aa r X i v : . [ m a t h . N T ] A p r CONGRUENCES FOR q -BINOMIAL COEFFICIENTS WADIM ZUDILIN
To George Andrews, with warm q -wishes and well-looking q -congruences Abstract.
We discuss q -analogues of the classical congruence (cid:0) apbp (cid:1) ≡ (cid:0) ab (cid:1) (mod p ),valid for primes p >
3, as well as its generalisations. In particular, we prove relatedcongruences for ( q -analogues of) integral factorial ratios. Introduction
For a non-negative integer, a standard q -environment includes q -numbers [ a ] =[ a ] q = (1 − q a ) / (1 − q ) ∈ Z [ q ], q -factorials [ a ]! = [1][2] · · · [ a ] ∈ Z [ q ] and q -binomialcoefficients (cid:20) ab (cid:21) = (cid:20) ab (cid:21) q = [ a ]![ b ]! [ a − b ]! ∈ Z [ q ] , where b = 0 , , . . . , a. One also adopts the cyclotomic polynomialsΦ n ( q ) = n Y j =1( j,n )=1 ( q − e πij/n ) ∈ Z [ q ]as q -analogues of prime numbers, because these are the only factors of the q -numberswhich are irreducible over Q .Arithmetically significant relations often possess several q -analogues. While look-ing for q -extensions of the classical (Wolstenholme–Ljunggren) congruence (cid:18) apbp (cid:19) ≡ (cid:18) ab (cid:19) (mod p ) for any prime p > , (1)more precisely, at a ‘ q -microscope setup’ (when q -congruences for truncated hyper-geometric 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], (cid:20) anbn (cid:21) q ≡ (cid:20) ab (cid:21) q n − b ( a − b ) (cid:18) ab (cid:19) n −
124 ( q n − (mod Φ n ( q ) ) , (2) Date : 23 January 2019.
Revised : 1 April 2019.2010
Mathematics Subject Classification.
Key words and phrases.
Congruence; q -binomial coefficient; cyclotomic polynomial; radialasymptotics. this author accidentally arrived at (cid:20) anbn (cid:21) σ bn q ( bn ) ≡ (cid:18) a − b (cid:19) + (cid:18) a − a − b (cid:19) σ an q ( an ) (mod Φ n ( q ) ) , (3)where the notation σ n = ( − n − 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 -extensionof (1), (cid:20) anbn (cid:21) q ≡ σ b ( a − b ) n q b ( a − b ) ( n ) (cid:20) ab (cid:21) q n (mod Φ n ( q ) ) (4)for any n >
1. This is given by Andrews in [2] for primes n = p > p ( q ) , a complimentary result from [2] demonstrates that (1) in itsfull modulo p strength can be derived from (4). More directly, Pan [7] shows that(4) can be generalised further to (cid:20) anbn (cid:21) q ≡ σ b ( a − b ) n q b ( a − b ) ( n ) (cid:20) ab (cid:21) q n + ab ( a − b ) (cid:18) ab (cid:19) n −
124 ( q n − (mod Φ n ( q ) ) . (5)It is worth mentioning that the transition from Φ n ( q ) to Φ n ( q ) (or, from p to p )is significant because the former has a simple combinatorial proof (resulting fromthe ( q -)Chu–Vandermonde identity) whereas no combinatorial proof is known forthe latter.Since q ( bn ) ∼ σ bn as q → ζ , a primitive n -th root of unity, the congruence (3) isseen to be an extension of the trivial ( q -Lucas) congruence (cid:20) anbn (cid:21) ≡ (cid:18) ab (cid:19) = (cid:18) a − b (cid:19) + (cid:18) a − a − b (cid:19) (mod Φ n ( q )) . The principal goal of this note is to provide a modulo Φ n ( q ) 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 (cid:18) apbp (cid:19) ≡ (cid:18) ab (cid:19) + ab ( a − b ) (cid:18) ab (cid:19) p p − X k =1 k (mod p ) for prime p > . The latter can be continued further to higher powers of primes [6], and our ‘mechan-ical’ approach here suggests that one may try — with a lot of effort! — to deducecorresponding q -analogues. Theorem 1.
The congruence (cid:20) anbn (cid:21) q ≡ (cid:20) ab (cid:21) q n − b ( a − b ) (cid:18) ab (cid:19) ( q n − (cid:18) a n − X k =1 q k − q k + a ( n − a + 1)( n − q n −
1) + ( b ( a − b ) n − a − n − q n − (cid:19) (6) holds modulo Φ n ( q ) for any n > . ONGRUENCES FOR q -BINOMIAL COEFFICIENTS 3 Theorem 2.
For any n > , we have the congruence (cid:20) anbn (cid:21) q ≡ σ b ( a − b ) n q b ( a − b ) ( n ) (cid:20) ab (cid:21) q n − ab ( a − b ) (cid:18) ab (cid:19) ( q n − (cid:18) n − X k =1 q k − q k + n − − ( b ( a − b ) n − n − q n − (cid:19) (mod Φ n ( q ) ) . (7)We point out that a congruence A ( q ) ≡ A ( q ) (mod P ( q )) for rational functions A ( q ) , A ( q ) ∈ Q ( q ) and a polynomial P ( q ) ∈ Q [ q ] is understood as follows: thepolynomial P ( q ) is relatively prime with the denominators of A ( q ) and A ( q ), and P ( q ) divides the numerator A ( q ) of the difference A ( q ) − A ( q ). The latter isequivalent to the condition that for each zero α ∈ C of P ( q ) of multiplicity k , thepolynomial ( q − α ) k divides A ( q ) in C [ q ]; in other words, A ( q ) − A ( q ) = O (cid:0) ( q − α ) k (cid:1) as q → α . This latter interpretation underlies our argument in proving the results.For example, the congruence (3) can be established by verifying that (cid:20) anbn (cid:21) q (1 − ε )( bn ) = (cid:18) a − b (cid:19) + (cid:18) a − a − b (cid:19) σ an (1 − ε )( an ) + O ( ε ) as ε → + , (8)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 developedby Gorodetsky in [4], who reads off the asymptotic information of binomial sums atroots of unity through q -Gauss congruences. It does not seem straightforward to usbut Gorodetsky’s method may be capable of proving Theorems 1 and 2. Further-more, 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 ofunity in Section 2 (essentially, the O ( ε )-extension of (8)), we perform a similarasymptotic analysis for q -harmonic sums in Section 3. The information gathered isthen applied in Section 4 to proving Theorems 1 and 2. Finally, in Section 5 wegeneralise the congruences (2) and (5) in a different direction, to integral factorialratios. 2. Expansions of q -binomials at roots of unity This section is exclusively devoted to an asymptotical result, which forms thegrounds of our later arithmetic analysis. We moderate its proof by highlightingprincipal ingredients (and difficulties) of derivation and leaving some technical de-tails to the reader.
Lemma 1.
Let ζ be a primitive n -th root of unity. Then, as q = ζ (1 − ε ) → ζ radially, (cid:20) anbn (cid:21) q σ bn q ( bn ) − (cid:18) a − b (cid:19) − (cid:18) a − a − b (cid:19) σ an q ( an )= b ( a − b ) (cid:18) ab (cid:19)(cid:0) − ε n ρ ( a, n ) + ε n ρ ( a, b, n ) + ε anS n − ( ζ ) (cid:1) + O ( ε ) , (9) WADIM ZUDILIN where ρ ( a, n ) = 3( an − − an − ,ρ ( a, b, n ) = abn ( an − an − n −
2) + ( an + 2)( an − ( an −
3) + an + a + 248 and S n − ( q ) = 12 n − X k =1 kq k (( k + 1) q k + k − − q k ) . Proof.
It follows from the q -binomial theorem [3, Chap. 10] that( x ; q ) N = N X k =0 (cid:20) Nk (cid:21) q ( − x ) k q ( k ) . (10)Taking N = an , for a primitive n -th root of unity ζ = ζ n , we have1 n n X j =1 ( ζ j x ; q ) an = an X k =0 n | k (cid:20) ank (cid:21) ( − x ) k q k ( k − / = a X b =0 (cid:20) anbn (cid:21) ( − x ) bn q bn ( bn − / . (11)When q = ζ (1 − ε ), we get d / d ε = − ζ (d / d q ). If f ( q ) = ( x ; q ) an and g ( q ) = dd q log f ( q ) = − an − X ℓ =1 ℓq ℓ − x − q ℓ x , then f ( q ) | ε =0 = (1 − x n ) a andd f d q = f g, d f d q = f (cid:18) g + d g d q (cid:19) , d f d q = f (cid:18) g + 3 g d g d q + d g d q (cid:19) . In particular,d f d ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = (1 − x n ) a an − X ℓ =1 ℓζ ℓ x − ζ ℓ x , d f d ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = (1 − x n ) a (cid:18)(cid:18) an − X ℓ =1 ℓζ ℓ x − ζ ℓ x (cid:19) − an − X ℓ =1 (cid:18) ℓ ζ ℓ x (1 − ζ ℓ x ) + ℓ ( ℓ − ζ ℓ x − ζ ℓ x (cid:19)(cid:19) and d f d ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = (1 − x n ) a (cid:18)(cid:18) an − X ℓ =1 ℓζ ℓ x − ζ ℓ x (cid:19) − an − X ℓ =1 ℓζ ℓ x − ζ ℓ x an − X ℓ =1 (cid:18) ℓ ζ ℓ x (1 − ζ ℓ x ) + ℓ ( ℓ − ζ ℓ x − ζ ℓ x (cid:19) + an − X ℓ =1 (cid:18) ℓ ζ ℓ x (1 − ζ ℓ x ) + 3 ℓ ( ℓ − ζ ℓ x (1 − ζ ℓ x ) + ℓ ( ℓ − ℓ − ζ ℓ x − ζ ℓ x (cid:19)(cid:19) . ONGRUENCES FOR q -BINOMIAL COEFFICIENTS 5 Now observe the following summation formulae:1 n n X j =1 x − x (cid:12)(cid:12)(cid:12)(cid:12) x ζ j x = x n − x n , n n X j =1 (cid:18) x − x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x ζ j x = nx n (1 − x n ) − x n − x n , n n X j =1 x − x ζ k x − ζ k x (cid:12)(cid:12)(cid:12)(cid:12) x ζ j x = − x n − x n for k n ) , n n X j =1 (cid:18) x − x (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) x ζ j x = n x n (1 + x n )2(1 − x n ) − nx n − x n ) + x n − x n , n n X j =1 x − x ζ k x − ζ k x ζ ℓ x − ζ ℓ x (cid:12)(cid:12)(cid:12)(cid:12) x ζ j x = x n − x n for k , ℓ , k ℓ (mod n ) , and 1 n n X j =1 (cid:18) x − x (cid:19) ζ k x − ζ k x (cid:12)(cid:12)(cid:12)(cid:12) x ζ j x = nx n (1 − x n ) ζ k − ζ k + x n − x n for k n ) . Implementing this information into (11) we obtain a X b =0 (cid:20) anbn (cid:21) ( − x ) bn q bn ( bn − / (cid:12)(cid:12)(cid:12)(cid:12) q = ζ (1 − ε ) = (1 − x n ) a ε x n − x n an − X ℓ =1 ℓ − ε x n − x n (cid:18) an − X ℓ =1 ℓ (cid:19) + ε nx n (1 − x n ) an − X ℓ ,ℓ =1 ℓ ≡ ℓ (mod n ) ℓ ℓ − ε (cid:18) nx n (1 − x n ) − x n − x n (cid:19) an − X ℓ =1 ℓ − ε x n − x n an − X ℓ =1 ℓ ( ℓ − · · · + ε nx n (1 − x n ) an − X ℓ ,ℓ ,ℓ =1 ℓ ≡ ℓ ℓ (mod n ) ℓ ℓ ℓ ζ ℓ − ℓ − ζ ℓ − ℓ + · · · + · · · − ε nx n (1 − x n ) an − X ℓ ,ℓ =1 ℓ ℓ (mod n ) ℓ ℓ ζ ℓ − ℓ − ζ ℓ − ℓ + · · · ! + O ( ε ) , where we intentionally omit all ordinary ε -terms — those that sum up to polyno-mials in a and n multiplied by powers of x n / (1 − x n ), like the ones appearing as ε - WADIM ZUDILIN and ε -terms. The exceptional ε -summands are computed separately: an − X ℓ ,ℓ ,ℓ =1 ℓ ≡ ℓ ℓ (mod n ) ℓ ℓ ℓ ζ ℓ − ℓ − ζ ℓ − ℓ = − a n − X k =1 kζ k (( k + 1) ζ k + k − − ζ k ) − a n ( n − an ( an − an −
2) + n ( an − a − − an − X ℓ ,ℓ =1 ℓ ℓ (mod n ) ℓ ℓ ζ ℓ − ℓ − ζ ℓ − ℓ = − a n − X k =1 kζ k (( k + 1) ζ k + k − − ζ k ) − a n ( n − an ( an − an − − an − . The finale of our argument is comparison of the coefficients of powers of x n on bothsides of the relation obtained; this way we arrive at the asymptotics in (9). (cid:3) A q -harmonic sum Again, the notation ζ is reserved for a primitive n -th root of unity. For the sum H n − ( q ) = n − X k =1 q k − q k , we have d H n − d q = n − X k =1 kq k − (1 − q k ) , d H n − d q = n − X k =1 kq k − (( k + 1) q k + k − − q k ) = 2 q − S n − ( q ) , where S n − ( q ) is defined in Lemma 1. It follows that, for q = ζ (1 − ε ), H n − ( q ) = n − X k =1 ζ k − ζ k − ε n − X k =1 kζ k (1 − ζ k ) + ε n − X k =1 kζ k (( k + 1) ζ k + k − − ζ k ) + O ( ε )= − n −
12 + ( n − n ε + S n − ( ζ ) ε + O ( ε )= − n − − n −
124 ( q n −
1) + ( n − n − n ( q n − + 1 n S n − ( ζ )( q n − + O ( ε ) (12)as ε →
0, where we use ε = − n ( q n −
1) + n − n ( q n − + O ( ε ) as ε → . ONGRUENCES FOR q -BINOMIAL COEFFICIENTS 7 The latter asymptotics implies that n − X k =1 q k − q k ≡ − n − − n −
124 ( q n −
1) + ( n − n − n ( q n − + ( q n − n n − X k =1 kq k (( k + 1) q k + k − − q k ) (mod Φ n ( q ) ) , which may be viewed as an extension of n − X k =1 q k − q k ≡ − n − − n −
124 ( q n −
1) (mod Φ n ( q ) )recorded, for example, in [6].A different consequence of (12) is the following fact. Lemma 2.
The term ε S n − ( ζ ) appearing in the expansion (9) can be replaced with H n − ( q ) + n −
12 + n −
124 ( q n − − ( n − n − n ( q n − + O ( ε ) when q = ζ (1 − ε ) and ε → . Proof of the theorems
In order to prove Theorems 1 and 2 we need to produce ‘matching’ asymptoticsfor (cid:20) ab (cid:21) q n and σ b ( a − b ) n q b ( a − b ) ( n ) (cid:20) ab (cid:21) q n , respectively. These happen to be easier than that from Lemma 1 because q n =(1 − ε ) n and q n = (1 − ε ) n do not depend on the choice of primitive n -th root ofunity ζ when q = ζ (1 − ε ). Lemma 3. As q = ζ (1 − ε ) → ζ radially, (cid:20) ab (cid:21) q n σ bn q ( bn ) − (cid:18) a − b (cid:19) − (cid:18) a − a − b (cid:19) σ an q ( an )= b ( a − b ) (cid:18) ab (cid:19)(cid:0) − ε n ˆ ρ ( a, n ) + ε n ˆ ρ ( a, b, n ) (cid:1) + O ( ε ) , where ˆ ρ ( a, n ) = 3( an − − ( a + 1) n , ˆ ρ ( a, b, n ) = bn ( an − an − − ( a + 1) n )48+ an ( an − − an − + 2( a + 1) n . WADIM ZUDILIN
Proof.
For N = a in (10), take x n q ( n ) and q n for x and q :( x n q ( n ); q n ) a = a X b =0 (cid:20) ab (cid:21) q n σ bn ( − x ) bn q ( bn ) . Then, for q = ζ (1 − ε ), we write y = σ n x n to obtain( x n q ( n ); q n ) a = ( y (1 − ε )( n ); ( ε ) n ) a = a − Y ℓ =0 (cid:0) − y (1 − ε ) ℓn + ( n ) (cid:1) = (1 − y ) a a − Y ℓ =0 (cid:18) − y − y ℓn + ( n ) X i =1 (cid:18) ℓn + (cid:0) n (cid:1) i (cid:19) ( − ε ) i (cid:19) . To conclude, we apply the same argument as in the proof of Lemma 1. (cid:3)
Proof of Theorem . Combining the expansions in Lemmas 1–3 we find out that (cid:20) anbn (cid:21) q σ bn q ( bn ) − (cid:20) ab (cid:21) q n σ bn q ( bn ) = − b ( a − b ) (cid:18) ab (cid:19) ( q n − (cid:18) a n − X k =1 q k − q k + a ( n − a + 1)( n − q n −
1) + ( b ( a − b ) n − a − n − q n − (cid:19) + O ( ε )as q = ζ (1 − ε ) → ζ radially. This means that the difference of both sides is divisibleby ( q − ζ ) for any n -th primitive root of unity ζ , hence by Φ n ( q ) . The latterproperty is equivalent to the congruence (6). (cid:3) Proof of Theorem . We first use Lemma 1 with n = 1: (cid:20) ab (cid:21) q q ( b ) − (cid:18) a − b (cid:19) − (cid:18) a − a − b (cid:19) q ( a )= b ( a − b ) (cid:18) ab (cid:19)(cid:0) − (1 − q ) ρ ( a,
1) + (1 − q ) ρ ( a, b, (cid:1) + O (cid:0) (1 − q ) (cid:1) as q →
1. Now, take n > q replaced with q n ,where q = ζ (1 − ε ), 0 < ε < ζ is a primitive n -th root of unity: σ b ( a − b ) n q b ( a − b ) ( n ) (cid:20) ab (cid:21) q n = (cid:18) a − b (cid:19) (1 − ε ) b ( a − b ) ( n ) − ( b ) n + (cid:18) a − a − b (cid:19) (1 − ε ) b ( a − b ) ( n ) + ( a ) n − ( b ) n + b ( a − b ) (cid:18) ab (cid:19)(cid:0) − (1 − (1 − ε ) n ) ρ ( a,
1) + ε n ρ ( a, b, (cid:1) × (1 − ε ) b ( a − b ) ( n ) − ( b ) n + O ( ε ) ONGRUENCES FOR q -BINOMIAL COEFFICIENTS 9 = (cid:18) a − b (cid:19) (1 − ε ) − ( bn ) + ab ( n ) + (cid:18) a − a − b (cid:19) (1 − ε )( an ) − ( bn ) − a ( a − b ) ( n )+ b ( a − b ) (cid:18) ab (cid:19)(cid:0) − ε n ρ ( a,
1) + ε n (cid:0) ( n − ρ ( a,
1) + nρ ( a, b, (cid:1)(cid:1) × (cid:18) − (( a − b ) n − a + 1) bn ε + O ( ε ) (cid:19) + O ( ε )as ε →
0. At the same time, from Lemma 1 we have (cid:20) anbn (cid:21) q = (cid:18) a − b (cid:19) (1 − ε ) − ( bn ) + (cid:18) a − a − b (cid:19) (1 − ε )( an ) − ( bn )+ b ( a − b ) (cid:18) ab (cid:19)(cid:0) − ε n ρ ( a, n ) + ε n ρ ( a, b, n ) + ε anS n − ( ζ ) (cid:1) × (cid:18) (cid:18) bn (cid:19) ε + O ( ε ) (cid:19) + O ( ε )as ε →
0. Using(1 − ε ) N = 1 − N ε + (cid:18) N (cid:19) ε − (cid:18) N (cid:19) ε + O ( ε ) as ε → N = − (cid:0) bn (cid:1) + ab (cid:0) n (cid:1) , (cid:0) an (cid:1) − (cid:0) bn (cid:1) − a ( a − b ) (cid:0) n (cid:1) , − (cid:0) bn (cid:1) and (cid:0) an (cid:1) − (cid:0) bn (cid:1) we deducefrom the two expansions and Lemma 2 that, for q = ζ (1 − ε ), (cid:20) anbn (cid:21) q − σ b ( a − b ) n q b ( a − b ) ( n ) (cid:20) ab (cid:21) q n = − ab ( a − b ) (cid:18) ab (cid:19) ( q n − (cid:18) n − X k =1 q k − q k + n − − ( b ( a − b ) n − n − q n − (cid:19) + O ( ε )as ε →
0. This implies the congruence in (7). (cid:3) q -rious congruences In this final part, we look at the binomial coefficients as particular instances ofintegral ratios of factorials, also known as Chebyshev–Landau factorial ratios. Inthe q -setting these are defined by D n ( q ) = D n ( a , b ; q ) = [ a n ]! · · · [ a r n ]![ b n ]! · · · [ b s n ]! , where a = ( a , . . . , a r ) and b = ( b , . . . , b s ) are positive integers satisfying a + · · · + a r = b + · · · + b s (13)and ⌊ a x ⌋ + · · · + ⌊ a r x ⌋ ≥ ⌊ b x ⌋ + · · · + ⌊ b s x ⌋ for all x > (see, for example, [10]), ⌊ · ⌋ denotes the integer part of a number. Then D n ( q ) ∈ Z [ q ]are polynomials with values D n (1) = ( a n )! · · · ( a r n )!( b n )! · · · ( b s n )!at q = 1, and the congruences (2) and (5) generalise as follows. Theorem 3.
In the notation c i = c i ( a , b ) = (cid:18) a i (cid:19) + · · · + (cid:18) a r i (cid:19) − (cid:18) b i (cid:19) − · · · − (cid:18) b s i (cid:19) for i = 2 , , the congruences D n ( q ) ≡ D ( q n ) − D (1) c ( n − q n − (mod Φ n ( q ) ) (15) and D n ( q ) ≡ σ c n q c ( n ) D ( q n ) + D (1) ( c + c ) ( n − q n − (mod Φ n ( q ) ) (16) are valid for any n ≥ . Observe that when n = p > q →
1, one recovers from any of these two thecongruences D p (1) ≡ D (1) (mod p ) , of which (1) is a special case. Furthermore, it is tempting to expect that these twofamilies of q -congruences may be generalised even further in the spirit of Theorems 1and 2, and that the polynomials D n ( q ) satisfy q -Gauss relations from [4]. We donot pursue this line here. Proof of Theorem . Though the congruences (15) and (16) are between polynomials rather than rational functions, we prove the theorem without assumption (14): inother words, the congruences remain true for the rational functions D n ( q ) providedthat the balancing condition (13) (equivalently, c ( a , b ) = 1 in the above notationfor c i ) is satisfied. In turn, this more general statement follows from its validity forparticular cases D n ( q ) = [ an ]![ bn ]! [( a − b ) n ]! and ˜ D n ( q ) = [ bn ]! [( a − b ) n ]![ an ]!by induction (on r + s , say). Indeed, the inductive step exploits the property ofboth (15) and (16) to imply the congruence for the product D n ( a , b ; q ) D n (˜ a , ˜ b ; q )whenever it is already known for the individual factors; we leave this simple fact tothe reader and only discuss its other appearance when dealing with ˜ D n ( q ) below.Notice that (cid:2) anbn (cid:3) ≡ (cid:0) anbn (cid:1) n ( q )), so that ˜ D n ( q ) = (cid:2) anbn (cid:3) − is well definedmodulo any power of Φ n ( q ).For D n ( q ) = (cid:2) anbn (cid:3) we have c = b ( a − b ) and c + c = ab ( a − b ) /
2, hence (15) and(16) follow from (2) and (5), respectively.
ONGRUENCES FOR q -BINOMIAL COEFFICIENTS 11 Turning to q = ζ (1 − ε ), where 0 < ε < ζ is a primitive n -th root of unity,write the congruences (2) and (5) as the asymptotic relation (cid:20) anbn (cid:21) = B ( q ) + cB (1) ε + O ( ε ) as ε → , in which B ( q ) = (cid:20) ab (cid:21) q n , c = − b ( a − b ) n ( n − B ( q ) = σ b ( a − b ) n q b ( a − b ) ( n ) (cid:20) ab (cid:21) q n , c = ab ( a − b ) n ( n − . Then ˜ D n ( q ) = (cid:20) anbn (cid:21) − = B ( q ) − (cid:0) cB (1) B ( q ) − ε + O ( ε ) (cid:1) − = B ( q ) − − cB (1) B ( q ) − ε + O ( ε )= B ( q ) − − cB (1) − ε + O ( ε ) , because we have B ( q ) = B (1) + O ( ε ) as ε → B ( q ). The resultingexpansion implies the truth of (15) and (16) for ˜ D n ( q ) = D n (( b, a − b ) , ( a ); q ) inview of c i (( b, a − b ) , ( a )) = − c i (( a ) , ( b, a − b )) for i = 2 , . As explained above, this also establishes the general case of (15) and (16). (cid:3)
For related Lucas-type congruences satisfied by the q -factorial ratios D n ( q ) see [1]. Acknowledgements.
I would like to thank Armin Straub for encouraging me tocomplete this project and for supply of available knowledge on the topic. I amgrateful to one of the referees whose feedback was terrific and helped me improvingthe exposition. Further, I thank Victor Guo for valuable comments on parts of thiswork.
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