SOME CONGRUENCES ONq-FRANEL NUMBERS ANDq-CATALAN NUMBERS
Hongcun Zhai∗, Bing He∗∗,∗∗∗∗and Long Li∗∗∗,∗∗∗∗
∗Department of Mathematics, Luoyang Normal University, Luoyang 471934,
People’s Republic of China
∗∗School of Mathematics and Statistics, Central South University, Changsha 410083,
Hunan, People’s Republic of China
∗∗∗School of Mathematical Sciences, Huaiyin Normal University, Huai’an, Jiangsu 223300,
People’s Republic of China
∗∗∗∗Department of Mathematics, East China Normal University, 500 Dongchuan Road,
Shanghai 200241, People’s Republic of China
e-mails: zhai−hc@163.com; yuhe001@foxmail.com; yuhelingyun@foxmail.com;
lilong6820@126.com
(Received 11 May 2017; after final revision 21 March 2018;
accepted 11 April 2018)
We present certainq-analogues of Franel numbers and Catalan numbers, and establish several congruences on theseq-numbers.
Key words :q-Franel numbers;q-Catalan numbers;q-congruence.
1. INTRODUCTION
Theq-analogue of a positive integerncan be defined by
[n]q= 1−q n
1−q .
It is clear thatlim
q→1[n]q =n.Supposing thata≡b (modn),we have
[a]q = 1−qa 1−q =
1−qb+qb(1−qa−b)
1−q ≡
1−qb
The Gaussian binomial coefficients, also called theq-binomial coefficients, are q-analogues of the
binomial coefficients, which are given by
·
n k
¸
q
=
( (q;q)
n
(q;q)k(q;q)n−k, if0≤k≤n,
0, otherwise,
where(a;q)kis defined by
(a;q)0 = 1, (a;q)k=
k−1Y
j=0
(1−aqj)fork≥1.
Franel in [3] noted that the numbers
fn= n
X
k=0
µ
n k
¶3
satisfy the recurrence relation:
(n+ 1)2fn+1 = (7n2+ 7n+ 2)fn+ 8n2fn−1
for n = 1,2, . . . .Now such numbers are usually called Franel numbers. The generalized Franel
numbers are defined by
fn(r)=
n
X
k=0
µ
n k
¶r
.
Then fn(3) = fn are the Franel numbers. In [2, Proposition 3]C Calkin proved the following
congruence:
fn(2r) ≡0 (mod p),
wherepis a prime such that mn < p < n+1m +m(2mr−1)n+1−m for some positive integerm.Guo and Zeng [5, Theorem 4.4] established that, for any positive integern,
fn(2r)≡0 (mod n+ 1).
It was Sun [10] who initiated the systematic investigation of fundamental congruences for the
Franel numbers. In particular, he obtained the following congruence: for any primep >3,
fp−1 ≡1 + 3pqp(2) + 3p2qp(2)2 (modp3), (1.1)
whereqp(2) = 2 p−1−1
In this paper, we will establish aq-analogue of (1.1) in the modulusp2.We define theq-analogues
of the generalized Franel numbers as
fn(2r+1)(q) :=
n
X
k=0
·
n k
¸2r+1
q
q(2r+1)(k+12 ), f(2r) n (q) :=
n
X
k=0
·
n k
¸2r
q
q2r(k+12 )+k.
Theorem 1.1 — Letpbe an odd prime andra nonnegative integer. Then
fp−1(2r)(q)≡(1−2r+ 2rp)[p]q (mod [p]2q),
fp−1(2r+1)(q)≡1 + (2r+ 1)Qp(2, q)[p]q (mod [p]2q).
In [11] Sun proposed several conjectured congruences on the Catalan numbers. For example, let p >3be a prime, then
p−1 2
X
k=0
CkCk+1
16k ≡8 (mod p2) (1.2)
and ifp≡1 (mod 4),then
p−1
X
k=0
¡2k
k
¢¡2k
k+1
¢
8k ≡0 (mod p). (1.3)
Here and below, the notationA ≡B (modpl)denotes that (A−B)/pl is ap-adic integer for A, B∈Q.The above congruences were confirmed by Zhang [12].
In this paper, we will give aq-analogue of (1.2) in the moduluspcase. Before stating our main
results, we need to mention the followingq-analogue of Catalan numbers:
Cn(q) =
£2n
n
¤
q
[n+ 1]q2
.
Theorem 1.2 — Letp≥5be a prime. Then
p−1 2
X
k=0
Ck(q)Ck+1(q)
(−q;q)2
k(−q2;q)2k
q4k+3≡(1 +q)3 (mod [p]q).
In addition, we set up a more generalq-analogue of (1.3).
Theorem 1.3 — Letpbe an odd prime andd∈ {0,1, . . . ,p−12 }.Then
p−1
X
k=0
£2k
k
¤
q
£2k
k+d
¤
q2(−q2(k+1);q2)p−1 2 −k
(−q;q)2 k
q2k+(p−1)(p4 −3)
≡
0 if p−12 6≡d (mod 2)
(−1)p−21q (p−1)2
4 −d2
£ p−1 2 p−1−2d
4
¤
q4 if p−1
2 ≡d (mod 2)
Throughout this article, we use the notationP(q)≡Q(q) (mod [p]q),whereP(q)andQ(q)are
rational functions inq,to denote that P(q)−Q(q)[p]q = B(q)A(q) for some polynomials A(q)andB(q) with rational coefficients andgcd(B(q),[p]q) = 1.
In the next section, we will provide some lemmas which are crucial in the proofs of Theorems
1.1-1.3. Section 3 is devoted to our proofs of Theorems 1.1-1.3.
2. SOMEAUXILIARYRESULTS
In order to prove Theorems 1.1-1.3, we need the following results.
Lemma 2.1 — Letpbe an odd prime. Then for any integerk∈ {1,2, . . . , p−1},we have
(−1)k
·
p−1 k
¸
q
q(k+12 ) ≡1−[p]qHk(q) + [p]2q
X
1≤i<j≤k
1
[i]q[j]q (mod [p] 3 q),
whereHk(q)is defined by
Hk(q) = k
X
j=1
1 [j]q.
PROOF: Fork∈ {1,2, . . . , p−1},
(−1)k
·
p−1 k
¸
q
q(k+12 ) = (q
p−q)(qp−q2)· · ·(qp−qk)
(q;q)k
= (1−[p]q)
µ
1−[p]q
[2]q
¶
· · ·
µ
1−[p]q
[k]q
¶
.
Then
(−1)k
·
p−1 k
¸
q
q(k+12 ) ≡1−[p]qHk(q) + [p]2q
X
1≤i<j≤k
1 [i]q[j]q
(mod [p]3q).
This proves Lemma 2.1. 2
The following result (see [9, Theorem 1.1] for a generalization) is also very important.
Lemma 2.2 — Letpbe an odd prime. Then
p−1 2
X
j=1
1 [2j]q
≡ −Qp(2, q) (mod [p]q),
Lemma 2.3 — (q-binomial Theorem, see [1]). Letnbe a positive integer andz, qtwo complex
numbers with|z|<1and|q|<1.Then
(z;q)n=
n
X
k=0
(−1)k
·
n k
¸
q
q(k2)zk.
Lemma 2.4 — (q-Lucas Theorem, see [8]). Letm, k, dbe positive integers withm=ad+band k=rd+s,where0≤b, s≤d−1.LetΦd(q)be thed-th cyclotomic polynomial inq.Then
·
m k
¸
q ≡
µ
a r
¶·
b s
¸
q
(mod Φd(q)).
We also need the following result which follows from direct computations.
Lemma 2.5 — Letpbe an odd prime. Then for0≤k≤(p−1)/2,
·
(p−1)/2 k
¸
q2
≡ (−1)kq−k 2
(−q;q)2 k
·
2k k
¸
q
(mod [p]q).
3. PROOF OFTHEOREMS1.1-1.3
PROOF OFTHEOREM1.1 : By Lemma 2.1, we have
fp−1(2r+1)(q)−1 =
p−1
X
k=1
·
p−1 k
¸2r+1
q
q(2r+1)(k+12 )
≡ p−1
X
k=1
(−1)k(1−[p]qHk(q))2r+1
≡ p−1
X
k=1
(−1)k(1−(2r+ 1)[p]qHk(q))
=−(2r+ 1)[p]q p−1
X
k=1
(−1)kHk(q) (mod [p]2q).
Since
p−1
X
k=1
(−1)kHk(q) =
p−1
X
k=1
(−1)k
k
X
j=1
1 [j]q
=
p−1
X
j=1
1 [j]q
p−1
X
k=j
(−1)k=
p−1 2
X
j=1
1 [2j]q
,
we use Lemma 2.2 to get
Similarly,
fp−1(2r)(q)−1 =
p−1
X
k=1
·
p−1 k
¸2r
q
q2r(k+12 )+k
≡ p−1
X
k=1
(1−[p]qHk(q))2rqk≡ p−1
X
k=1
(1−2r[p]qHk(q))qk
= q(1−qp−1)
1−q −2r[p]q
p−1
X
k=1
Hk(q)qk (mod [p]2q).
Since
p−1
X
k=1
Hk(q)qk=
p−1 X k=1 qk k X j=1 1 [j]q =
p−1 X j=1 1 [j]q p−1 X k=j qk = p−1 X j=1 1 [j]q
qj −qp
1−q ≡1−p (mod [p]q),
we see that
fp−1(2r)(q)≡(1−2r+ 2rp)[p]q (mod [p]2q).
This concludes the proof of Theorem 1.1. 2
PROOF OF THEOREM 1.2 : We denote[zl]f(z)as the coefficient ofzlin the polynomialf(z). By theq-binomial theorem, we have
(−z;q2)n+1 =
n+1
X
k=0
·
n+ 1 k
¸
q2
qk(k−1)zk,
(−zq2(n+1);q2)n+1 = n+1
X
k=0
·
n+ 1 k
¸
q2
qk(k−1)+2(n+1)kzk,
(−z;q2)2n+2 =
2n+2X
k=0
·
2n+ 2 k
¸
q2
qk(k−1)zk.
Then
·
2n+ 2 n
¸
q2
qn(n−1) = [zn](−z;q2)2n+2
= [zn]{(−z;q2)n+1·(−zq2(n+1);q2)n+1}
=
n
X
k=0
÷
n+ 1 k
¸
q2
qk(k−1)+2(n+1)k·
·
n+ 1 n−k
¸ q2 q(n−k)(n−k−1) ! = n X k=0 ·
n+ 1 k
¸
q2
·
n+ 1 n−k
¸
q2
namely,
·
2n+ 2 n ¸ q2 = n X k=0 ·
n+ 1 k
¸
q2
·
n+ 1 k+ 1
¸
q2
q2k2+2k. (3.1)
Below we setn= p−12 .By theq-Lucas theorem,
·
2n+ 2 n+ 1
¸
q
≡0 (mod [p]q) (3.2)
and ·
2n+ 2 n
¸
q2
≡0 (mod [p]q)forp≥5. (3.3)
It follows from (3.2) and Lemma 2.5 that
n
X
k=0
Ck(q)Ck+1(q)
(−q;q)2
k(−q2;q)2k
q4k+3 ≡ −(1 +q)2
n−1 X k=0 £n k ¤ q2 £ n k+1 ¤ q2
[k+ 1]q2[k+ 2]q2q
2k2+6k+4
=−(1 +q)2
[n+ 1]2 q2
n−1
X
k=0
·
n+ 1 k+ 1
¸
q2
·
n+ 1 k+ 2
¸
q2
q2k2+6k+4
≡ −(1 +q)4
n
X
k=1
·
n+ 1 k
¸
q2
·
n+ 1 k+ 1
¸
q2
q2k2+2k (mod [p]q).
From (3.1) and (3.3), we know that
n
X
k=1
·
n+ 1 k
¸
q2
·
n+ 1 k+ 1
¸
q2
q2k2+2k=
n
X
k=0
·
n+ 1 k
¸
q2
·
n+ 1 k+ 1
¸
q2
q2k2+2k−[n+ 1]q2
=
·
2n+ 2 n
¸
q2
−[n+ 1]q2
≡ − 1
1 +q (mod [p]q).
Hence
n
X
k=0
Ck(q)Ck+1(q) (−q;q)2
k(−q2;q)2k
q4k+3≡(1 +q)3 (mod [p]q).
This finishes the proof of Theorem 1.2. 2
PROOF OFTHEOREM 1.3 : By theq-Lucas theorem,
·
2k k
¸
q
Then by Lemma 2.5 and [6, (4.1)],
p−1
X
k=0
£2k
k
¤
q
£2k
k+d
¤
q2(−q2(k+1);q2)n−k
(−q;q)2 k
q2k+(p−1)(p4 −3)
≡ n
X
k=0
(−1)kq2(n−2k)
·
n k
¸
q2
·
2k k+d
¸
q2
(−q2(k+1);q2)n−k
=
0 ifn6≡d (mod 2)
(−1)nqn2−d2£ n n−d
2
¤
q4 ifn≡d (mod 2)
(mod [p]q).
This ends the proof of Theorem 1.3. 2
Remark 3.1 : Similarly, we can use the method which is used in the proof of Theorem 1.3 and the
identities (4.2), (4.7) and (4.8) in [6] to deduce another threeq-analogues of (1.3).
ACKNOWLEDGEMENT
The authors would like to thank the referee for his/her helpful comments. The first author was partially
supported by the National Natural Science Foundation of China (Grant No. 11371184) and the
Nat-ural Science Foundation of Henan Province (Grant No. 162300410086, 2016B259, 172102410069).
The second author was partially supported by the National Natural Science Foundation of China
(Grant No. 11801451) and the Natural Science Basic Research Plan in Shaanxi Province of China
(No. 2017JQ1001).
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