x)n (1 − x)m
d
= lim
x→−k dx (x)2 (x + k + 1)
k
n−k (x + k + 1)m−k
("
= lim
x→−k
+
m
X
(x)2k (x
+ k + 1)n−k (x + k + 1)m−k
−1
(−x + s)
+
s=1
"
=
#"
(1 − x)n (1 − x)m
n−k
X
−1
(x + k + s)
s=1
(1 + k)n (1 + k)m
(−k)2k (1)n−k (1)m−k
#"
1+k
+
m−k
X
1−x
−1
(x + k + s)
=
m+k
k
m
k
+2
s=1
n
X
s=1
(k + s)−1 +
k−1
X
!#)
−1
(x + s)
s=0
m
X
(k + s)−1 +
s=1
m−k
X
s=1
(−x + s)−1
s=1
+
n
X
n−k
X
(s)−1
s=1
k−1
X
(s)−1 + 2
!#
(−k + s)−1
s=0
n+k
n
(1)
(1)
(1)
(1)
(1)
1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk
.
k
k
BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES
5
Similarly, for n + 1 ≤ k ≤ m,
x(1 − x)n (1 − x)m
x→−k (x)n+1 (x)k (x + k + 1)m−k
Dk = lim (x + k)f (x) = lim
x→−k
=
−k(k + 1)n (k + 1)m
(−k)n+1 (−k)k (1)m−k
= (−1)k−n
m+k
k
m
k
n+k . k−1
.
k
n
Theorem 2.2. Let x be an indeterminate and let l, m, n be positive integers with l > m ≥ n ≥
and c1 , c2 ∈ Q some constants. Then
( n
X
1
m+k
m
n+k
n
(1)
(1)
c1 Hk+n − Hk+l−n−1
x+k
k
k
k
k
k=0
x
(1)
(1)
(1)
(1)
(1)
(1)
(1)
+ k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk
+ c2 Hk+m − Hk+l−m−1 ·
x+k
)
(2)
(2)
(2)
(2)
− k c1 Hk+n − Hk+l−n−1 + c2 Hk+m − Hk+l−m−1
+
l
2
m
X
(−1)k−n m + k
m
n+k . k−1
x+k
k
k
k
n
k=n+1
· c1
(1)
Hk+n
−
(1)
Hk+l−n−1
+ c2
(1)
Hk+m
−
(1)
Hk+l−m−1
"
#
n
m
X
X
x(1 − x)n (1 − x)m
=
c1
(−x + s)−1 + c2
(−x + s)−1 . (2.2)
(x)n+1 (x)m+1
s=l−n
s=l−m
Proof. Using partial fraction decomposition we can write
"
#
n
m
X
X
x(1 − x)n (1 − x)m
c1
(−x + s)−1 + c2
(−x + s)−1
f (x) : =
(x)n+1 (x)m+1
s=l−n
s=l−m
n m
X
A X
Bk
Ck
Dk
= +
+
+
2
x
(x + k)
x+k
x+k
k=1
k=n+1
for some A, Bk , Ck and Dk ∈ Q. As in the proof of Theorem 2.1, we isolate the coefficients
A, Bk , Ck and Dk by taking various limits of f (x). For brevity, we first let
6
DERMOT McCARTHY
n
X
Ta(r) := c1
(a + s)−r + c2
s=l−n
m
X
(a + s)−r
s=l−m
and
(r)
(r)
(r)
(r)
U (r) := c1 Hk+n − Hk+l−n−1 + c2 Hk+m − Hk+l−m−1 .
Then we have
n
X
A = lim xf (x) = c1 lim
x→0
x→0
s=l−n
n
X
= c1
m
X
(1 − x)n (1 − x)m
(1 − x)n (1 − x)m
+ c2 lim
x→0
(1 + x)n (1 + x)m (s − x)
(1 + x)n (1 + x)m (s − x)
s=l−m
m
X
s−1 + c2
s=l−n
s−1
s=l−m
(1)
(1)
(1)
= c1 Hn(1) − Hl−n−1 + c2 Hm
− Hl−m−1 .
For 1 ≤ k ≤ n,
x(1 − x)n (1 − x)m
Bk = lim (x + k)2 f (x) = lim
x→−k (x)2 (x
k
x→−k
=
(1)
+ k + 1)n−k (x + k + 1)m−k
−k(k + 1)n (k + 1)m
T−x
(1)
T
(−k)2k (1)n−k (1)m−k k
m+k
m
n+k
n
= −k
U (1)
k
k
k
k
and
"
#
d
2
Ck = lim
(x + k) f (x)
x→−k dx
"
#
x(1 − x)n (1 − x)m
d
(1)
= lim
T−x
x→−k dx (x)2 (x + k + 1)
k
n−k (x + k + 1)m−k
("
#"
(1 − x)n (1 − x)m
(2)
(1)
(1)
= lim
x T−x + T−x − x T−x
2
x→−k
(x)k (x + k + 1)n−k (x + k + 1)m−k
·
n
m
n−k
X
X
X
−1
−1
(−x + s) +
(−x + s) +
(x + k + s)−1
s=1
s=1
s=1
+
m−k
X
s=1
(x + k + s)
−1
+2
k−1
X
s=0
!#)
(x + s)
−1
BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES
"
=
(1 + k)n (1 + k)m
#"
(2)
(1)
−kTk + Tk
(−k)2k (1)n−k (1)m−k
1+k
+
(s)−1 +
s=1
=
n
m
X
X
(k + s)−1 +
(k + s)−1
s=1
n−k
X
m−k
X
7
s=1
(s)−1 + 2
s=1
k−1
X
!!#
(−k + s)−1
s=0
m+k
m
n+k
n
k
k
k
k
(1)
(1)
(1)
(1)
(1)
(2)
(1)
.
· −k U + 1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk
U
For n + 1 ≤ k ≤ m,
Dk = lim (x + k)f (x) = lim
x→−k
x→−k
=
x(1 − x)n (1 − x)m
(1)
T
(x)n+1 (x)k (x + k + 1)m−k −x
−k(k + 1)n (k + 1)m (1)
T
(−k)n+1 (−k)k (1)m−k k
k−n
= (−1)
U
(1)
m+k
k
m
k
n+k . k−1
.
k
n
Proofs of Theorems 1.2 and 1.3. Multiply both sides of (2.1) and (2.2) respectively by x and
take the limit as x → ∞.
References
[1] S. Ahlgren, S. B. Ekhad, K. Ono, D. Zeilberger, A binomial coefficient identity associated to a conjecture of
Beukers, Electron. J. Combin. 5 (1998), Research Paper 10, 1.
[2] S. Ahlgren, K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine
Angew. Math. 518 (2000), 187–212.
[3] W. Chu, A binomial coefficient identity associated with Beukers’ conjecture on Apéry numbers, Electron. J.
Combin. 11 (2004), no. 1, Note 15, 3.
[4] D. McCarthy, p-adic hypergeometric series and supercongruences, Ph.D thesis, University College Dublin,
2010.
[5] D. McCarthy, Extending Gaussian hypergeometric series to the p-adic setting, Int. J. Number Theory,
accepted for publication, 24 pages.
[6] D. McCarthy, On a supercongruence conjecture of Rodriguez-Villegas, Proc. Amer. Math. Soc. 140 (2012),
2241–2254.
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
E-mail address: [email protected]
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