Congruences for \({\varvec{q}}\)-Binomial Coefficients

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

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One also adopts the cyclotomic polynomials

$$\begin{aligned} \Phi _n(q)=\prod _{\begin{array}{c} j=1\\ (j,n)=1 \end{array}}^n(q-e^{2\pi ij/n})\in {\mathbb {Z}}[q] \end{aligned}$$

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

$$\begin{aligned} \left( {\begin{array}{c}ap\\ bp\end{array}}\right) \equiv \left( {\begin{array}{c}a\\ b\end{array}}\right) \pmod {p^3} \quad \text {for any prime}\; p>3, \end{aligned}$$

(1.1)

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],

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(1.2)

this author accidentally arrived at

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(1.3)

where the notation

$$\begin{aligned} \sigma _n=(-1)^{n-1} \end{aligned}$$

is implemented. Notice that the expression on the right-hand side is a sum of two q-monomials. The q-congruence (1.3) may be compared with another q-extension of (1.1),

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(1.4)

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 (1.1) in its full modulo \(p^3\) strength can be derived from (1.4). More directly, Pan [7] shows that (1.4) can be generalised further to

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(1.5)

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 \(q^{\left( {\begin{array}{c}bn\\ 2\end{array}}\right) }\sim \sigma _n^b\) as \(q\rightarrow \zeta \), a primitive n-th root of unity, the congruence (1.3) is seen to be an extension of the trivial (q-Lucas) congruence

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The principal goal of this note is to provide a modulo \(\Phi _n(q)^4\) extension of (1.3) (see Lemma 2.1 below) as well as to use the result for extending the congruences (1.2) and (1.5). In this way, our theorems provide two q-extensions of the congruence

$$\begin{aligned} \left( {\begin{array}{c}ap\\ bp\end{array}}\right) \equiv \left( {\begin{array}{c}a\\ b\end{array}}\right) +ab(a-b)\left( {\begin{array}{c}a\\ b\end{array}}\right) p\sum _{k=1}^{p-1}\frac{1}{k}\pmod {p^4} \quad \text {for prime}~ p>3. \end{aligned}$$

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 1.1

The congruence

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(1.6)

holds modulo \(\Phi _n(q)^4\) for any \(n>1\).

Theorem 1.2

For any \(n>1\), we have the congruence

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(1.7)

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 \(q\rightarrow \alpha \). This latter interpretation underlies our argument in proving the results. For example, the congruence (1.3) can be established by verifying that

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(1.8)

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 the q-Gauss congruences. It does not seem straightforward to us but Gorodetsky’s method may be capable of proving Theorems 1.1 and 1.2. Furthermore, the part [4, Sect. 2.3] contains a survey on q-analogues of (1.1).

After proving an asymptotical expansion for q-binomial coefficients at roots of unity in Sect. 2 [essentially, the \(O(\varepsilon ^4)\)-extension of (1.8)], we perform a similar asymptotic analysis for q-harmonic sums in Sect. 3. The information gathered is then applied in Sect. 4 to proving Theorems 1.1 and 1.2. Finally, in Sect. 5, we generalise the congruences (1.2) and (1.5) in a different direction, to integral factorial ratios.

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 2.1

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

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(2.1)

where

$$\begin{aligned} \rho _0(a,n)&=\frac{3(an-1)^2-an^2-1}{24},\\ \rho _1(a,b,n)&=\frac{abn^2(an-1)(an-n-2)+(an+2)(an-1)^2(an-3)+an^2+a+2}{48} \end{aligned}$$

and

$$\begin{aligned} S_{n-1}(q)=\frac{1}{2}\sum _{k=1}^{n-1}\frac{kq^k((k+1)q^k+k-1)}{(1-q^k)^3}. \end{aligned}$$

Proof

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

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(2.2)

Taking \(N=an\), for a primitive n-th root of unity \(\zeta =\zeta _n\), we have

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(2.3)

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

$$\begin{aligned} f(q)=(x;q)_{an} \quad \text {and}\quad g(q)=\frac{{\mathrm d}}{{\mathrm d}q}\log f(q)=-\sum _{\ell =1}^{an-1}\frac{\ell q^{\ell -1}x}{1-q^\ell x}, \end{aligned}$$

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

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

In particular,

$$\begin{aligned} \frac{{\mathrm d}f}{{\mathrm d}\varepsilon }\bigg |_{\varepsilon =0}&=(1-x^n)^a\sum _{\ell =1}^{an-1}\frac{\ell \zeta ^\ell x}{1-\zeta ^\ell x}, \\ \frac{{\mathrm d}^2f}{{\mathrm d}\varepsilon ^2}\bigg |_{\varepsilon =0}&=(1-x^n)^a\biggl (\biggl (\sum _{\ell =1}^{an-1}\frac{\ell \zeta ^\ell x}{1-\zeta ^\ell x}\biggr )^2 -\sum _{\ell =1}^{an-1}\biggl (\frac{\ell ^2\zeta ^{2\ell }x^2}{(1-\zeta ^\ell x)^2}+\frac{\ell (\ell -1)\zeta ^\ell x}{1-\zeta ^\ell x}\biggr )\biggr ) \end{aligned}$$

and

$$\begin{aligned} \frac{{\mathrm d}^3f}{{\mathrm d}\varepsilon ^3}\bigg |_{\varepsilon =0}&=(1-x^n)^a\biggl (\biggl (\sum _{\ell =1}^{an-1}\frac{\ell \zeta ^\ell x}{1-\zeta ^\ell x}\biggr )^3 \\&\quad -3\sum _{\ell =1}^{an-1}\frac{\ell \zeta ^\ell x}{1-\zeta ^\ell x} \sum _{\ell =1}^{an-1}\biggl (\frac{\ell ^2\zeta ^{2\ell }x^2}{(1-\zeta ^\ell x)^2}+\frac{\ell (\ell -1)\zeta ^\ell x}{1-\zeta ^\ell x}\biggr ) \\&\quad +\sum _{\ell =1}^{an-1}\biggl (\frac{2\ell ^3\zeta ^{3\ell }x^3}{(1-\zeta ^\ell x)^3}+\frac{3\ell ^2(\ell -1)\zeta ^{2\ell }x^2}{(1-\zeta ^\ell x)^2}+\frac{\ell (\ell -1)(\ell -2)\zeta ^\ell x}{1-\zeta ^\ell x}\biggr )\biggr ). \end{aligned}$$

Now observe the following summation formulae:

$$\begin{aligned} \frac{1}{n}\sum _{j=1}^n\frac{x}{1-x}\bigg |_{x\mapsto \zeta ^jx}&=\frac{x^n}{1-x^n}, \\ \frac{1}{n}\sum _{j=1}^n\biggl (\frac{x}{1-x}\biggr )^2\bigg |_{x\mapsto \zeta ^jx}&=\frac{nx^n}{(1-x^n)^2}-\frac{x^n}{1-x^n}, \\ \frac{1}{n}\sum _{j=1}^n\frac{x}{1-x}\,\frac{\zeta ^kx}{1-\zeta ^kx}\bigg |_{x\mapsto \zeta ^jx}&=-\frac{x^n}{1-x^n} \quad \text {for}\; k\not \equiv 0\pmod n, \\ \frac{1}{n}\sum _{j=1}^n\biggl (\frac{x}{1-x}\biggr )^3\bigg |_{x\mapsto \zeta ^jx}&=\frac{n^2x^n(1+x^n)}{2(1-x^n)^3}-\frac{3nx^n}{2(1-x^n)^2}+\frac{x^n}{1-x^n}, \\ \frac{1}{n}\sum _{j=1}^n\frac{x}{1-x}\,\frac{\zeta ^kx}{1-\zeta ^kx}\,\frac{\zeta ^\ell x}{1-\zeta ^\ell x}\bigg |_{x\mapsto \zeta ^jx}&=\frac{x^n}{1-x^n} \\&\qquad \text {for}\; k\not \equiv 0,\; \ell \not \equiv 0,\; k\not \equiv \ell \pmod n, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{n}\sum _{j=1}^n\biggl (\frac{x}{1-x}\biggr )^2\frac{\zeta ^kx}{1-\zeta ^kx}\bigg |_{x\mapsto \zeta ^jx}&=\frac{nx^n}{(1-x^n)^2}\,\frac{\zeta ^k}{1-\zeta ^k}+\frac{x^n}{1-x^n} \\&\quad \text {for}~ k\not \equiv 0\pmod n. \end{aligned}$$

Implementing this information in (2.3), we obtain

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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:

$$\begin{aligned} \sum _{\begin{array}{c} \ell _1,\ell _2,\ell _3=1\\ \ell _1\equiv \ell _2\not \equiv \ell _3\pmod n \end{array}}^{an-1}\ell _1\ell _2\ell _3\,\frac{\zeta ^{\ell _3-\ell _1}}{1-\zeta ^{\ell _3-\ell _1}} =-a^3\sum _{k=1}^{n-1}\frac{k\zeta ^k((k+1)\zeta ^k+k-1)}{(1-\zeta ^k)^3} \\ \quad -\frac{a^3n(n-1)(3an(an-1)(an-2)+n^2(an-2a-1)-2)}{48} \end{aligned}$$

and

$$\begin{aligned} \sum _{\begin{array}{c} \ell _1,\ell _2=1\\ \ell _1\not \equiv \ell _2\pmod n \end{array}}^{an-1} \ell _1^2\ell _2\,\frac{\zeta ^{\ell _2-\ell _1}}{1-\zeta ^{\ell _2-\ell _1}} =-a^2\sum _{k=1}^{n-1}\frac{k\zeta ^k((k+1)\zeta ^k+k-1)}{(1-\zeta ^k)^3} \\ \quad -\frac{a^2n(n-1)(an(an-1)(2an-3)-an^2-1)}{24}. \end{aligned}$$

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 (2.1). \(\square \)

A different consequence of (3.2) is the following fact.

Proof of Theorem 5.1

Though the congruences (5.3) and (5.4) are between polynomials rather than rational functions, we prove the theorem without the assumption (5.2): in other words, the congruences remain true for the rational functions \(D_n(q)\) provided that the balancing condition (5.1) (equivalently, \(c_1(\varvec{a},\varvec{b})=0\) in the above notation for \(c_i\)) is satisfied. In turn, this more general statement follows from its validity for particular cases

$$\begin{aligned} D_n(q)=\frac{[an]!}{[bn]!\,[(a-b)n]!} \quad \text {and}\quad \tilde{D}_n(q)=\frac{[bn]!\,[(a-b)n]!}{[an]!} \end{aligned}$$

by induction (on \(r+s\), say). Indeed, the inductive step exploits the property of both (5.3) and (5.4) to imply the congruence for the product

$$\begin{aligned} D_n(\varvec{a},\varvec{b};q)D_n(\tilde{\varvec{a}},\tilde{\varvec{b}};q) \end{aligned}$$

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 Open image in new window, so that Open image in new window is well-defined modulo of any power of \(\Phi _n(q)\).

For Open image in new window, we have \(c_2=b(a-b)\) and \(c_2+c_3=ab(a-b)/2\); hence, (5.3) and (5.4) follow from (1.2) and (1.5), respectively.

Turning to \(q=\zeta (1-\varepsilon )\), where \(0<\varepsilon <1\) and \(\zeta \) is a primitive n-th root of unity, write the congruences (1.2) and (1.5) as the asymptotic relation

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in which

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and

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Then

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because we have \(B(q)=B(1)+O(\varepsilon )\) as \(\varepsilon \rightarrow 0\) for our choices of B(q). The resulting expansion implies the truth of (5.3) and (5.4) for \(\tilde{D}_n(q)=D_n((b,a-b),(a);q)\) in view of

$$\begin{aligned} c_i((b,a-b),(a))=-c_i((a),(b,a-b)) \quad \text {for}~ i=2,3. \end{aligned}$$

As explained above, this also establishes the general case of (5.3) and (5.4).\(\square \)

For related Lucas-type congruences satisfied by the q-factorial ratios \(D_n(q)\), see [1].