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

- 63 Downloads

## Abstract

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

## Keywords

Congruence*q*-Binomial coefficient Cyclotomic polynomial Radial asymptotics

## Mathematics Subject Classification

Primary 11B65 Secondary 05A10 11A07## 1 Introduction

*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 coefficientsOne also adopts the cyclotomic polynomials

*q*-analogues of prime numbers, because these are the only factors of the

*q*-numbers which are irreducible over \({\mathbb {Q}}\).

*q*-analogues. While looking for

*q*-extensions of the classical (Wolstenholme–Ljunggren) congruence

*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],

*q*-monomials. The

*q*-congruence (1.3) may be compared with another

*q*-extension of (1.1),

*q*-Chu–Vandermonde identity) whereas no combinatorial proof is known for the latter.

*n*-th root of unity, the congruence (1.3) is seen to be an extension of the trivial (

*q*-Lucas) congruenceThe 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

*q*-analogues.

### Theorem 1.1

### Theorem 1.2

*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

*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.

## 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 2.1

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

### Proof

*q*-binomial theorem [3, Chap. 10] that

*n*-th root of unity \(\zeta =\zeta _n\), we have

*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:

## 3 A *q*-Harmonic Sum

*n*-th root of unity. For the sum

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

### Lemma 3.1

## 4 Proof of the Theorems

*n*-th root of unity \(\zeta \) when \(q=\zeta (1-\varepsilon )\).

### Lemma 4.1

### Proof

*x*and

*q*:Then, for \(q=\zeta (1-\varepsilon )\), we write \(y=\sigma _nx^n\) to obtain

### Proof of Theorem 1.1

*n*-th primitive root of unity \(\zeta \), hence by \(\Phi _n(q)^4\). The latter property is equivalent to the congruence (1.6). \(\square \)

### Proof

*q*replaced with \(q^n\), where \(q=\zeta (1-\varepsilon )\), \(0<\varepsilon <1\) and \(\zeta \) is a primitive

*n*-th root of unity:as \(\varepsilon \rightarrow 0\). At the same time, from Lemma 2.1, we haveas \(\varepsilon \rightarrow 0\). Using

## 5 *q*-Rious Congruences

*q*-setting, these are defined by

### Theorem 5.1

*q*-congruences may be generalised even further in the spirit of Theorems 1.1 and 1.2, and that the polynomials \(D_n(q)\) satisfy the

*q*-Gauss relations from [4]. We do not pursue this line here.

### Proof of Theorem 5.1

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.

*n*-th root of unity, write the congruences (1.2) and (1.5) as the asymptotic relationin whichandThenbecause 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

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

## Notes

### Acknowledgements

I would like to thank Armin Straub for encouraging me to complete this project and for the 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.

## References

- 1.Adamczewski, B., Bell, J.P., Delaygue, É., Jouhet, F.: Congruences modulo cyclotomic polynomials and algebraic independence for \(q\)-series. Sém. Lothar. Combin. 78B, #A54 (2017)Google Scholar
- 2.Andrews, G.E.: \(q\)-Analogs of the binomial coefficient congruences of Babbage, Wolstenholme and Glaisher. Discrete Math. 204(1-3), 15–25 (1999)MathSciNetCrossRefGoogle Scholar
- 3.Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)CrossRefGoogle Scholar
- 4.Gorodetsky, O.: \(q\)-Congruences, with applications to supercongruences and the cyclic sieving phenomenon. Intern. J. Number Theory 15(9), 1919–1968 (2019)MathSciNetCrossRefGoogle Scholar
- 5.Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)MathSciNetCrossRefGoogle Scholar
- 6.Meštrović, R.: Wolstenholme’s theorem: its generalizations and extensions in the last hundred and fifty years (1862–2012). arXiv:1111.3057 (2011)
- 7.Pan, H.: Factors of some lacunary \(q\)-binomial sums. Monatsh. Math. 172(4), 387–398 (2013)MathSciNetCrossRefGoogle Scholar
- 8.Straub, A.: A \(q\)-analog of Ljunggren’s binomial congruence. In: DMTCS Proceedings: 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, pp. 897–902. Assoc. Discrete Math. Theor. Comput. Sci., Nancy (2011)Google Scholar
- 9.Straub, A.: Supercongruences for polynomial analogs of the Apéry numbers. Proc. Amer. Math. Soc. 147(3), 1023–1036 (2019)MathSciNetCrossRefGoogle Scholar
- 10.Warnaar, S.O., Zudilin, W.: A \(q\)-rious positivity. Aequationes Math. 81(1-2), 177–183 (2011)MathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.