Journal of Number Theory 151 (2015) 223–229
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Journal of Number Theory
www.elsevier.com/locate/jnt
A connection between Jacobi–Stirling numbers and
Bernoulli polynomials
Mircea Merca
Department of Mathematics, University of Craiova, A. I. Cuza 13,
Craiova, 200585, Romania
a r t i c l e
i n f o
Article history:
Received 8 October 2014
Received in revised form 22
December 2014
Accepted 23 December 2014
Available online 7 February 2015
Communicated by David Goss
a b s t r a c t
A finite discrete convolution involving the Jacobi–Stirling
numbers of both kinds is expressed in this paper in terms
of the Bernoulli polynomials.
© 2015 Elsevier Inc. All rights reserved.
MSC:
05E05
11B68
11B75
Keywords:
Bernoulli polynomials
Jacobi–Stirling numbers
Legendre–Stirling numbers
Chebyshev–Stirling numbers
1. Introduction
The first objects of our investigations are the Jacobi–Stirling numbers of both kinds
that were discovered in 2007 as a result of a problem involving the spectral theory of
E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jnt.2014.12.024
0022-314X/© 2015 Elsevier Inc. All rights reserved.
224
M. Merca / Journal of Number Theory 151 (2015) 223–229
powers !of the
" classical second-order Jacobi differential expression [10]. There is an infinite
n
(n, k ∈ Z) of polynomials in γ satisfying the recurrence relation
family
k γ
!
"
! "
!
"
n−1
n
n−1
+ (n − 1)(n + 2γ − 2)
,
=
k
k γ
k−1 γ
γ
with the initial conditions
! "
n
= δn,0 ,
0 γ
! "
0
= δk,0 ,
k γ
where δi,j is the Kronecker delta. If n and k are non-negative integers and γ is positive
then these polynomials
# are
$ the Jacobi–Stirling numbers of the first kind. Similarly, there
n
is an infinite family
(n, k ∈ Z) of polynomials in γ satisfying the recurrence
k γ
relation
# $
$
#
$
#
n
n−1
n−1
,
+ k(k + 2γ − 1)
=
k γ
k−1 γ
k
γ
with the initial conditions
# $
n
= δn,0 ,
0 γ
# $
0
= δk,0 .
k γ
These polynomials are the Jacobi–Stirling numbers of the second kind when n and k are
non-negative integers and γ is positive. In the last decade, the Jacobi–Stirling numbers
of both kinds received considerable attention especially in combinatorics and graph theory. The application of these numbers in these areas is natural since they have many
properties similar to those of the classical Stirling numbers, see, e.g., [2–4,7,11–13,16,18,
19].
The second objects are the Bernoulli polynomials which are denoted in this paper
by Bn (x). Bernoulli [6, p. 97] introduced these polynomials in 1690 in parallel with the
discovery of numbers Bn related to the calculation of the sum of powers of the first m
non-negative integers,
m−1
%
k=0
'
n &
1 % n+1
Bk mn+1−k ,
k =
n+1
k
n
k=0
when he set
m−1
%
k=0
kn =
Bn+1 (m) − Bn+1 (0)
.
n+1
(1)
M. Merca / Journal of Number Theory 151 (2015) 223–229
225
The Bernoulli polynomials admit a variety of different representations. The explicit formula
n " #
!
n
Bn (x) =
Bk xn−k
k
k=0
shows that Bn (x) is a polynomial of degree n. The numbers Bn = Bn (0) (n = 0, 1, 2, . . .)
are called Bernoulli numbers. Many properties of the Bernoulli polynomials have been
studied later in the classical papers by Appell [5], Euler [8], Lehmer [14] and Lucas [15].
These polynomials have an important role in various expansions and approximation
formulas which are useful both in the analytic theory of numbers and in classical and
numerical analysis.
Recently, Merca [17] proved that the classical Stirling numbers of both kinds and the
Bernoulli polynomials are related by
k
!
j−1
(−1)
j=1
$
n+1
j
n+1−j
%&
n+k−j
n
'
=
Bk+1 (n + 1) − Bk+1 (0)
.
k+1
(2)
In this paper, motivated by this identity, we prove the following analogous result.
Theorem 1. The Jacobi–Stirling numbers of both kinds and the Bernoulli polynomials are
related by
k
!
j−1
(−1)
j=1
=
$
n+1
j
n+1−j
% &
γ
n+k−j
n
'
γ
" #
k
!
Bk+j+1 (n + 1) − Bk+j+1 (0) k
j=0
k+j+1
j
(2γ − 1)k−j .
It is well known that the Jacobi–Stirling numbers of both kinds may be reduced to the
Legendre–Stirling numbers of both kinds by setting γ = 1. The Legendre–Stirling numbers were discovered in 2002 as a result of a problem involving the spectral theory of powers of the classical second-order Legendre differential expression [9]. Many properties of
these numbers have been studied later in [2,3]. In this case Theorem 1 can be rewritten as
Corollary 1.1. The Legendre–Stirling numbers of both kinds and the Bernoulli polynomials
are related by
$
k
!
j−1
(−1) j
j=1
n+1
n+1−j
% &
1
n+k−j
n
'
1
" #
k
!
(−1)k
Bk+j+1 (n + 1) k
.
=
(
)+
k+j+1
j
(k + 1) 2k+2
k+1
j=0
226
M. Merca / Journal of Number Theory 151 (2015) 223–229
The case γ = 1/2 of the Jacobi–Stirling numbers is known in the literature as
the Chebyshev–Stirling numbers. Some properties of these numbers have been studied
recently by Gawronski, Littlejohn and Neuschel [11]. Another property of the Chebyshev–
Stirling numbers can be easily derived from Theorem 1.
Corollary 1.2. The Chebyshev–Stirling numbers of both kinds and the Bernoulli polynomials are related by
k
!
j=1
j−1
(−1)
"
n+1
j
n+1−j
#
1/2
$
n+k−j
n
%
=
1/2
B2k+1 (n + 1) − B2k+1 (0)
.
2k + 1
In fact, this result suggests a new way to compute the sum of the kth powers of the
first n positive integers when k is even. According to (1) and (2), the sum
n
!
j 2k
j=1
can be written as a finite discrete convolution with 2k terms involving the classical
Stirling numbers of both kinds. By Corollary 1.2, we see that the sum of the 2kth
powers of the first n positive integers can be rewritten as a finite discrete convolution
with only k terms.
2. Proofs
Recently, Mongelli [19] has shown that the Jacobi–Stirling numbers are specializations
of the elementary and complete homogeneous symmetric functions of the numbers 2γ, 2 +
4γ, . . . , n(n − 1 + 2γ). This allows us to consider the following two identities of formal
power series in t for the Jacobi–Stirling numbers:
E(t) =
n
&
(1 + k(k + 2γ − 1) · t) =
k=1
∞ "
!
k=0
n+1
n+1−k
#
tk
γ
and
H(t) =
n
&
k=1
with |t| <
show that
1
n(n+2γ−1) .
−1
(1 − k(k + 2γ − 1) · t)
=
%
∞ $
!
n+k
k=0
n
tk ,
γ
Taking into account that E(−t)H(t) = 1, it is an easy exercise to
d
ln H(t) = E ′ (−t)H(t),
dt
M. Merca / Journal of Number Theory 151 (2015) 223–229
227
where
"
∞
!
E (−t) =
k
′
k=1
In addition, for |t| <
1
n(n+2γ−1)
n+1
n+1−k
#
(−t)k−1 .
γ
we have
n
n
k=1
k=1
∞
!
!!
d
k(k + 2γ − 1)
ln H(t) =
=
kj (k + 2γ − 1)j tj−1 .
dt
1 − k(k + 2γ − 1) · t
j=1
Moreover, we can write
n
∞ !
!
j k (j + 2γ − 1)k tk−1 =
k=1 j=1
$
"
∞
!
k
k=1
n+1
n+1−k
#
(−t)k−1
γ
%$
'
∞ &
!
n+k
k=0
n
γ
tk
%
.
Equating coefficients of tk−1 on each side and rearranging gives the identity
n
!
j=1
"
k
!
j−1
(−1) j
j (j + 2γ − 1) =
k
k
j=1
n+1
n+1−j
# &
γ
n+k−j
n
'
.
γ
Taking into account the formula (1), the left side of this identity can be written as
n
!
k
k
j (j + 2γ − 1) =
n !
k ( )
!
k
j=1 i=0
j=1
i
j k+i (2γ − 1)k−i
( )
k
!
Bk+j+1 (n + 1) − Bk+j+1 (0) k
(2γ − 1)k−j .
=
k
+
j
+
1
j
j=0
The proof of Theorem 1 is finished.
According to Agoh and Dilcher [1], for the positive integers k, m and a, we have the
following identity
m+1
(−1)
( )
( )
k
m
!
!
Bm+j+1 k k−j
Bk+j+1 m m−j
k+1
+ (−1)
a
a
m
+
j
+
1
k
+
j
+
1
j
j
j=0
j=0
a
!
k!m!
k+m+1
a
=
−
j k (a − j)m
(k + m + 1)!
j=0
(3)
for the Bernoulli numbers. The case k = m of this identity can be written as
( )
a
k
!
(−1)k+1
(−1)k+1 ! k
Bk+j+1 k k−j
2k+1
=
j (a − j)k .
−
a
*2k+2+ a
k+j+1 j
2
(k + 1) k+1
j=0
j=0
(4)
228
M. Merca / Journal of Number Theory 151 (2015) 223–229
Now the proof of Corollary 1.1 follows easily from Theorem 1 taking into account the
last relation.
3. Concluding remarks
The relationships of the complete and elementary symmetric functions to the Jacobi–
Stirling numbers have been used in this paper to discover and prove a connection between
these numbers and Bernoulli polynomials.
There are three natural questions arising from this work:
1. Is it possible to extend the identity (3) for the Bernoulli numbers to the Bernoulli
polynomials? For example, can the expression
m+1
(−1)
" #
k
!
Bm+j+1 (n) k
j=0
m+j+1
j
a
k−j
k+1
+ (−1)
" #
m
!
Bk+j+1 (n) m m−j
a
k
+
j
+
1
j
j=0
be rewritten in a simplified form?
2. Is it possible to obtain another generalization for the identity (4)? We want to get a
simpler form for the sum
" #
k
!
Bk+j+1 (n + 1) k
(2γ − 1)k−j .
k
+
j
+
1
j
j=0
3. When k is even, the sum of the kth powers of the first n positive integers is represented in this paper as a finite discrete convolution with k/2 terms. Is it possible to
obtain a similar result for k odd?
Acknowledgments
The author appreciates the anonymous referees for their comments on the original
version of this paper. The author expresses his gratitude to Takashi Agoh and Karl
Dilcher for their support. Special thanks go to Dr. Oana Merca for the careful reading
of the manuscript and helpful remarks.
References
[1] T. Agoh, K. Dilcher, Convolution identities and lacunary recurrences for Bernoulli numbers, J. Number Theory 124 (2007) 105–122.
[2] G.E. Andrews, L.L. Littlejohn, A combinatorial interpretation of the Legendre–Stirling numbers,
Proc. Amer. Math. Soc. 137 (8) (2009) 2581–2590.
[3] G.E. Andrews, W. Gawronski, L.L. Littlejohn, The Legendre–Stirling numbers, Discrete Math. 311
(2011) 1255–1272.
[4] G.E. Andrews, E.S. Egge, W. Gawronski, L.L. Littlejohn, The Jacobi–Stirling numbers, J. Combin.
Theory Ser. A 120 (2013) 288–303.
[5] P.E. Appell, Sur une classe de polynomes, Ann. Sci. Éc. Norm. Supér. (2) 9 (1882) 119–144.
M. Merca / Journal of Number Theory 151 (2015) 223–229
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
229
J. Bernoulli, Ars conjectandi, Basel, 1713, posthumously published.
E.S. Egge, Legendre–Stirling permutations, European J. Combin. 31 (2010) 1735–1750.
L. Euler, Methodus generalis summandi progressiones, Comment. Acad. Sci. Petrop. 6 (1738).
W.N. Everitt, L.L. Littlejohn, R. Wellman, Legendre polynomials, Legendre–Stirling numbers, and
the left-definite analysis of the Legendre differential expression, J. Comput. Appl. Math. 148 (1)
(2002) 213–238.
W.N. Everitt, K.H. Kwon, L.L. Littlejohn, R. Wellman, G.J. Yoon, Jacobi–Stirling numbers, Jacobi
polynomials, and the left-definite analysis of the classical Jacobi differential expression, J. Comput.
Appl. Math. 208 (2007) 29–56.
W. Gawronski, L.L. Littlejohn, T. Neuschel, Asymptotics of Stirling and Chebyshev–Stirling numbers of the second kind, Stud. Appl. Math. 133 (2014) 1–17.
Y. Gelineau, J. Zeng, Combinatorial interpretations of the Jacobi–Stirling numbers, Electron. J.
Combin. 17 (2010) R70.
I.M. Gessel, Z. Lin, J. Zeng, Jacobi–Stirling polynomials and P-partitions, European J. Combin. 33
(2012) 1987–2000.
D.H. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly 95 (1988) 905–911.
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891.
M. Merca, A note on the Jacobi–Stirling numbers, Integral Transforms Spec. Funct. 25 (3) (2014)
196–202.
M. Merca, A new connection between r-Whitney numbers and Bernoulli polynomials, Integral
Transforms Spec. Funct. 25 (12) (2014) 937–942.
P. Mongelli, Total positivity properties of Jacobi–Stirling numbers, Adv. in Appl. Math. 48 (2012)
354–364.
P. Mongelli, Combinatorial interpretations of particular evaluations of complete and elementary
symmetric functions, Electron. J. Combin. 19 (1) (2012) #P60.