Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun

For all positive integers n, we prove the following divisibility properties: $$(2n+3){2n\choose n} | 3{6n\choose 3n}{3n\choose n}, and (10n+3){3n\choose n} | 21{15n\choose 5n} {5n\choose n}.$$ This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers m and n, the product $am{am+bm-1\choose am}{an+bn\choose an}$ is divisible by m+n. In fact, the latter result can be generalized to the q-binomial coefficients and q-integers case, which generalizes the positivity of q-Catalan numbers. We also propose several related conjectures.

Abstract. For all positive integers n, we prove the following divisibility properties:
Some similar divisibility results were later obtained by Guo [9] and Guo and Krattenthaler [10]. It should be mentioned that Bober [6] has completely described when ratios of factorial products of the form (a 1 n)! · · · (a k n)! (b 1 n)! · · · (b k+1 n)! with a 1 + · · · + a k = b 1 + · · · + b k+1 are always integers. Let In this paper we first prove the following two results conjectured by Z.-W. Sun [16,17]. We shall also give more congruences for S n and t n as follows.
Theorem 1.3 Let n be a positive integer. Then (1.5) 43263t n ≡ 0 (mod 10n + 9). (1.6) Let Z denote the set of integers. Another result in this paper is the following. In the next section, we give some lemmas. The proofs of Theorems 1.1-1.3 will be given in Sections 3-5 respectively. A proof of the q-analogue of Theorem 1.4 will be given in Section 6. We close our paper with some further remarks and open problems in Section 7.

Some lemmas
For the p-adic order of n!, there is a known formula where ⌊x⌋ denotes the greatest integer not exceeding x. In this section, we give some results on the floor function ⌊x⌋. It follows that Therefore, the identity (2.5) is true for any positive integer m 5.

Lemma 2.3
Let m and n be two positive integers such that m|10n + 3 and m 9. Then Proof. It is easy to see that (2.6) is equivalent to Now suppose that m|10n + 3 and m 9. We have It is easy to check that and so the identity (2.7) holds.

A q-analogue of Theorem 1.4
Recall that the q-binomial coefficients are defined by We begin with the announced strengthening of Theorem 1.4. It is easily seen that Theorem 1.4 can be obtained upon letting q → 1 in Corollary 6.2. is a polynomial in q with non-negative integer coefficients.
Corollary 6.2 Let a, b, m, n 1. Then is a polynomial in q with non-negative integer coefficients.
It is clear that, when a = b = m = 1, the numbers (6.2) reduce to the q-Catalan numbers It is well known that the q-Catalan numbers C n (q) are polynomials with non-negative integer coefficients (see [2,3,5,7]). There are many different q-analogues of the Catalan numbers (see Fürlinger and Hofbauer [7]). For the so-called q, t-Catalan numbers, see [8,11,12].
Recall that a polynomial P (q) = d i=0 p i q i in q of degree d is called reciprocal if p i = p d−i for all i, and that it is called unimodal if there is an integer r with 0 r d and 0 p 0 · · · p r · · · p d 0. An elementary but crucial property of reciprocal and unimodal polynomials is the following. Lemma 6.3 If A(q) and B(q) are reciprocal and unimodal polynomials, then so is their product A(q)B(q). Lemma 6.3 is well known and its proof can be found, e.g., in [1]  Lemma 6.4 Let P (q) be a reciprocal and unimodal polynomial and m and n positive integers with m n. Furthermore, assume that A(q) = 1−q m 1−q n P (q) is a polynomial in q. Then A(q) has non-negative coefficients.
Proof of Theorem 6.1. It is well known that the q-binomial coefficients are reciprocal and unimodal polynomials in q (cf. [15,Ex. 7.75.d]), and by Lemma 6.3, so is the product of two q-binomial coefficients. In view of Lemma 6.4, for proving Theorem 6.1 it is enough to show that the expression (6.1) is a polynomial in q. We shall accomplish this by a count of cyclotomic polynomials.
Recall the well-known fact that where Φ d (q) denotes the d-th cyclotomic polynomial in q. Consequently, where χ(S) = 1 if S is true and χ(S) = 0 otherwise. This is clearly non-negative, unless d | m + n and d ∤ gcd(am, m + n). So, let us assume that d | m + n and d ∤ gcd(am, m + n), which means that d ∤ am and therefore am d = am − 1 d .
Note that, when d | m + n, we have and so e d = 0 is also non-negative in this case. This completes the proof of polynomiality of (6.1).
Proof of Corollary 6.2. This follows immediately from Theorem 6.1 and the fact that gcd(am, m + n) | am.

Concluding remarks and open problems
On January 2, 2014 T. Amdeberhan (personal communication) found the following generalization of Theorem 1.1. It would be interesting to give a proof of it.  Conjecture 7.6 The two polynomials in Theorem 7.5 have non-negative integer coefficients.