Congruences for $q$-binomial coefficients

Wadim Zudilin

arXiv:1901.07843·math.NT·December 16, 2019

Congruences for $q$-binomial coefficients

Wadim Zudilin

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TL;DR

This paper explores advanced congruences related to $q$-binomial coefficients, extending classical results and establishing new congruences for $q$-analogues of factorial ratios, with implications for number theory.

Contribution

It introduces new $q$-congruences generalizing classical binomial coefficient congruences and proves related results for $q$-factorial ratios.

Findings

01

Established $q$-analogues of classical binomial congruences

02

Proved new congruences for $q$-factorial ratios

03

Extended classical results to broader $q$-analogues

Abstract

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

Equations152

[b a] = [b a]_{q} = \frac{[ a ]!}{[ b ]! [ a - b ]!} \in Z [q], where b = 0, 1, \dots, a .

[b a] = [b a]_{q} = \frac{[ a ]!}{[ b ]! [ a - b ]!} \in Z [q], where b = 0, 1, \dots, a .

Φ_{n} (q) = j = 1 (j, n) = 1 \prod n (q - e^{2 π ij / n}) \in Z [q]

Φ_{n} (q) = j = 1 (j, n) = 1 \prod n (q - e^{2 π ij / n}) \in Z [q]

(b p a p) \equiv (b a) (mod p^{3}) for any prime p > 3,

(b p a p) \equiv (b a) (mod p^{3}) for any prime p > 3,

[bn an]_{q} \equiv [b a]_{q^{n^{2}}} - b (a - b) (b a) \frac{n ^{2} - 1}{24} (q^{n} - 1)^{2} (mod Φ_{n} (q)^{3}),

[bn an]_{q} \equiv [b a]_{q^{n^{2}}} - b (a - b) (b a) \frac{n ^{2} - 1}{24} (q^{n} - 1)^{2} (mod Φ_{n} (q)^{3}),

[bn an] σ_{n}^{b} q^{(2 bn)} \equiv (b a - 1) + (a - b a - 1) σ_{n}^{a} q^{(2 an)} (mod Φ_{n} (q)^{2}),

[bn an] σ_{n}^{b} q^{(2 bn)} \equiv (b a - 1) + (a - b a - 1) σ_{n}^{a} q^{(2 an)} (mod Φ_{n} (q)^{2}),

σ_{n} = (- 1)^{n - 1}

σ_{n} = (- 1)^{n - 1}

[bn an]_{q} \equiv σ_{n}^{b (a - b)} q^{b (a - b) (2 n)} [b a]_{q^{n}} (mod Φ_{n} (q)^{2})

[bn an]_{q} \equiv σ_{n}^{b (a - b)} q^{b (a - b) (2 n)} [b a]_{q^{n}} (mod Φ_{n} (q)^{2})

[bn an]_{q} \equiv σ_{n}^{b (a - b)} q^{b (a - b) (2 n)} [b a]_{q^{n}} + ab (a - b) (b a) \frac{n ^{2} - 1}{24} (q^{n} - 1)^{2} (mod Φ_{n} (q)^{3}) .

[bn an]_{q} \equiv σ_{n}^{b (a - b)} q^{b (a - b) (2 n)} [b a]_{q^{n}} + ab (a - b) (b a) \frac{n ^{2} - 1}{24} (q^{n} - 1)^{2} (mod Φ_{n} (q)^{3}) .

[bn an] \equiv (b a) = (b a - 1) + (a - b a - 1) (mod Φ_{n} (q)) .

[bn an] \equiv (b a) = (b a - 1) + (a - b a - 1) (mod Φ_{n} (q)) .

(b p a p) \equiv (b a) + ab (a - b) (b a) p k = 1 \sum p - 1 \frac{1}{k} (mod p^{4}) for prime p > 3.

(b p a p) \equiv (b a) + ab (a - b) (b a) p k = 1 \sum p - 1 \frac{1}{k} (mod p^{4}) for prime p > 3.

[bn an]_{q}

[bn an]_{q}

\displaystyle\quad+\frac{(a+1)(n^{2}-1)}{24}(q^{n}-1)+\frac{(b(a-b)n-a-2)(n^{2}-1)}{48}\,(q^{n}-1)^{2}\biggr{)}

[bn an]_{q}

[bn an]_{q}

\displaystyle\quad-\frac{(b(a-b)n-1)(n^{2}-1)}{48}\,(q^{n}-1)^{2}\biggr{)}\pmod{\Phi_{n}(q)^{4}}.

[bn an]_{q} (1 - ε)^{(2 bn)} = (b a - 1) + (a - b a - 1) σ_{n}^{a} (1 - ε)^{(2 an)} + O (ε^{2}) as ε \to 0^{+},

[bn an]_{q} (1 - ε)^{(2 bn)} = (b a - 1) + (a - b a - 1) σ_{n}^{a} (1 - ε)^{(2 an)} + O (ε^{2}) as ε \to 0^{+},

[bn an]_{q} σ_{n}^{b} q^{(2 bn)} - (b a - 1) - (a - b a - 1) σ_{n}^{a} q^{(2 an)}

[bn an]_{q} σ_{n}^{b} q^{(2 bn)} - (b a - 1) - (a - b a - 1) σ_{n}^{a} q^{(2 an)}

\displaystyle\quad=b(a-b)\binom{a}{b}\bigl{(}-\varepsilon^{2}n^{2}\rho_{0}(a,n)+\varepsilon^{3}n^{2}\rho_{1}(a,b,n)+\varepsilon^{3}\,anS_{n-1}(\zeta)\bigr{)}+O(\varepsilon^{4}),

ρ_{0} (a, n)

ρ_{0} (a, n)

ρ_{1} (a, b, n)

S_{n - 1} (q) = \frac{1}{2} k = 1 \sum n - 1 \frac{k q ^{k} (( k + 1 ) q ^{k} + k - 1 )}{( 1 - q ^{k} ) ^{3}} .

S_{n - 1} (q) = \frac{1}{2} k = 1 \sum n - 1 \frac{k q ^{k} (( k + 1 ) q ^{k} + k - 1 )}{( 1 - q ^{k} ) ^{3}} .

(x; q)_{N} = k = 0 \sum N [k N]_{q} (- x)^{k} q^{(2 k)} .

(x; q)_{N} = k = 0 \sum N [k N]_{q} (- x)^{k} q^{(2 k)} .

\frac{1}{n} j = 1 \sum n (ζ^{j} x; q)_{an} = k = 0 n ∣ k \sum an [k an] (- x)^{k} q^{k (k - 1) /2} = b = 0 \sum a [bn an] (- x)^{bn} q^{bn (bn - 1) /2} .

\frac{1}{n} j = 1 \sum n (ζ^{j} x; q)_{an} = k = 0 n ∣ k \sum an [k an] (- x)^{k} q^{k (k - 1) /2} = b = 0 \sum a [bn an] (- x)^{bn} q^{bn (bn - 1) /2} .

f (q) = (x; q)_{an} and g (q) = \frac{d}{d q} lo g f (q) = - ℓ = 1 \sum an - 1 \frac{ℓ q ^{ℓ - 1} x}{1 - q ^{ℓ} x},

f (q) = (x; q)_{an} and g (q) = \frac{d}{d q} lo g f (q) = - ℓ = 1 \sum an - 1 \frac{ℓ q ^{ℓ - 1} x}{1 - q ^{ℓ} x},

\frac{{\mathrm{d}}f}{{\mathrm{d}}q}=fg,\quad\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}}q^{2}}=f\biggl{(}g^{2}+\frac{{\mathrm{d}}g}{{\mathrm{d}}q}\biggr{)},\quad\frac{{\mathrm{d}}^{3}f}{{\mathrm{d}}q^{3}}=f\biggl{(}g^{3}+3g\frac{{\mathrm{d}}g}{{\mathrm{d}}q}+\frac{{\mathrm{d}}^{2}g}{{\mathrm{d}}q^{2}}\biggr{)}.

\frac{{\mathrm{d}}f}{{\mathrm{d}}q}=fg,\quad\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}}q^{2}}=f\biggl{(}g^{2}+\frac{{\mathrm{d}}g}{{\mathrm{d}}q}\biggr{)},\quad\frac{{\mathrm{d}}^{3}f}{{\mathrm{d}}q^{3}}=f\biggl{(}g^{3}+3g\frac{{\mathrm{d}}g}{{\mathrm{d}}q}+\frac{{\mathrm{d}}^{2}g}{{\mathrm{d}}q^{2}}\biggr{)}.

\displaystyle\frac{{\mathrm{d}}f}{{\mathrm{d}}\varepsilon}\bigg{|}_{\varepsilon=0}

\displaystyle\frac{{\mathrm{d}}f}{{\mathrm{d}}\varepsilon}\bigg{|}_{\varepsilon=0}

\displaystyle\frac{{\mathrm{d}}^{2}f}{{\mathrm{d}}\varepsilon^{2}}\bigg{|}_{\varepsilon=0}

\displaystyle\frac{{\mathrm{d}}^{3}f}{{\mathrm{d}}\varepsilon^{3}}\bigg{|}_{\varepsilon=0}

\displaystyle\qquad-3\sum_{\ell=1}^{an-1}\frac{\ell\zeta^{\ell}x}{1-\zeta^{\ell}x}\sum_{\ell=1}^{an-1}\biggl{(}\frac{\ell^{2}\zeta^{2\ell}x^{2}}{(1-\zeta^{\ell}x)^{2}}+\frac{\ell(\ell-1)\zeta^{\ell}x}{1-\zeta^{\ell}x}\biggr{)}

\displaystyle\qquad+\sum_{\ell=1}^{an-1}\biggl{(}\frac{2\ell^{3}\zeta^{3\ell}x^{3}}{(1-\zeta^{\ell}x)^{3}}+\frac{3\ell^{2}(\ell-1)\zeta^{2\ell}x^{2}}{(1-\zeta^{\ell}x)^{2}}+\frac{\ell(\ell-1)(\ell-2)\zeta^{\ell}x}{1-\zeta^{\ell}x}\biggr{)}\biggr{)}.

\displaystyle\frac{1}{n}\sum_{j=1}^{n}\frac{x}{1-x}\bigg{|}_{x\mapsto\zeta^{j}x}

\displaystyle\frac{1}{n}\sum_{j=1}^{n}\frac{x}{1-x}\bigg{|}_{x\mapsto\zeta^{j}x}

\displaystyle\frac{1}{n}\sum_{j=1}^{n}\biggl{(}\frac{x}{1-x}\biggr{)}^{2}\bigg{|}_{x\mapsto\zeta^{j}x}

\displaystyle\frac{1}{n}\sum_{j=1}^{n}\frac{x}{1-x}\,\frac{\zeta^{k}x}{1-\zeta^{k}x}\bigg{|}_{x\mapsto\zeta^{j}x}

\displaystyle\frac{1}{n}\sum_{j=1}^{n}\biggl{(}\frac{x}{1-x}\biggr{)}^{3}\bigg{|}_{x\mapsto\zeta^{j}x}

\displaystyle\frac{1}{n}\sum_{j=1}^{n}\frac{x}{1-x}\,\frac{\zeta^{k}x}{1-\zeta^{k}x}\,\frac{\zeta^{\ell}x}{1-\zeta^{\ell}x}\bigg{|}_{x\mapsto\zeta^{j}x}

\displaystyle\frac{1}{n}\sum_{j=1}^{n}\biggl{(}\frac{x}{1-x}\biggr{)}^{2}\frac{\zeta^{k}x}{1-\zeta^{k}x}\bigg{|}_{x\mapsto\zeta^{j}x}

for k \neq \equiv 0 (mod n) .

\displaystyle\sum_{b=0}^{a}{\genfrac{[}{]}{0.0pt}{}{an}{bn}}(-x)^{bn}q^{bn(bn-1)/2}\bigg{|}_{q=\zeta(1-\varepsilon)}=(1-x^{n})^{a}\Biggl{(}1+\varepsilon\,\frac{x^{n}}{1-x^{n}}\sum_{\ell=1}^{an-1}\ell

\displaystyle\sum_{b=0}^{a}{\genfrac{[}{]}{0.0pt}{}{an}{bn}}(-x)^{bn}q^{bn(bn-1)/2}\bigg{|}_{q=\zeta(1-\varepsilon)}=(1-x^{n})^{a}\Biggl{(}1+\varepsilon\,\frac{x^{n}}{1-x^{n}}\sum_{\ell=1}^{an-1}\ell

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Full text

Congruences for $q$ -binomial coefficients

Wadim Zudilin

Department of Mathematics, IMAPP, Radboud University, PO Box 9010, 6500 GL Nijmegen, Netherlands

[email protected]

To George Andrews, with warm $q$ -wishes and well-looking $q$ -congruences

(Date: 23 January 2019. Revised: 1 April 2019)

Abstract.

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.

Key words and phrases:

Congruence; $q$ -binomial coefficient; cyclotomic polynomial; radial asymptotics.

2010 Mathematics Subject Classification:

11B65 (Primary), 05A10, 11A07 (Secondary)

1. Introduction

For $a$ non-negative integer, a standard $q$ -environment includes $q$ -numbers $[a]=[a]_{q}=(1-q^{a})/(1-q)\in\mathbb{Z}[q]$ , $q$ -factorials $[a]!=[1][2]\dotsb[a]\in\mathbb{Z}[q]$ and $q$ -binomial coefficients

[TABLE]

One also adopts the cyclotomic polynomials

[TABLE]

as $q$ -analogues of prime numbers, because these are the only factors of the $q$ -numbers which are irreducible over $\mathbb{Q}$ .

Arithmetically significant relations often possess several $q$ -analogues. While looking for $q$ -extensions of the classical (Wolstenholme–Ljunggren) congruence

[TABLE]

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],

[TABLE]

this author accidentally arrived at

[TABLE]

where the notation

[TABLE]

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),

[TABLE]

for any $n>1$ . This is given by Andrews in [2] for primes $n=p>3$ only; though proved modulo $\Phi_{p}(q)^{2}$ , 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

[TABLE]

It is worth mentioning that the transition from $\Phi_{n}(q)^{2}$ to $\Phi_{n}(q)^{3}$ (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 $q^{\binom{bn}{2}}\sim\sigma_{n}^{b}$ as $q\to\zeta$ , a primitive $n$ -th root of unity, the congruence (3) is seen to be an extension of the trivial ( $q$ -Lucas) congruence

[TABLE]

The principal goal of this note is to provide a modulo $\Phi_{n}(q)^{4}$ 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

[TABLE]

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 1.

The congruence

[TABLE]

holds modulo $\Phi_{n}(q)^{4}$ for any $n>1$ .

Theorem 2.

For any $n>1$ , we have the congruence

[TABLE]

We point out that a congruence $A_{1}(q)\equiv A_{2}(q)\pmod{P(q)}$ for rational functions $A_{1}(q),A_{2}(q)\in\mathbb{Q}(q)$ and a polynomial $P(q)\in\mathbb{Q}[q]$ 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 $\alpha\in\mathbb{C}$ of $P(q)$ of multiplicity $k$ , the polynomial $(q-\alpha)^{k}$ divides $A(q)$ in $\mathbb{C}[q]$ ; in other words, $A_{1}(q)-A_{2}(q)=O\bigl{(}(q-\alpha)^{k}\bigr{)}$ as $q\to\alpha$ . This latter interpretation underlies our argument in proving the results. For example, the congruence (3) can be established by verifying that

[TABLE]

when $q=\zeta(1-\varepsilon)$ and $\zeta$ 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 $O(\varepsilon^{4})$ -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.

2. 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 $\zeta$ be a primitive $n$ -th root of unity. Then, as $q=\zeta(1-\varepsilon)\to\zeta$ radially,

[TABLE]

where

[TABLE]

and

[TABLE]

Proof.

It follows from the $q$ -binomial theorem [3, Chap. 10] that

[TABLE]

Taking $N=an$ , for a primitive $n$ -th root of unity $\zeta=\zeta_{n}$ , we have

[TABLE]

When $q=\zeta(1-\varepsilon)$ , we get ${\mathrm{d}}/{\mathrm{d}}\varepsilon=-\zeta\,({\mathrm{d}}/{\mathrm{d}}q)$ . If

[TABLE]

then $f(q)|_{\varepsilon=0}=(1-x^{n})^{a}$ and

[TABLE]

In particular,

[TABLE]

Now observe the following summation formulae:

[TABLE]

Implementing this information into (11) we obtain

[TABLE]

where we intentionally omit all ordinary $\varepsilon^{3}$ -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 $\varepsilon$ - and $\varepsilon^{2}$ -terms. The exceptional $\varepsilon^{3}$ -summands are computed separately:

[TABLE]

and

[TABLE]

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). ∎

3. A $q$ -harmonic sum

Again, the notation $\zeta$ is reserved for a primitive $n$ -th root of unity. For the sum

[TABLE]

we have

[TABLE]

where $S_{n-1}(q)$ is defined in Lemma 1. It follows that, for $q=\zeta(1-\varepsilon)$ ,

[TABLE]

as $\varepsilon\to 0$ , where we use

[TABLE]

The latter asymptotics implies that

[TABLE]

which may be viewed as an extension of

[TABLE]

recorded, for example, in [6].

A different consequence of (12) is the following fact.

Lemma 2.

The term $\varepsilon^{2}S_{n-1}(\zeta)$ appearing in the expansion (9) can be replaced with

[TABLE]

when $q=\zeta(1-\varepsilon)$ and $\varepsilon\to 0$ .

4. Proof of the theorems

In order to prove Theorems 1 and 2 we need to produce ‘matching’ asymptotics for

[TABLE]

respectively. These happen to be easier than that from Lemma 1 because $q^{n^{2}}=(1-\varepsilon)^{n^{2}}$ and $q^{n}=(1-\varepsilon)^{n}$ do not depend on the choice of primitive $n$ -th root of unity $\zeta$ when $q=\zeta(1-\varepsilon)$ .

Lemma 3.

As $q=\zeta(1-\varepsilon)\to\zeta$ radially,

[TABLE]

where

[TABLE]

Proof.

For $N=a$ in (10), take $x^{n}q^{\binom{n}{2}}$ and $q^{n^{2}}$ for $x$ and $q$ :

[TABLE]

Then, for $q=\zeta(1-\varepsilon)$ , we write $y=\sigma_{n}x^{n}$ to obtain

[TABLE]

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

[TABLE]

as $q=\zeta(1-\varepsilon)\to\zeta$ radially. This means that the difference of both sides is divisible by $(q-\zeta)^{4}$ for any $n$ -th primitive root of unity $\zeta$ , hence by $\Phi_{n}(q)^{4}$ . The latter property is equivalent to the congruence (6). ∎

Proof of Theorem 2.

We first use Lemma 1 with $n=1$ :

[TABLE]

as $q\to 1$ . Now, take $n>1$ arbitrary and apply this relation for $q$ replaced with $q^{n}$ , where $q=\zeta(1-\varepsilon)$ , $0<\varepsilon<1$ and $\zeta$ is a primitive $n$ -th root of unity:

[TABLE]

as $\varepsilon\to 0$ . At the same time, from Lemma 1 we have

[TABLE]

as $\varepsilon\to 0$ . Using

[TABLE]

for $N=-\binom{bn}{2}+ab\binom{n}{2}$ , $\binom{an}{2}-\binom{bn}{2}-a(a-b)\binom{n}{2}$ , $-\binom{bn}{2}$ and $\binom{an}{2}-\binom{bn}{2}$ we deduce from the two expansions and Lemma 2 that, for $q=\zeta(1-\varepsilon)$ ,

[TABLE]

as $\varepsilon\to 0$ . This implies the congruence in (7). ∎

5. $q$ -rious congruences

In this final part, we look at the binomial coefficients as particular instances of integral ratios of factorials, also known as Chebyshev–Landau factorial ratios. In the $q$ -setting these are defined by

[TABLE]

where $\boldsymbol{a}=(a_{1},\dots,a_{r})$ and $\boldsymbol{b}=(b_{1},\dots,b_{s})$ are positive integers satisfying

[TABLE]

and

[TABLE]

(see, for example, [10]), $\lfloor\,\cdot\,\rfloor$ denotes the integer part of a number. Then $D_{n}(q)\in\mathbb{Z}[q]$ are polynomials with values

[TABLE]

at $q=1$ , and the congruences (2) and (5) generalise as follows.

Theorem 3.

In the notation

[TABLE]

the congruences

[TABLE]

are valid for any $n\geq 1$ .

Observe that when $n=p>3$ and $q\to 1$ , one recovers from any of these two the congruences

[TABLE]

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, $c_{1}(\boldsymbol{a},\boldsymbol{b})=1$ in the above notation for $c_{i}$ ) is satisfied. In turn, this more general statement follows from its validity for particular cases

[TABLE]

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 $D_{n}(\boldsymbol{a},\boldsymbol{b};q)D_{n}(\tilde{\boldsymbol{a}},\tilde{\boldsymbol{b}};q)$ 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 with $\tilde{D}_{n}(q)$ below. Notice that $\genfrac{[}{]}{0.0pt}{}{an}{bn}\equiv\binom{an}{bn}\not\equiv 0\pmod{\Phi_{n}(q)}$ , so that $\tilde{D}_{n}(q)={\genfrac{[}{]}{0.0pt}{}{an}{bn}}^{-1}$ is well defined modulo any power of $\Phi_{n}(q)$ .

For $D_{n}(q)=\genfrac{[}{]}{0.0pt}{}{an}{bn}$ 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.

Turning to $q=\zeta(1-\varepsilon)$ , where $0<\varepsilon<1$ and $\zeta$ is a primitive $n$ -th root of unity, write the congruences (2) and (5) as the asymptotic relation

[TABLE]

in which

[TABLE]

Then

[TABLE]

because we have $B(q)=B(1)+O(\varepsilon)$ as $\varepsilon\to 0$ for our choices of $B(q)$ . The resulting expansion implies the truth of (15) and (16) for $\tilde{D}_{n}(q)=D_{n}((b,a-b),(a);q)$ in view of

[TABLE]

As explained above, this also establishes the general case of (15) and (16). ∎

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 to complete this project and for supply of available knowledge on the topic. I am grateful to one of the referees whose feedback was terrific and helped me improving the exposition. Further, I thank Victor Guo for valuable comments on parts of this work.

Bibliography10

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1[1] B. Adamczewski , J. P. Bell , Ë. Delaygue and F. Jouhet , Congruences modulo cyclotomic polynomials and algebraic independence for q 𝑞 q -series, Sém. Lotharingien Combin. 78B (2017), Article #54, 12 pages.
2[2] G. E. Andrews , q 𝑞 q -analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher, Discrete Math. 204 (1999), 15–25.
3[3] G. E. Andrews , R. Askey and R. Roy , Special functions , Encyclopedia Math. Appl. 71 (Cambridge University Press, Cambridge, 1999).
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TL;DR

Contribution

Findings

Abstract

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Congruences for qqq-binomial coefficients

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

Theorem 1**.**

Theorem 2**.**

2. Expansions of qqq-binomials at roots of unity

Lemma 1**.**

Proof.

3. A qqq-harmonic sum

Lemma 2**.**

4. Proof of the theorems

Lemma 3**.**

Proof.

Proof of Theorem 1.

Proof of Theorem 2.

5. qqq-rious congruences

Theorem 3**.**

Proof of Theorem 3.

Congruences for $q$ -binomial coefficients

Theorem 1.

Theorem 2.

2. Expansions of $q$ -binomials at roots of unity

Lemma 1.

3. A $q$ -harmonic sum

Lemma 2.

Lemma 3.

5. $q$ -rious congruences

Theorem 3.