Open Access

A generalization of Morley’s congruence

Advances in Difference Equations20152015:254

https://doi.org/10.1186/s13662-015-0568-6

Received: 5 April 2015

Accepted: 8 July 2015

Published: 16 August 2015

Abstract

We establish an explicit formula for q-analog of Morley’s congruence.

Keywords

Morley’s congruence q-analogbinomial sums

MSC

11B6811B6511B83

1 Introduction

For arbitrary positive integer n, let
$$[n]_{q}=\frac{1-q^{n}}{1-q}=1+q+q^{2}+\cdots+q^{n-1}, $$
which is the q-analog of an integer n since \(\lim_{q\to1}(1-q^{n})/(1-q)=n\). Also, for \(n, m\in \mathbb{Z}\), define the q-binomial coefficients by
$$\left .\begin{bmatrix} {n}\\ {m} \end{bmatrix} \right ._{q}= \frac{[n]_{q}[n-1]_{q}\cdots [n-m+1]_{q}}{[m]_{q}[m-1]_{q}\cdots[1]_{q}} $$
when \(m\geqslant0\), and if \(m<0\) we set \(\bigl [{\scriptsize\begin{matrix}{}n\cr m\end{matrix}} \bigr ]_{q}=0\). It is easy to check that
$$\left .\begin{bmatrix} {n+1}\\ {m} \end{bmatrix} \right ._{q}=q^{m} \left .\begin{bmatrix} {n}\\ {m} \end{bmatrix} \right ._{q}+\left .\begin{bmatrix} {n}\\ {m-1} \end{bmatrix} \right ._{q}. $$
Some combinatorial and arithmetical properties of the binomial sums
$$\sum_{k=0}^{n}\binom{n}{k}^{a} \quad\mbox{and}\quad\sum_{k=0}^{n}(-1)^{k} \binom{n}{k}^{a} $$
have been investigated by several authors (e.g., Calkin [1], Cusick [2], McIntosh [3], Perlstadt [4]). Indeed, we know (cf. [5], equations (3.81) and (6.6)) that
$$ \sum_{k=0}^{2n}(-1)^{k} \binom{2n}{k}^{2}=(-1)^{n}\binom{2n}{n} $$
(1.1)
and
$$ \sum_{k=0}^{2n}(-1)^{k} \binom{2n}{k}^{3}=(-1)^{n}\binom{2n}{n} \binom{3n}{n}. $$
(1.2)
However, by using asymptotic methods, de Bruijn [6] has showed that no closed form exists for the sum \(\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}^{a}\) when \(a\geqslant4\). Wilf proved (in a personal communication with Calkin; see [1]) that the sum \(\sum_{k=0}^{n}\binom{n}{k}^{a}\) has no closed form provided that \(3\leqslant a\leqslant9\).
As a q-analog of (1.1), we have
$$ \sum_{k=0}^{2n}(-1)^{k}q^{(n-k)^{2}} \left .\begin{bmatrix} {2n}\\ {k} \end{bmatrix} \right ._{q}^{2}=(-1)^{n} \left .\begin{bmatrix} {2n}\\ {n} \end{bmatrix} \right ._{q^{2}}. $$
(1.3)
Indeed, from the well-known q-binomial theorem (cf. Corollary 10.2.2 of [7])
$$\sum_{k=0}^{n}\left .\begin{bmatrix} {n}\\ {k} \end{bmatrix} \right ._{q}(-1)^{k}q^{\binom{k}{2}}x^{k}=(x;q)_{n}, $$
where
$$\begin{aligned} (x;q)_{n}= \textstyle\begin{cases} (1-x) (1-xq)\cdots (1-xq^{n-1} ),&\mbox{if }n\geq1, \\ 1,&\mbox{if } n=0, \end{cases}\displaystyle \end{aligned}$$
it follows that
$$\begin{aligned} \bigl(x^{2};q^{2} \bigr)_{2n}&=(x;q)_{2n}(-x;q)_{2n}= \Biggl(\sum_{k=0}^{2n}\left .\begin{bmatrix} {2n}\\ {k} \end{bmatrix} \right ._{q}(-1)^{k}q^{\binom{k}{2}}x^{k} \Biggr) \Biggl(\sum_{k=0}^{2n}\left .\begin{bmatrix} {2n}\\ {k} \end{bmatrix} \right ._{q}q^{\binom{k}{2}}x^{k} \Biggr) \\ &=\sum_{m=0}^{4n}x^{m}\sum _{k=0}^{2n}\left .\begin{bmatrix} {2n}\\ {k} \end{bmatrix} \right ._{q}\left .\begin{bmatrix} {2n}\\ {m-k} \end{bmatrix} \right ._{q}(-1)^{k}q^{\binom{k}{2}+\binom{m-k}{2}}, \end{aligned}$$
whence (1.3) is derived by comparing the coefficients of \(x^{2n}\) in the equation above.
As early as 1895, with the help of De Moivre’s theorem, Morley [8] proved that
$$ (-1)^{\frac{p-1}{2}}\binom{p-1}{(p-1)/2}\equiv4^{p-1} \ \bigl(\operatorname{mod}\ p^{3}\bigr). $$
(1.4)
In [9], Pan gave a q-analog of Morley’s congruence and showed that
$$ (-1)^{\frac{n-1}{2}}q^{\frac{n^{2}-1}{4}}\left .\begin{bmatrix} {n-1}\\ {(n-1)/2} \end{bmatrix} \right ._{q^{2}}\equiv(-q;q)_{n-1}^{2}- \frac{n^{2}-1}{24}(1-q)^{2}[n]_{q}^{2} \ \bigl( \operatorname{mod}\ \Phi_{n}(q)^{3}\bigr), $$
(1.5)
where
$$\Phi_{n}(q)=\mathop{\prod_{1\leqslant j\leqslant n}}_{(j,n)=1} \bigl(q-e^{2\pi ij/n} \bigr) $$
is the nth cyclotomic polynomial. In this section, we shall establish a generalization of Morley’s congruence (1.4) proved by Cai and Granville [10], Theorem 6:
$$ \sum_{k=0}^{p-1}(-1)^{(a-1)k} \binom{p-1}{k}^{a}\equiv2^{a(p-1)}\ \bigl(\operatorname{mod}\ p^{3}\bigr) $$
(1.6)
for any prime \(p\geqslant5\) and positive integer a. We also shall obtain a generalization of (1.5) in view of (1.3).

Theorem 1.1

Let n be a positive odd integer. Then
$$\begin{aligned} &\sum_{k=0}^{n-1}(-1)^{(a-1)k}q^{a\binom{k+1}{2}} \left .\begin{bmatrix} {n-1}\\ {k} \end{bmatrix} \right ._{q}^{a} \\ &\quad\equiv(-q;q)_{n-1}^{a}+\frac{a(a-1)(n^{2}-1)}{24}(1-q)^{2}[n]_{q}^{2} \ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3}\bigr). \end{aligned}$$
(1.7)
Furthermore, we have
$$\begin{aligned} &q^{a(n^{2}-1)/4}\sum_{k=0}^{n-1}(-1)^{k}q^{a((n-1)/2-k)^{2}} \left .\begin{bmatrix} {n-1}\\ {k} \end{bmatrix} \right ._{q}^{2a} \\ &\quad\equiv(-q;q)_{n-1}^{2a}+\frac{a(a-2)(n^{2}-1)}{24}(1-q)^{2}[n]_{q}^{2} \ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3}\bigr). \end{aligned}$$
(1.8)

Remark

Clearly (1.6) is the special case of (1.7) in the limiting case \(q->1\) for \(n=p\).

2 Some lemmas

In this section, the following lemmas will be used in the proof of Theorem 1.1.

Lemma 2.1

$$ q^{kn}\equiv1-k(1-q)[n]_{q}+ \frac{k(k-1)}{2}(1-q)^{2}[n]_{q}^{2} \ \bigl( \operatorname{mod}\ [n]_{q}^{3}\bigr). $$
(2.1)

Proof

$$ q^{kn}=\sum_{j=0}^{k}(-1)^{j} \binom{k}{j}(1-q)^{j}[n]_{q}^{j}\equiv 1-k(1-q)[n]_{q}+\frac{k(k-1)}{2}(1-q)^{2}[n]_{q}^{2} \ \bigl(\operatorname{mod}\ [n]_{q}^{3}\bigr). $$
 □

Lemma 2.2

Let n be a positive odd integer. Then
$$\begin{aligned} \sum_{j=1}^{(n-1)/2} \frac{1}{[2j]_{q}^{2}} &=\sum_{j=1}^{(n-1)/2} \frac{q^{2j}}{[2j]_{q}^{2}}+(1-q)\sum_{j=1}^{(n-1)/2} \frac{1}{[2j]_{q}} \\ &\equiv-\frac{n^{2}-1}{24}(1-q)^{2}-\mathrm{Q}_{n}(2,q) (1-q)\ \bigl(\operatorname{mod}\ \Phi_{n}(q)\bigr), \end{aligned}$$
(2.2)
where the q-Fermat quotient is defined by
$$\mathrm{Q}_{n}(m,q)=\frac{(q^{m};q^{m})_{n-1}/(q;q)_{n-1}-1}{[n]_{q}}. $$

Lemma 2.3

Let n be a positive odd integer. Then
$$\begin{aligned} &2\sum_{j=1}^{(n-1)/2} \frac{1}{[2j]_{q}}+2\mathrm{Q}_{n}(2,q)-\mathrm{Q}_{n}(2,q)^{2}[n]_{q} \\ &\quad\equiv \biggl(\mathrm{Q}_{n}(2,q) (1-q)+\frac{n^{2}-1}{8}(1-q)^{2} \biggr)[n]_{q}\ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{2} \bigr). \end{aligned}$$
(2.3)

When n is an odd prime, the above two lemmas have been proved in [9], equation (2.7) and [9], Theorem 1.1, respectively. Of course, clearly the same discussions are also valid for general odd n.

3 Proofs of Theorem 1.1

In this section, we shall prove (1.7) and (1.8).

Proof

By the properties of the q-binomial coefficients, we know that
$$\begin{aligned} (-1)^{k}\left .\begin{bmatrix} {n-1}\\ {k} \end{bmatrix} \right ._{q}&= \prod_{j=1}^{k} \frac{[j]_{q}-[n]_{q}}{q^{j}[j]_{q}}\\ &\equiv q^{-\binom{k+1}{2}} \Biggl(1-\sum_{j=1}^{k} \frac {[n]_{q}}{[j]_{q}}+\sum_{1\leqslant i< j\leqslant k}\frac {[n]_{q}^{2}}{[i]_{q}[j]_{q}} \Biggr)\ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3}\bigr). \end{aligned}$$
Thus
$$\begin{aligned} &(-1)^{ak}q^{a\binom{k+1}{2}}\left .\begin{bmatrix}{n-1}\\{k} \end{bmatrix} \right ._{q}^{a} \\ &\quad\equiv1-a\sum_{j=1}^{k} \frac{[n]_{q}}{[j]_{q}}+a\sum_{1\leqslant i< j\leqslant k}\frac{[n]_{q}^{2}}{[i]_{q}[j]_{q}}+ \binom{a}{2} \Biggl(\sum_{j=1}^{k} \frac{[n]_{q}}{[j]_{q}} \Biggr)^{2}\ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3}\bigr). \end{aligned}$$
Noting that
$$\Biggl(\sum_{j=1}^{k}\frac{1}{[j]_{q}} \Biggr)^{2}=2\sum_{1\leqslant i< j\leqslant k}\frac{1}{[i]_{q}[j]_{q}}+ \sum_{j=1}^{k}\frac{1}{[j]_{q}^{2}}, $$
we have
$$\begin{aligned} &\sum_{k=1}^{n-1}(-1)^{(a-1)k}q^{a\binom{k+1}{2}} \left .\begin{bmatrix}{n-1}\\{k} \end{bmatrix} \right ._{q}^{a} \\ &\quad\equiv\sum_{k=1}^{n-1}(-1)^{k} \Biggl(1-a\sum_{j=1}^{k}\frac {[n]_{q}}{[j]_{q}}+a \sum_{1\leqslant i< j\leqslant k}\frac {[n]_{q}^{2}}{[i]_{q}[j]_{q}}+\binom{a}{2} \Biggl(\sum_{j=1}^{k}\frac {[n]_{q}}{[j]_{q}} \Biggr)^{2} \Biggr) \\ &\quad= \Biggl(-a\sum_{j=1}^{n-1} \frac{[n]_{q}}{[j]_{q}}+a^{2}\sum_{1\leqslant i< j\leqslant n-1} \frac{[n]_{q}^{2}}{[i]_{q}[j]_{q}}+\binom{a}{2}\sum_{j=1}^{n-1} \frac{[n]_{q}^{2}}{[j]_{q}^{2}} \Biggr)\sum_{k=j}^{n-1}(-1)^{k} \\ &\quad=-a\mathop{\sum_{j=1}}_{2\mid j}^{n-1}\frac{[n]_{q}}{[j]_{q}}+a^{2} \mathop{\sum_{1\leqslant i< j\leqslant n-1}}_{2\mid j}\frac {[n]_{q}^{2}}{[i]_{q}[j]_{q}}+\binom{a}{2}\mathop{\sum _{j=1}}_{2\mid j}^{n-1}\frac{[n]_{q}^{2}}{[j]_{q}^{2}}\ \bigl( \operatorname{mod}\ \Phi_{n}(q)^{3}\bigr). \end{aligned}$$
(3.1)
Thus letting \(a=1\) in (3.1), we get
$$ \mathop{\sum_{1\leqslant i< j\leqslant n-1}}_{2\mid j}\frac {[n]_{q}^{2}}{[i]_{q}[j]_{q}} \equiv(-q;q)_{n-1}-1+\mathop{\sum_{j=1}}_{2\mid j}^{n-1} \frac{[n]_{q}}{[j]_{q}}\ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3} \bigr), $$
(3.2)
whence by (2.3) and (3.2)
$$\mathop{\sum_{1\leqslant i< j\leqslant n-1}}_{2\mid j}\frac {1}{[i]_{q}[j]_{q}} \equiv \frac{\mathrm{Q}_{n}(2,q)^{2}}{2}+\frac{\mathrm{Q}_{n}(2,q)}{2}(1-q)+\frac {n^{2}-1}{16}(1-q)^{2} \ \bigl(\operatorname{mod}\ \Phi_{n}(q)\bigr). $$
Recalling (2.2) and (2.3), then we obtain
$$\begin{aligned} &\sum_{k=0}^{n-1}(-1)^{(a-1)k}q^{a\binom{k+1}{2}} \left .\begin{bmatrix}{n-1}\\{k} \end{bmatrix} \right ._{q}^{a} \\ &\quad\equiv1+a[n]_{q}\mathrm{Q}_{n}(2,q)+\binom{a}{2}[n]_{q}^{2} \mathrm{Q}_{n}(2,q)^{2}+\binom {a}{2}\frac{n^{2}-1}{12}(1-q)^{2}[n]_{q}^{2} \\ &\quad\equiv\sum_{j=0}^{a}\binom{a}{j}[n]_{q}^{j} \mathrm{Q}_{n}(2,q)^{j}+\binom {a}{2}\frac{n^{2}-1}{12}(1-q)^{2}[n]_{q}^{2} \\ &\quad=(-q;q)_{n-1}^{a}+\frac{a(a-1)(n^{2}-1)}{24}(1-q)^{2}[n]_{q}^{2} \ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3}\bigr). \end{aligned}$$
Let us turn to (1.8). Similarly
$$\begin{aligned} &\sum_{k=0}^{n-1}(-1)^{k}q^{2a\binom{k+1}{2}} \left .\begin{bmatrix}{n-1}\\{k} \end{bmatrix} \right ._{q}^{2a}-q^{a(n^{2}-1)/4} \sum_{k=0}^{n-1}(-1)^{k}q^{a((n-1)/2-k)^{2}} \left .\begin{bmatrix}{n-1}\\{k} \end{bmatrix} \right ._{q}^{2a} \\ &\quad=1-q^{a\binom{n}{2}}+\sum_{k=1}^{n-1}(-1)^{k}q^{2a\binom {k+1}{2}} \bigl(1-q^{a(\binom{n}{2}-nk)} \bigr)\left .\begin{bmatrix}{n-1}\\{k} \end{bmatrix} \right ._{q}^{2a} \\ &\quad\equiv1-q^{a\binom{n}{2}}+\sum_{k=1}^{n-1}(-1)^{k} \bigl(1-q^{a(\binom {n}{2}-nk)} \bigr) \Biggl(1-2a\sum_{j=1}^{k} \frac{[n]_{q}}{[j]_{q}} \Biggr)\ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3} \bigr). \end{aligned}$$
Recalling (2.1), then we have
$$1-q^{a(\binom{n}{2}-nk)}\equiv{} a \biggl(\frac{n-1}{2}-k \biggr) \bigl(1-q^{n} \bigr)+\binom{a((n-1)/2-k)}{2} \bigl(1-q^{n} \bigr)^{2}\ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3} \bigr), $$
therefore
$$\begin{aligned} &\sum_{k=1}^{n-1}(-1)^{k} \bigl(1-q^{a(\binom{n}{2}-nk)} \bigr) \Biggl(1-2a\sum_{j=1}^{k} \frac{[n]_{q}}{[j]_{q}} \Biggr) \\ &\quad\equiv\frac{q^{a\binom{n}{2}}-q^{-a\binom {n}{2}}}{1+q^{an}}-2a^{2}\sum_{j=1}^{n-1} \frac{[n]_{q}}{[j]_{q}}\sum_{k=j}^{n-1}(-1)^{k} \biggl(\frac{n-1}{2}-k \biggr) \bigl(1-q^{n} \bigr) \\ &\quad=\frac{q^{a\binom{n}{2}}-q^{-a\binom {n}{2}}}{1+q^{an}}+a^{2} \bigl(1-q^{n} \bigr)[n]_{q} \Biggl(\mathop{\sum_{j=1}}_{2\mid j}^{n-1} \frac{j}{[j]_{q}}+\mathop{\sum_{j=1}}_{2\nmid j}^{n-1} \frac {n-j}{[j]_{q}} \Biggr) \ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3} \bigr). \end{aligned}$$
Since
$$q^{an}= \bigl(1- \bigl(1-q^{n} \bigr) \bigr)^{a} \equiv1-a \bigl(1-q^{n} \bigr)+\binom{a}{2} \bigl(1-q^{n} \bigr)^{2}\ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{3} \bigr) $$
and
$$\frac{1}{2-a(1-q^{n})}=\frac{1}{2}\frac{1}{1-a(1-q^{n})/2}\equiv\frac {1}{2} \bigl(1+a \bigl(1-q^{n} \bigr)/2 \bigr)\ \bigl(\operatorname{mod}\ \Phi_{n}(q)^{2}\bigr), $$
we have
$$\begin{aligned} \frac{q^{a\binom{n}{2}}-q^{-a\binom{n}{2}}}{1+q^{an}}&\equiv-\frac {a(n-1)(1-q^{n})+a(n-1)/2\cdot(1-q^{n})^{2}}{2-a(1-q^{n})} \\ &\equiv-\frac{a(n-1)}{2} \biggl( \bigl(1-q^{n} \bigr)+ \frac{a+1}{2} \bigl(1-q^{n} \bigr)^{2} \biggr)\ \bigl( \operatorname{mod}\ \Phi_{n}(q)^{3}\bigr). \end{aligned}$$
Noting that
$$\begin{aligned} \mathop{\sum_{j=1}}_{2\mid j}^{n-1}\frac{j}{[j]_{q}}+\mathop{\sum _{j=1}}_{2\nmid j}^{n-1}\frac{n-j}{[j]_{q}} &=\mathop{\sum _{j=1}}_{2\mid j}^{n-1}\frac{j}{[j]_{q}}+\mathop{\sum _{j=1}}_{2\mid j}^{n-1}\frac{j}{[n-j]_{q}} \\ &=\mathop{\sum_{j=1}}_{2\mid j}^{n-1} \biggl(\frac{j}{[j]_{q}}+ \frac {jq^{j}}{[n]_{q}-[j]_{q}} \biggr) \equiv\frac{n^{2}-1}{4}(1-q)\ \bigl( \operatorname{mod}\ \Phi_{n}(q)\bigr), \end{aligned}$$
we have
$$\begin{aligned} &\sum_{k=0}^{n-1}(-1)^{k}q^{2a\binom{k+1}{2}} \left .\begin{bmatrix}{n-1}\\{k} \end{bmatrix} \right ._{q}^{2a}-q^{a(n^{2}-1)/4} \sum_{k=0}^{n-1}(-1)^{k}q^{a((n-1)/2-k)^{2}} \left .\begin{bmatrix}{n-1}\\{k} \end{bmatrix} \right ._{q}^{2a} \\ &\quad\equiv \biggl(-\binom{a(n-1)/2}{2}-\frac{a(a+1)(n-1)}{4}+\frac {a^{2}(n^{2}-1)}{4} \biggr) \bigl(1-q^{n} \bigr)^{2} \\ &\quad=\frac{a^{2}(n^{2}-1)}{8} \bigl(1-q^{n} \bigr)^{2}\ \bigl( \operatorname{mod}\ \Phi_{n}(q)^{3}\bigr). \end{aligned}$$
In view of (1.7), this concludes the proof of (1.8). □

Declarations

Acknowledgements

The authors thank the anonymous referee for his (her) valuable comments and suggestions. The first and the third authors are supported by the foundation of Jiangsu Educational Committee (No. 14KJB110008) and National Natural Science Foundation of China (Grant No. 11401301). The second author is also supported by NNSF (Grant No. 11271185).

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.

Authors’ Affiliations

(1)
Department of Teaching Affairs, Nanjing Institute of Technology, Nanjing, People’s Republic of China
(2)
Department of Mathematics, Nanjing University, Nanjing, People’s Republic of China
(3)
Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, People’s Republic of China

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© Liu et al. 2015