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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(1qa−b)

1−q

1−qb

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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

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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)(12r+ 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+(p1)(p4 3)

 

0 if p−12 6≡d (mod 2)

(1)p−21q (p1)2

4 −d2

£ p−1 2 p−12d

4

¤

q4 if p−1

2 ≡d (mod 2)

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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, . . . , p1},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, . . . , p1},

(1)k

·

p−1 k

¸

q

q(k+12 ) = (q

pq)(qpq2)· · ·(qpqk)

(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),

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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≤(p1)/2,

·

(p1)/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

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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

(12r[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)(12r+ 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

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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

(8)

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+(p1)(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−dn 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).

REFERENCES

1. G. E. Andrews, The theory of partitions, Cambridge University Press, Cambridge, 1998.

2. N. J. Calkin, Factors of sums of powers of binomial coefficients, Acta Arith., 86 (1998), 17-26.

3. J. Franel, On a question of Laisant, L’Interm´ediaire des Math´ematiciens, 1 (1894), 45-47.

4. Victor J. W. Guo, Proof of two conjectures of Sun on congruences for Franel numbers, Integral Trans-forms Spec. Funct., 24 (2013), 532-539.

5. Victor J. W. Guo and J. Zeng, New congruences for sums involving Ap´ery numbers or central Delannoy numbers, Int. J. Number Theory, 8 (2012), 2003-2016.

6. Victor J. W. Guo and J. Zeng, Some congruences involving centralq-binomial coefficients, Adv. in Appl. Math., 45 (2010), 303-316.

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8. G. Olive, Generalized powers, Amer. Math. Monthly, 72 (1965), 619-627.

9. H. Pan, Aq-analogue of Lehmer’s congruence, Acta Arith., 128 (2007), 303-318.

10. Z.-W. Sun, Congruences for Franel numbers, Adv. in Appl. Math., 51 (2013), 524-535.

11. Z.-W. Sun, Open conjectures on congruences, arXiv.org/abs/0911.5665.

References

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