EVALUATION OF TWO DETERMINANTS INVOLVING

Preprint

EVALUATION OF TWO DETERMINANTS
INVOLVING q-INTEGERS

ZHI-WEI SUN
arXiv:2605.16240v1 [math.CO] 15 May 2026




Abstract. The q-analogue of an integer m is given by [m]q = (1 −
q m )/(1 − q). Let a be an integer, and let n be a positive odd integer.
Via discrete Fourier transforms, we establish the following two identities:
"  #  
aj − (a + 1)k a(a + 1)
det =− q (1−3n)/2
n q n
1⩽j,k⩽n

and
"  #  
(a + 1)j − ak a(a + 1)
det = q (n−1)/2 ,
n q n
1⩽j,k⩽n

where ( n· ) denotes the Jacobi symbol.




1. Introduction
For any real number x, let ⌊x⌋ denote the largest integer not exceeding x.
The function ⌊·⌋ is called the floor function. In 2021, the author [8] evaluated
some permanents and determinants with entries involving the floor function.
In 2025, S. Fu, Z. Lin and the author [2] proved that for any positive integer
n the permanent of the matrix [⌊ 2j−k n ⌋]1⩽j,k⩽n is 2(2
n+1 − 1)B
n+1 , where
B0 , B1 , B2 , . . . are the Bernoulli numbers.
For any integer m, its q-analogue is given by
1 − qm 1 − q −m
[m]q := = −q m .
1−q 1−q
In particular, [0]q = 0, [1]q = 1, and [2]q = 1 + q. Note that limq→1 [m]q = m
for all m ∈ Z.
For a matrix A = [ajk ]1⩽j,k⩽n over a field, we denote the determinant
of A by det(A) or det[ajk ]1⩽j,k⩽n . In this paper, we mainly study certain
determinants involving q-integers.
Now we state our first theorem.


Key words and phrases. Determinants, q-integers, the floor function, the ceiling func-
tion, Jacobi symbols.
2020 Mathematics Subject Classification. Primary 05A30, 11C20; Secondary 05A19,
11A15, 15A15.
Supported by the Natural Science Foundation of China (grant 12371004).
1

, 2 ZHI-WEI SUN

Theorem 1.1. Let a be an integer and let n be a positive odd integer. Then
we have
"  #  
aj − (a + 1)k a(a + 1) (1−3n)/2
det =− q , (1.1)
n q n
1⩽j,k⩽n

where ( n· ) is the Jacobi symbol.
Remark 1.1. Jacobi symbols play important roles in the theory of quadratic
residues modulo primes. For their definition and basic properties, one may
consult [4, pp. 56-57].
Letting q tend to 1 or 2, we obtain from Theorem 1.1 the following corol-
lary.
Corollary 1.1. For any integer a and positive odd integer n, we have
   
aj − (a + 1)k a(a + 1)
det =− (1.2)
n 1⩽j,k⩽n n
and  
h aj−(a+1)k i a(a + 1)
det 2⌊ n
⌋
−1 =− 2(1−3n)/2 . (1.3)
1⩽j,k⩽n n
The ceiling function ⌈·⌉ is defined as follows: For any real number x, ⌈x⌉
denotes the least integer not smaller than x. Our second theorem involve
q-integers and the ceiling function.
Theorem 1.2. Let a be an integer and let n be a positive odd integer. Then
we have
"  #  
(a + 1)j − ak a(a + 1) (n−1)/2
det = q . (1.4)
n q n
1⩽j,k⩽n

Letting q tend to 1 or 2, we obtain from Theorem 1.1 the following corol-
lary.
Corollary 1.2. For any integer a and positive odd integer n, we have
   
(a + 1)j − ak a(a + 1)
det = (1.5)
n 1⩽j,k⩽n n
and  
h
⌈
(a+1)j−ak
⌉
i a(a + 1)
det 2 n −1 = 2(n−1)/2 . (1.6)
1⩽j,k⩽n n
Both Theorems 1.1 and 1.2 were conjectured by the author in 2021 (cf.
[7] and [8]). We will deduce an auxiliary proposition in Section 2 via the
discrete Fourier transforms, and then prove Theorems 1.1 and 1.2 in Section
3. Actually, before proving the above two theorems, we need to establish
the following result in Section 3.