Two Nice Determinantal Expressions and A Recurrence Relation for the Apostol--Bernoulli Polynomials | Qi | Journal of the Indonesian Mathematical Society 274 947 1 PB

J. Indones. Math. Soc.
Vol. 23, No. 1 (2017), pp. 81–87.

TWO NICE DETERMINANTAL EXPRESSIONS AND A
RECURRENCE RELATION FOR THE
APOSTOL–BERNOULLI POLYNOMIALS
Feng Qi1,2,3 and Bai-Ni Guo4
1 Institute

of Mathematics, Henan Polytechnic University,
Jiaozuo City, Henan Province, 454010, China
2 College of Mathematics, Inner Mongolia University for Nationalities,
Tongliao City, Inner Mongolia Autonomous Region, 028043, China
3 Department of Mathematics, College of Science, Tianjin Polytechnic
University, Tianjin City, 300387, China
E-mail: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com
4 School of Mathematics and Informatics, Henan Polytechnic University,
Jiaozuo City, Henan Province, 454010, China
E-mail: bai.ni.guo@gmail.com, bai.ni.guo@hotmail.com

Abstract. In the paper, the authors establish two nice determinantal expressions

and a recurrence relation for the Apostol–Bernoulli polynomials.
Key words and Phrases: Apostol–Bernoulli polynomial; determinantal expression;
recurrence relation; determinant; derivative of a ratio between two functions.

Abstrak. Pada makalah ini, para penulis menyajikan dua pernyataan berbentuk
determinan dan sebuah relasi rekurensi untuk suku banyak Apostol-Bernoulli.
Kata kunci: Suku banyak Apostol–Bernoulli, ekspresi berbentuk determinan, relasi
rekurensi, determinan, turunan dari rasio dua fungsi.

1. Introduction
It is well-known that the Bernoulli numbers Bk , the Bernoulli polynomials
Bk (u), and the Apostol–Bernoulli polynomials Bk (u, z) for k ≥ 0 can be generated
respectively by
∞
∞
X
x
xk
x2k
x X

B
B2k
=
=
1
−
+
, |x| < 2π,
k
x
e −1
k!
2
(2k)!
k=0

k=1

2000 Mathematics Subject Classification: Primary 11B68; Secondary 11B83, 11C20, 15A15, 26A06,
26A09, 33B10.

Received: 01-08-2016, revised: 03-03-2017, accepted: 04-03-2017.
81

F. Qi & B.-N. Guo

82
∞

X
xk
xeux
B
(u)
=
,
k
ex − 1
k!

|x| < 2π,

k=0

and
∞

X
xk
xeux
Bk (u, z) ,
=
x
ze − 1
k!

|x| <

k=0

(

2π,
| ln z|,

z = 1;
z 6= 1.

(1)

It is clear that these notions have the relations
Bk = Bk (0)

and

Bk (u) = Bk (u, 1).

In [1, 2], Apostol connected special values of the Lerch zeta functions with
the Apostol–Bernoulli polynomials Bk (u, z). In [8], Luo gave a relation between the
λ-power sums and the Apostol–Bernoulli polynomials Bk (u, z), which generalize J.
Bernoulli’s formula on the representation of power sums in terms of the Bernoulli

polynomials Bk (u). In [7], Kim and Hu obtained the sums of products identity for
the Apostol–Bernoulli numbers Bk (u, z), which is an analogue of the classical sums
of products identity for the Bernoulli numbers Bk dating back to Euler.
Let p = p(x) and q = q(x) 6= 0 be two differentiable functions. Then


 p
q
0
···
0 
 ′
 p
q′
q
···
0 


k 

k  p′′
q ′′
2q ′
···
0 
(−1) 
p(x)
d
= k+1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , k ≥ 0. (2)
d z k q(x)
q


 (k−1)

k−1 (k−2)
p
q (k−1)
···
q

1
q


k (k−1)
k
′
 p(k)
q (k)
···
1 q
k−1 q

See [3, p. 40]. We can rewrite the formula (2) as



(−1)k 
dk p(x)
W(k+1)×(k+1) (x),
=
k
k+1

d x q(x)
q
(x)

(3)

where |W(k+1)×(k+1) (x)| denotes the determinant of the (k + 1) × (k + 1) matrix


W(k+1)×(k+1) (x) = U(k+1)×1 (x) V(k+1)×k (x) ,

the quantity U(k+1)×1 (x) is a (k + 1) × 1 matrix whose elements uℓ,1 (x) = p(ℓ−1) (x)
for 1 ≤ ℓ ≤ k + 1, and V(k+1)×k (x) is a (k + 1) × k matrix whose elements


 i − 1 q (i−j) (x), i − j ≥ 0
j−1
vi,j (x) =

0,
i−j