Congruences for $q$-binomial coefficients

We discuss $q$-analogues of the classical congruence $\binom{ap}{bp}\equiv\binom{a}{b}\pmod{p^3}$, valid for primes $p>3$, as well as its generalisations. In particular, we prove related congruences for ($q$-analogues of) integral factorial ratios.

One also adopts the cyclotomic polynomials Φ n (q) = n j=1 (j,n)=1 as q-analogues of prime numbers, because these are the only factors of the q-numbers which are irreducible over Q.
Arithmetically significant relations often possess several q-analogues. While looking for q-extensions of the classical (Wolstenholme-Ljunggren) congruence more precisely, at a 'q-microscope setup' (when q-congruences for truncated hypergeometric sums are read off from the asymptotics of their non-terminating versions, usually equipped with extra parameters, at roots of unity -see [5]) for Straub's q-congruence [8], [9,Theorem 2.2], this author accidentally arrived at an bn σ b n q ( bn 2 ) ≡ a − 1 b + a − 1 a − b σ a n q ( an 2 ) (mod Φ n (q) 2 ), where the notation σ n = (−1) n−1 is implemented. Notice that the expression on the right-hand side is a sum of two q-monomials. The q-congruence (3) may be compared with another q-extension of (1), for any n > 1. This is given by Andrews in [2] for primes n = p > 3 only; though proved modulo Φ p (q) 2 , a complimentary result from [2] demonstrates that (1) in its full modulo p 3 strength can be derived from (4). More directly, Pan [7] shows that (4) can be generalised further to It is worth mentioning that the transition from Φ n (q) 2 to Φ n (q) 3 (or, from p 2 to p 3 ) is significant because the former has a simple combinatorial proof (resulting from the (q-)Chu-Vandermonde identity) whereas no combinatorial proof is known for the latter. Since q ( bn 2 ) ∼ σ b n as q → ζ, a primitive n-th root of unity, the congruence (3) is seen to be an extension of the trivial (q-Lucas) congruence The principal goal of this note is to provide a modulo Φ n (q) 4 extension of (3) (see Lemma 1 below) as well as to use the result for extending the congruences (2) and (5). In this way, our theorems provide two q-extensions of the congruence The latter can be continued further to higher powers of primes [6], and our 'mechanical' approach here suggests that one may try -with a lot of effort! -to deduce corresponding q-analogues.
Theorem 2. For any n > 1, we have the congruence We point out that a congruence A 1 (q) ≡ A 2 (q) (mod P (q)) for rational functions A 1 (q), A 2 (q) ∈ Q(q) and a polynomial P (q) ∈ Q[q] is understood as follows: the polynomial P (q) is relatively prime with the denominators of A 1 (q) and A 2 (q), and P (q) divides the numerator A(q) of the difference A 1 (q) − A 2 (q). The latter is equivalent to the condition that for each zero α ∈ C of P (q) of multiplicity k, the polynomial (q−α) k divides A(q) in C[q]; in other words, A 1 (q)−A 2 (q) = O (q−α) k as q → α. This latter interpretation underlies our argument in proving the results. For example, the congruence (3) can be established by verifying that when q = ζ(1 − ε) and ζ is any primitive n-th root of unity. Our approach goes in line with [5] and shares similarities with the one developed by Gorodetsky in [4], who reads off the asymptotic information of binomial sums at roots of unity through q-Gauss congruences. It does not seem straightforward to us but Gorodetsky's method may be capable of proving Theorems 1 and 2. Furthermore, the part [4, Sect. 2.3] contains a survey on q-analogues of (1).
After proving an asymptotical expansion for q-binomial coefficients at roots of unity in Section 2 (essentially, the O(ε 4 )-extension of (8)), we perform a similar asymptotic analysis for q-harmonic sums in Section 3. The information gathered is then applied in Section 4 to proving Theorems 1 and 2. Finally, in Section 5 we generalise the congruences (2) and (5) in a different direction, to integral factorial ratios.

Expansions of q-binomials at roots of unity
This section is exclusively devoted to an asymptotical result, which forms the grounds of our later arithmetic analysis. We moderate its proof by highlighting principal ingredients (and difficulties) of derivation and leaving some technical details to the reader. Lemma 1. Let ζ be a primitive n-th root of unity. Then, as q = ζ(1 − ε) → ζ radially, Proof. It follows from the q-binomial theorem [3, Chap. 10] that Taking N = an, for a primitive n-th root of unity ζ = ζ n , we have In particular, Now observe the following summation formulae: Implementing this information into (11) we obtain a b=0 an bn (−x) bn q bn(bn−1)/2 where we intentionally omit all ordinary ε 3 -terms -those that sum up to polynomials in a and n multiplied by powers of x n /(1 − x n ), like the ones appearing as ε-and ε 2 -terms. The exceptional ε 3 -summands are computed separately: The finale of our argument is comparison of the coefficients of powers of x n on both sides of the relation obtained; this way we arrive at the asymptotics in (9).

A q-harmonic sum
Again, the notation ζ is reserved for a primitive n-th root of unity. For the sum where S n−1 (q) is defined in Lemma 1. It follows that, for q = ζ(1 − ε), as ε → 0, where we use The latter asymptotics implies that which may be viewed as an extension of recorded, for example, in [6]. A different consequence of (12) is the following fact.

Proof of the theorems
In order to prove Theorems 1 and 2 we need to produce 'matching' asymptotics for a b q n 2 and σ b(a−b) respectively. These happen to be easier than that from Lemma 1 because q n 2 = (1 − ε) n 2 and q n = (1 − ε) n do not depend on the choice of primitive n-th root of unity ζ when q = ζ(1 − ε). Proof. For N = a in (10), take x n q ( n 2 ) and q n 2 for x and q: Then, for q = ζ(1 − ε), we write y = σ n x n to obtain To conclude, we apply the same argument as in the proof of Lemma 1.
Proof of Theorem 1. Combining the expansions in Lemmas 1-3 we find out that as q = ζ(1 − ε) → ζ radially. This means that the difference of both sides is divisible by (q − ζ) 4 for any n-th primitive root of unity ζ, hence by Φ n (q) 4 . The latter property is equivalent to the congruence (6).
Theorem 3. In the notation and are valid for any n ≥ 1.
Observe that when n = p > 3 and q → 1, one recovers from any of these two the congruences of which (1) is a special case. Furthermore, it is tempting to expect that these two families of q-congruences may be generalised even further in the spirit of Theorems 1 and 2, and that the polynomials D n (q) satisfy q-Gauss relations from [4]. We do not pursue this line here.
Proof of Theorem 3. Though the congruences (15) and (16) are between polynomials rather than rational functions, we prove the theorem without assumption (14): in other words, the congruences remain true for the rational functions D n (q) provided that the balancing condition (13) (equivalently, c 1 (a, b) = 1 in the above notation for c i ) is satisfied. In turn, this more general statement follows from its validity for particular cases by induction (on r + s, say). Indeed, the inductive step exploits the property of both (15) and (16) to imply the congruence for the product D n (a, b; q)D n (ã,b; q) whenever it is already known for the individual factors; we leave this simple fact to the reader and only discuss its other appearance when dealing withD n (q) below. Notice that an bn ≡ an bn ≡ 0 (mod Φ n (q)), so thatD n (q) = an bn −1 is well defined modulo any power of Φ n (q). For D n (q) = an bn we have c 2 = b(a − b) and c 2 + c 3 = ab(a − b)/2, hence (15) and (16) follow from (2) and (5), respectively.
For related Lucas-type congruences satisfied by the q-factorial ratios D n (q) see [1].