# Gaussian Binomial Coefficients with Negative Arguments

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## Abstract

Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients. Moreover, we demonstrate that several of the well-known arithmetic properties of binomial coefficients also hold in the case of negative entries. In particular, we show that Lucas’ theorem on binomial coefficients modulo *p* not only extends naturally to the case of negative entries, but even to the Gaussian case.

## Keywords

*q*-Binomial coefficients

*q*-Binomial theorem Lucas congruences

## Mathematics Subject Classification

05A10 05A30 11B65 11A07## 1 Introduction

Occasionally, the binomial coefficient \(\left( {\begin{array}{c}n\\ k\end{array}}\right) \), with integer entries *n* and *k*, is considered to be zero when \(k < 0\) (see Remark 1.9, where it is further indicated that the common extension, via the gamma function, of binomial coefficients to complex *n* and *k* does not immediately lend itself to the case of negative integers *k*). However, as demonstrated by Loeb [14], an alternative extension of the binomial coefficients to negative arguments is arguably more natural for many combinatorial or number theoretic applications. The *q*-binomial coefficients \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q\) (often also referred to as Gaussian polynomials) are a polynomial generalization of the binomial coefficients that occur naturally in varied contexts, including combinatorics, number theory, representation theory and mathematical physics. For instance, if *q* is a prime power, then they count the number of *k*-dimensional subspaces of an *n*-dimensional vector space over the finite field \({\mathbb {F}}_q\). We refer to the book [11] for a very pleasant introduction to the *q*-calculus. Yet, surprisingly, *q*-binomial coefficients with general integer entries have, to the best of our knowledge, not been studied in the literature (Gasper and Rahman define *q*-binomial coefficients with complex entries in [9, Ex. 1.2 and (I.40)], see Remark 1.8, but do not pursue the case of integer entries; Loeb [14] does briefly discuss such *q*-binomial coefficients but only in the case \(k \ge 0\)). The goal of this paper is to fill this gap, and to demonstrate that these generalized *q*-binomial coefficients are natural, by showing that they satisfy many of the fundamental combinatorial and arithmetic properties of the usual binomial coefficients. In particular, we extend Loeb’s interesting combinatorial interpretation [14] in terms of sets with negative numbers of elements. On the arithmetic side, we prove that Lucas’ theorem can be uniformly generalized to both binomial coefficients and *q*-binomial coefficients with negative entries.

*q*-series, it is common to introduce the

*q*-binomial coefficient, for \(n, k \ge 0\), as the quotient

*q*-Pochhammer symbol

*n*and

*k*, we employ the natural convention that, for all integers

*r*and

*s*,

*q*-Pochhammer symbol are equivalent and hold for all integers

*n*.

Note that \((q ; q)_n = \infty \) whenever \(n < 0\), so that (1.1) does not immediately extend to the case when *n* or *k* is negative. We, therefore, make the following definition, which clearly reduces to (1.1) when \(n, k \ge 0\):

### Definition 1.1

*n*and

*k*,

Though not immediately obvious from (1.4) when *n* or *k* is negative, these generalized *q*-binomial coefficients are Laurent polynomials in *q* with integer coefficients. In particular, upon setting \(q = 1\), we always obtain integers.

### Example 1.2

*n*and

*k*, the

*q*-binomial coefficients are also characterized by the Pascal relation:

Among the other basic properties of the generalized *q*-binomial coefficients are the following: All of these are well known in the classical case \(k \ge 0\) (see, for instance, [9, Appendix I]). That they extend uniformly to all integers *n* and *k* (though, as illustrated by (1.5) and item (c), some care has to be applied when generalizing certain properties) serves as a first indication that the generalized *q*-binomial coefficients are natural objects. For (c), the sign function \({\text {sgn}} (k)\) is defined to be 1 if \(k \ge 0\), and \(- 1\) if \(k < 0\).

### Lemma 1.3

*n*and

*k*,

- (a)
\(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q = q^{k (n - k)} \left( {\begin{array}{c}n\\ k\end{array}}\right) _{q^{- 1}}\),

- (b)
\(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q = \left( {\begin{array}{c}n\\ n - k\end{array}}\right) _q\),

- (c)
\(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q = (- 1)^k {\text {sgn}} (k) q^{\frac{1}{2} k (2 n - k + 1)} \left( {\begin{array}{c}k - n - 1\\ k\end{array}}\right) _q\),

- (d)
\(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q = \frac{1 - q^n}{1 - q^k} \left( {\begin{array}{c}n - 1\\ k - 1\end{array}}\right) _q\), if \(k \ne 0\).

Properties (b) and (d) follow directly from the definition (1.4), while property (a) is readily deduced from (1.5) combined with (b). In the classical case \(n, k \ge 0\), property (a) reflects the fact that the *q*-binomial coefficient is a self-reciprocal polynomial in *q* of degree \(k (n - k)\). In contrast to that and as illustrated in Example 1.2, the *q*-binomial coefficients with negative entries are Laurent polynomials, whose degrees are recorded in Corollary 3.3.

The reflection rule (c) is the subject of Sect. 3 and is proved in Theorem 3.1. Rule (c) reduced to the case \(q = 1\) is the main object in [19], where Sprugnoli observed the necessity of including the sign function when extending the binomial coefficient to negative entries. Sprugnoli further realized that the basic symmetry (b) and the negation rule (c) act on binomial coefficients as a group of transformations isomorphic to the symmetric group on three letters. In Sect. 3, we observe that the same is true for *q*-binomial coefficients.

*q*-binomial coefficient is a

*q*-hypergeometric term.

### Example 1.4

*k*,

In Sect. 4, we review the remarkable and beautiful observation of Loeb [14] that the combinatorial interpretation of binomial coefficients as counting subsets can be naturally extended to the case of negative entries. We then prove that this interpretation can be generalized to *q*-binomial coefficients. Theorem 4.5, our main result of that section, is a precise version of the following:

### Theorem 1.5

*n*and

*k*,

*k*-element subsets

*Y*of the

*n*-element set \(X_n\).

The notion of sets (and subsets) with a negative number of elements, as well as the definitions of \(\sigma \) and \(X_n\), are deferred to Sect. 4. In the previously known classical case \(n, k \ge 0\), the sign in that formula is positive, \(X_n = \{ 0, 1, 2, \ldots , n - 1 \}\), and \(\sigma (Y)\) is the sum of the elements of *Y*. As an application of Theorem 1.5, we demonstrate at the end of Sect. 4 how to deduce from it generalized versions of the Chu–Vandermonde identity as well as the (commutative) *q*-binomial theorem.

In Sect. 5, we discuss the binomial theorem, which interprets the binomial coefficients as coefficients in the expansion of \((x + y)^n\). Loeb showed that, by also considering expansions in inverse powers of *x*, one can extend this interpretation to the case of binomial coefficients with negative entries. Once more, we are able to show that the generalized *q*-binomial coefficients share this property in a uniform fashion.

### Theorem 1.6

*n*,

*k*,

Here, the operator \(\{ x^k y^{n - k} \}\), which is defined in Sect. 5, extracts the coefficient of \(x^k y^{n - k}\) in the appropriate expansion of what follows.

*p*is a prime, then

*p*-adic digits of the nonnegative integers

*n*and

*k*, respectively. In Sect. 6, we show that this congruence in fact holds for all integers

*n*and

*k*. In Sect. 7, we prove that generalized Lucas congruences uniformly hold for

*q*-binomial coefficients.

### Theorem 1.7

*n*and

*k*,

Here, \(\Phi _m (q)\) is the *m*-th cyclotomic polynomial. The classical special case \(n, k \ge 0\) of this result has been obtained by Olive [16] and Désarménien [7].

We conclude this introduction with some comments on alternative approaches to and conventions for binomial coefficients with negative entries. In particular, we remark on the current state of computer algebra systems and on how it could benefit from the generalized *q*-binomial coefficients introduced in this paper.

### Remark 1.8

*n*and

*k*such that \(n, k \not \in \{ - 1, - 2, \ldots \}\). This definition, however, does not immediately lend itself to the case of negative integers; though the structure of poles (and lack of zeros) of the underlying gamma function is well understood, the binomial function (1.6) has a subtle structure when viewed as a function of two variables. For a study of this function, as well as a historical account on binomials, we refer to [8]. For instance, let us note that, employing (1.6) as the definition of the binomial coefficients, we have

*n*and

*k*are now allowed to take any complex values. This is in fact the definition that Loeb [14] and Sprugnoli [19] adopt. (That the

*q*-binomial coefficients we introduce in (1.4) reduce to (1.7) when \(q = 1\) can be seen, for instance, from observing that the

*q*-Pascal relation (1.5) reduces to the Pascal relation established by Loeb for (1.7).)

*q*-binomial coefficient for complex arguments

*n*and

*k*(and \(| q | < 1\)) using the

*q*-gamma function as

*n*(for negative integers

*n*, formula (1.9) is compatible with (1.3)). When

*n*or

*k*is a negative integer, however, the right-hand side of (1.8) must be interpreted appropriately by cancelling matching zeros in the infinite products. Interpreting (1.8) in this way, it follows from (1.9) that definition (1.8) is necessarily equivalent to (1.4).

### Remark 1.9

Other conventions for binomial coefficients with negative integer entries exist and have their merit. Most prominently, if, for instance, one insists that Pascal’s relation (1.5) should hold for all integers *n* and *k*, then the resulting version of the binomial coefficients is zero when \(k < 0\) (see, for instance, [12, Section 1.2.6 (3)]). On the other hand, as illustrated by the results in [14] and this paper, it is reasonable and preferable for many purposes to extend the classical binomial coefficients (as well as its polynomial counterpart) to negative arguments as done here.

As an unfortunate consequence, both conventions are implemented in current computer algebra systems, which can be a source of confusion. For instance, SageMath currently (as of Version 8.0) uses the convention that all binomial coefficients with \(k < 0\) are evaluated as zero. On the other hand, recent versions of Mathematica (at least Version 9 and higher) and Maple (at least Version 18 and higher) evaluate binomial coefficients with negative entries in the way advertised in [14] and here.

In Version 7, Mathematica introduced the QBinomial[n,k,q] function; however, as of Version 11, this function evaluates the *q*-binomial coefficient as zero whenever \(k < 0\). Similarly, Maple implements these coefficients as QBinomial(n,k,q), but, as of Version 18, choosing \(k < 0\) results in a division-by-zero error. We hope that this paper helps to adjust these inconsistencies with the classical case \(q = 1\) by offering a natural extension of the *q*-binomial coefficient for negative entries.

## 2 Characterization via a *q*-Pascal Relation

*n*and

*k*that are not both zero [14, Proposition 4.4]. In this brief section, we show that the

*q*-binomial coefficients (with arbitrary integer entries), defined in (1.4), are also characterized by a

*q*-analog of the Pascal rule. It is well known that this is true for the familiar

*q*-binomial coefficients when \(n, k \ge 0\) (see, for instance, [11, Proposition 6.1]).

### Lemma 2.1

*n*and

*k*, the

*q*-binomial coefficients are characterized by

Observe that \(\left( {\begin{array}{c}0\\ 0\end{array}}\right) _q = 1\), while the corresponding right-hand side of (2.2) is \(\left( {\begin{array}{c}- 1\\ - 1\end{array}}\right) _q + q^0 \left( {\begin{array}{c}- 1\\ 0\end{array}}\right) _q = 2 \ne 1\), illustrating the need to exclude the case \((n, k) = (0, 0)\). It should also be noted that the initial conditions are natural but not minimal: the case \(\left( {\begin{array}{c}n\\ 0\end{array}}\right) _q\) with \(n \le - 2\) is redundant (but consistent).

### Proof of Lemma 2.1

*q*-binomial coefficient. It, therefore, only remains to show that (2.2) holds for the

*q*-binomial coefficient as defined in (1.4). For the purpose of this proof, let us write

*n*and

*k*,

*n*and

*k*. If \(n \ne k\), then

### Remark 2.2

*q*-binomial coefficient, we find that Pascal’s relation (2.2) is equivalent to the alternative form

## 3 Reflection Formulas

*k*is allowed to be negative. Instead, he shows that, for all integers

*n*and

*k*,

*q*-binomial coefficients. Observe that the result of Sprugnoli [19] is immediately obtained as the special case \(q = 1\).

### Theorem 3.1

*n*and

*k*,

### Proof

*n*and

*k*,

*n*,

*n*and

*k*. Suppose we have already shown that, for any integer

*n*,

*q*-binomial coefficients are different from zero (for more details on this argument, see [19, Theorem 2.2]).

It was observed in [19, Theorem 3.2] that the basic symmetry (Lemma 1.3(b)) and the negation rule (3.2) act on (formal) binomial coefficients as a group of transformations isomorphic to the symmetric group on three letters. The same is true for *q*-binomial coefficients. Since the argument is identical, we only record the resulting six forms for the *q*-binomial coefficients.

### Corollary 3.2

*n*and

*k*,

### Proof

*q*-binomial coefficients are different from zero (again, see [19, Theorem 2.2] for more details on this argument). \(\square \)

It follows directly from the definition (1.4) that the *q*-binomial coefficient \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q\) is zero if \(k > n \ge 0\) or if \(n \ge 0 > k\). The third equality in Corollary 3.2 then makes it plainly visible that the *q*-binomial coefficient also vanishes if \(0> k > n\). Moreover, we can read off from Corollary 3.2 that the *q*-binomial coefficient is nonzero otherwise, that is, it is nonzero precisely in the three regions \(0 \le k \le n\) (the classical case), \(n < 0 \le k\) and \(k \le n < 0\). More precisely, we have the following, of which the first statement is, of course, well known (see, for instance, [11, Corollary 6.1]).

### Corollary 3.3

- (a)
If \(0 \le k \le n\), then \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q\) is a polynomial of degree \(k (n - k)\).

- (b)
If \(n < 0 \le k\), then \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q\) is \(q^{\frac{1}{2} k (2 n - k + 1)}\) times a polynomial of degree \(k (- n - 1)\).

- (c)
If \(k \le n < 0\), then \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q\) is \(q^{\frac{1}{2} (n (n + 1) - k (k + 1))}\) times a polynomial of degree \((- n - 1) (n - k)\).

*n*and

*k*. Indeed, the three regions in which the binomial coefficients are nonzero are \(0 \le k \le n\), \(n < 0 \le k\) and \(k \le n < 0\), and these three are permuted by the transformations in Corollary 3.2.

## 4 Combinatorial Interpretation

For integers \(n, k \ge 0\), the binomial coefficient \(\left( {\begin{array}{c}n\\ k\end{array}}\right) \) counts the number of *k*-element subsets of a set with *n* elements. It is a remarkable and beautiful observation of Loeb [14] that this interpretation (up to an overall sign) can be extended to all integers *n* and *k* by a natural notion of sets with a negative number of elements. In this section, after briefly reviewing Loeb’s result, we generalize this combinatorial interpretation to the case of *q*-binomial coefficients.

*U*be a collection of elements (the “universe”). A set

*X*with elements from

*U*can be thought of as a map \(M_X:U \rightarrow \{ 0, 1 \}\) with the understanding that \(u \in X\) if and only if \(M_X (u) = 1\). Similarly, a multiset

*X*can be thought of as a map

*u*. In this spirit, Loeb introduces a

*hybrid set*

*X*as a map \(M_X :U \rightarrow {\mathbb {Z}}\). We will denote hybrid sets in the form \(\{ \ldots | \ldots \}\), where elements with a positive multiplicity are listed before the bar, and elements with a negative multiplicity after the bar.

### Example 4.1

The hybrid set \(\{ 1, 1, 4|2, 3, 3 \}\) contains the elements 1, 2, 3, 4 with multiplicities \(2, - 1, - 2, 1\), respectively.

A hybrid set *Y* is a subset of a hybrid set *X*, if one can repeatedly remove elements from *X* (here, removing means decreasing by one the multiplicity of an element with nonzero multiplicity) and thus obtain *Y* or have removed *Y*. We refer to [14] for a more formal definition and further discussion, including a proof that this notion of being a subset is a well-defined partial order (but not a lattice). The interested reader will find there also connections to symmetric functions and, in particular, the involutive relation between elementary and complete symmetric functions.

### Example 4.2

From the hybrid set \(\{ 1, 1, 4|2, 3, 3 \}\) we can remove the element 4 to obtain \(\{ 1, 1|2, 3, 3 \}\) (at which point, we cannot remove 4 again). We can further remove 2 twice to obtain \(\{ 1, 1|2, 2, 2, 3, 3 \}\). Consequently, \(\{ 4| \}\) and \(\{ 1, 1|2, 3, 3 \}\) as well as \(\{ 2, 2, 4| \}\) and \(\{ 1, 1|2, 2, 2, 3, 3 \}\) are subsets of \(\{ 1, 1, 4|2, 3, 3 \}\).

Following [14], a *new set* is a hybrid set such that either all multiplicities are 0 or 1 (a “positive set”) or all multiplicities are 0 or \(- 1\) (a “negative set”).

### Theorem 4.3

[14]. For all integers *n* and *k*, the number of *k*-element subsets of an *n*-element new set is \(\left| \left( {\begin{array}{c}n\\ k\end{array}}\right) \right| \).

### Example 4.4

*n*elements, by which we mean \(X_n = \{ 0, 1, \ldots , n - 1| \}\), if \(n \ge 0\), and \(X_n = \{ | - 1, - 2, \ldots , n \}\), if \(n < 0\). For a hybrid set \(Y \subseteq X_n\) with multiplicity function \(M_Y\), we write

*Y*is a classic set, then \(\sigma (Y)\) is just the sum of its elements. With this setup, we prove the following uniform generalization of [14, Theorem 5.2], which is well known in the case that \(n, k \ge 0\) (see, for instance, [11, Theorem 6.1]):

### Theorem 4.5

*n*and

*k*,

*k*-element subsets

*Y*of the

*n*-element set \(X_n\). If \(0 \le k \le n\), then \(\varepsilon = 1\). If \(n < 0 \le k\), then \(\varepsilon = (- 1)^k\). If \(k \le n < 0\), then \(\varepsilon = (- 1)^{n - k}\).

### Proof

The case \(n, k \ge 0\) is well known. A proof can be found, for instance, in [11, Theorem 6.1]. On the other hand, if \(n \ge 0 > k\), then both sides vanish.

*C*(

*n*,

*k*) is the collection of

*k*-element subsets of the

*n*-element set \(X_n^+ = \{ |0, 1, 2, \ldots , | n | - 1 \}\) (note that a natural bijection \(X_n \rightarrow X_n^+\) is given by \(x \mapsto | n | + x\)).

*n*,

*k*and suppose that (4.2) holds whenever

*n*and

*k*are replaced with \(n'\) and \(k'\) such that \(n< n' < 0\) or \(n = n'< 0 \le k' < k\). Then,

*D*(

*n*,

*k*) is the collection of

*k*-element subsets

*Y*of the

*n*-element set \(X_n = \{ | - 1, - 2, \ldots , n \}\). If \(n = - 1\), then (4.3) holds because the only contribution comes from \(Y = \{ | - 1, - 1, \ldots , - 1 \}\), with \(M_Y (- 1) = | k |\) and \(\sigma (Y) = - k\). If, on the other hand, \(k = - 1\), then (4.3) holds as well because a contributing

*Y*only exists if \(n = - 1\). Fix \(n, k < - 1\) and suppose that (4.3) holds whenever

*n*and

*k*are replaced with \(n'\) and \(k'\) such that \(k< k' < 0\) and \(n \le n' < 0\). Then the right-hand side of (4.3) equals

*n*from

*Y*(once) and, to make up for that, replace \(\sigma (Y)\) with \(\sigma (Y) - n\). Proceeding this way, we see that the right-hand side of (4.3) equals

### Remark 4.6

*k*elements from a set of

*n*elements with replacement is

*k*elements by lining them up in some order with elements of the same kind separated by dividers (since there are

*n*kinds of elements, we need \(n - 1\) dividers). The \(n - 1\) positions of the dividers among all \(k + n - 1\) positions then encode a choice of

*k*elements. Formula (4.2) is a

*q*-analog of this combinatorial fact.

### Example 4.7

In the remainder of this section, we consider two applications of Theorem 4.5. The first of these is the following extension of the classical Chu–Vandermonde identity:

### Lemma 4.8

*n*,

*m*and

*k*, with \(k \ge 0\),

### Proof

Throughout this proof, if *Y* is a *k*-element set, write \(\tau (Y) = \sigma (Y) - k (k - 1) / 2\).

*j*-element subset of \(X_n\), and \(Y_2\) a \((k - j)\)-element subset of \(X_m\). Let \(Y_2' = \{ y + n : y \in Y_2 \}\), so that \(Y = Y_1 \cup Y_2'\) is a

*k*-element subset of \(X_{n + m}\). Then, since

Similarly, one can deduce from Theorem 4.5 the following version for the case when *k* is a negative integer. It also holds if \(n, m \ge 0\), but the identity does not generally hold in the case when *n* and *m* have mixed signs.

### Lemma 4.9

*n*,

*m*and

*k*,

As another application of the combinatorial characterization in Theorem 4.5, we readily obtain the following identity. In the case \(n \ge 0\), this identity is well known and often referred to as the (commutative version of the) *q*-binomial theorem (in which case the sum only extends over \(k = 0, 1, \ldots , n\)). We will discuss the noncommutative *q*-binomial theorem in the next section.

### Theorem 4.10

*n*,

### Proof

*Y*we associate the product of the terms \(x^m q^{y m}\) where \(y \in Y\) and \(m = M_Y (y)\) is the multiplicity of

*y*. Therefore,

## 5 The Binomial Theorem

*n*and

*k*may be negative integers. In this section, we show that this extension can also be generalized to the case of

*q*-binomial coefficients.

*f*(

*x*) is a function with Laurent expansions

*f*(

*x*).

### Theorem 5.1

*n*and

*k*,

### Example 5.2

It is well known (see, for instance, [11, Theorem 5.1]) that, if *x* and *y* are noncommuting variables such that \(y x = q x y\), then the *q*-binomial coefficients arise from the expansion of \((x + y)^n\).

### Theorem 5.3

*q*-binomial Theorem 5.3 and Loeb’s Theorem 5.1. In analogy with the classical case, we consider expansions of \(f_n (x, y) = (x + y)^n\) in the two

*q*-commuting variables

*x*,

*y*. As before, we can expand \(f_n (x, y)\) in two different ways, that is,

### Theorem 5.4

*n*and

*k*,

### Proof

*q*-commutativity,

*n*. Then,

*n*. This implies the present claim in the case \(k \ge 0\). The case when \(k < 0\) can also be deduced from (5.3). Indeed, observe that \(x y = q^{- 1} y x\), so that, for any integer

*n*, by (5.2) and (5.3),

## 6 Lucas’ Theorem

*p*is a prime, then

*p*-adic digits of the nonnegative integers

*n*and

*k*, respectively. Our first goal is to prove that this congruence in fact holds for all integers

*n*and

*k*. The next section is then concerned with further extending these congruences to the polynomial setting.

### Example 6.1

*p*expansion of a negative integer is infinite. However, only finitely many digits are different from \(p - 1\). For instance, in base 7,

*p*, except in the boundary cases \(k = 0\) and \(k = p\).

### Theorem 6.2

*p*be a prime. Then, for any integers

*n*and

*k*,

### Proof

*p*,

*n*.

*k*as well (because \((1 + x)^{n_0}\) is a polynomial, in which case the expansions (5.1) around 0 and \(\infty \) agree). Thus,

## 7 A *q*-Analog of Lucas’ Theorem

Let \(\Phi _m (q)\) be the *m*-th cyclotomic polynomial. In this section, we prove congruences of the type \(A (q) \equiv B (q)\) modulo \({\Phi _m (q)}\), where *A*(*q*) and *B*(*q*) are Laurent polynomials. The congruence is to be interpreted in the natural sense that the difference \(A (q) - B (q)\) is divisible by \(\Phi _m (q)\).

### Example 7.1

In the case \(n, k \ge 0\), the following *q*-analog of Lucas’ classical binomial congruence has been obtained by Olive [16] and Désarménien [7]. A nice proof based on a group action is given by Sagan [17], who attributes the combinatorial idea to Strehl. We show that these congruences extend uniformly to all integers *n* and *k*. A minor difference to keep in mind is that the *q*-binomial coefficients in this extended setting are Laurent polynomials (see Example 7.1).

### Theorem 7.2

*n*and

*k*,

### Proof

*x*and

*y*satisfy \(y x = q x y\). It follows from the (noncommutative)

*q*-binomial Theorem 5.3 that, for nonnegative integers

*m*,

*n*.

In [2], Adamczewski, Bell, Delaygue and Jouhet consider congruences modulo cyclotomic polynomials for multidimensional *q*-factorial ratios and are thus able to generalize many Lucas-type congruences. In particular, specializing [2, Proposition 1.4] (the case \(q = 1\) of which had previously been proved in [1]) to \(d = 2\), \(u = 1\), \(v = 2\), \(e_1 = (1 ; 0)\), \(f_1 = (1 ; - 1)\) and \(f_2 = (0 ; 1)\), we obtain the classical case \(n, k \ge 0\) of Theorem 7.2. As pointed out by Adamczewski, Bell, Delaygue and Jouhet in private communication, an alternative, a little more tricky, proof of the general case of Theorem 7.2 can be obtained by reducing it, via Corollary 3.2, to the nonnegative case.

## 8 Conclusion

We believe (and hope that the results of this paper provide some evidence to that effect) that the binomial and *q*-binomial coefficients with negative entries are natural and beautiful objects. On the other hand, let us indicate an application, taken from [21], of binomial coefficients with negative entries.

### Example 8.1

*Apéry numbers*

*m*,

*r*. The definition of the Apéry numbers

*A*(

*n*) can be extended to all integers

*n*by setting

*m*replaced with \(- m\). By working with binomial coefficients with negative entries, the second author gave a uniform proof of both sets of congruences in [21]. In addition, the symmetry (8.5), which becomes visible when allowing negative indices, explains why other Apéry-like numbers satisfy (8.3) but not (8.2).

We illustrated that the Gaussian binomial coefficients can be usefully extended to the case of negative arguments. More general binomial coefficients, formed from an arbitrary sequence of integers, are considered, for instance, in [13] and it is shown by Hu and Sun [10] that Lucas’ theorem can be generalized to these. It would be interesting to investigate the extent to which these coefficients and their properties can be extended to the case of negative arguments. Similarly, an elliptic analog of the binomial coefficients has recently been introduced by Schlosser [18], who further obtains a general noncommutative binomial theorem of which Theorem 5.3 is a special case. It is natural to wonder whether these binomial coefficients have a natural extension to negative arguments as well.

In the last section, we showed that the generalized *q*-binomial coefficients satisfy Lucas congruences in a uniform fashion. It would be of interest to determine whether other well-known congruences for the *q*-binomial coefficients, such as those considered in [3] or [20], have similarly uniform extensions.

## Notes

### Acknowledgements

Part of this work was completed while the first author was supported by a Summer Undergraduate Research Fellowship (SURF) through the Office of Undergraduate Research (OUR) at the University of South Alabama. We are grateful to Wadim Zudilin for helpful comments on an earlier draft of this paper, as well as to the referee who provided useful historical remarks. We also thank Boris Adamczewski, Jason P. Bell, Éric Delaygue, and Frédéric Jouhet for pointing out the connection between Theorem 7.2 and the results in [2] (see the comments included after Theorem 7.2).

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