Some relationships between poly-Cauchy numbers and poly-Bernoulli numbers ∗
Takao Komatsu
a, Florian Luca
baGraduate School of Science and Technology Hirosaki University, Hirosaki, Japan
komatsu@cc.hirosaki-u.ac.jp
bFundación Marcos Moshinsky
Instituto de Ciencias Nucleares UNAM, Circuito Exterior, Mexico fluca@matmor.unam.mx
Abstract
In this paper, we show some relationships between poly-Cauchy numbers introduced by T. Komatsu and poly-Bernoulli numbers introduced by M.
Kaneko.
Keywords: Bernoulli numbers; Cauchy numbers; poly-Bernoulli numbers;
poly-Cauchy numbers MSC: 05A15, 11B75
1. Introduction
Letn and k be positive integers. Poly-Cauchy numbers of the first kind c(k)n are defined by
c(k)n = Z1
0
. . . Z1
| {z }0 k
(x1x2. . . xk)(x1x2. . . xk−1). . .(x1x2. . . xk−n+ 1)dx1dx2. . . dxk
∗The first author was supported in part by the Grant-in-Aid for Scientific research (C) (No.22540005), the Japan Society for the Promotion of Science. The second author worked on this project during a visit to Hirosaki in January and February of 2012 with a JSPS Fellowship (No.S-11021). This author thanks the Mathematics Department of Hirosaki University for its hospitality and JSPS for support.
Proceedings of the
15thInternational Conference on Fibonacci Numbers and Their Applications Institute of Mathematics and Informatics, Eszterházy Károly College
Eger, Hungary, June 25–30, 2012
99
(see in [7]). The concept of poly-Cauchy numbers is a generalization of that of the classical Cauchy numberscn=c(1)n defined by
cn = Z1
0
x(x−1). . .(x−n+ 1)dx
(see e.g. [2, 8]). The generating function of poly-Cauchy numbers ([7, Theorem 2]) is given by
Lifk(ln(1 +x)) = X∞ n=0
c(k)n xn n! , where
Lifk(z) = X∞ m=0
zm m!(m+ 1)k
is thek-th polylogarithm factorial function. An explicit formula for c(k)n ([7, The- orem 1]) is given by
c(k)n = (−1)n Xn
m=0
hn m
i (−1)m
(m+ 1)k (n≥0, k≥1), (1.1) where n
m
are the (unsigned) Stirling numbers of the first kind, arising as coeffi- cients of the rising factorial
x(x+ 1). . .(x+n−1) = Xn
m=0
hn m
ixm
(see e.g. [4]).
On the other hand, M. Kaneko ([6]) introduced the poly-Bernoulli numbers Bn(k) by
Lik(1−e−x) 1−e−x =
X∞ n=0
Bn(k)xn n! , where
Lik(z) = X∞ m=1
zm mk
is thek-th polylogarithm function. Whenk= 1,Bn=B(1)n is the classical Bernoulli number withB1(1)= 1/2, defined by the generating function
xex ex−1 =
X∞ n=0
Bnxn n! .
An explicit formula forB(k)n ([6, Theorem 1]) is given by
Bn(k)= (−1)n Xn
m=0
nn m
o(−1)mm!
(m+ 1)k (n≥0, k≥1), (1.2) wheren
m are the Stirling numbers of the second kind, determined by nn
m o= 1
m!
Xm
j=0
(−1)j m
j
(m−j)n
(see e.g. [4]).
In this paper, we show some relationships between poly-Cauchy numbers and poly-Bernoulli numbers.
2. Main result
Poly-Bernoulli numbers can be expressed by poly-Cauchy numbers ([7, Theorem 8]).
Theorem 2.1. Forn≥1 we have
Bn(k)= Xn
l=1
Xn
m=1
m!nn m
o m−1 l−1
c(k)l .
On the other hand, c(k)2 = 1
2!B(k)2 +3 2B1(k)
= 1
2!(B2(k)+ 3B1(k)), c(k)3 = 1
3!B(k)3 + 2B2(k)+23 6 B1(k)
= 1
3!(B3(k)+ 12B(k)2 + 23B(k)1 ), c(k)4 = 1
4!B(k)4 +5
4B3(k)+215
24 B2(k)+55 4 B1(k)
= 1
4!(B4(k)+ 30B(k)3 + 215B2(k)+ 330B1(k)), c(k)5 = 1
5!B(k)5 +1
2B4(k)+207
24 B3(k)+95
2 B2(k)+1901 30 B(k)1
= 1
5!(B5(k)+ 60B(k)4 + 1035B(k)3 + 5700B2(k)+ 7604B1(k)), c(k)6 = 1
6!B(k)6 + 7
48B5(k)+707
144B(k)4 +1015
16 B3(k)+13279
45 B(k)2 +4277 12 B1(k)
= 1
6!(B6(k)+ 105B5(k)+ 3535B4(k)+ 45675B3(k)+ 212464B2(k)+ 256620B1(k)). In general, we have the following identity, expressing poly-Cauchy numbersc(k)n
by using poly-Bernoulli numbersB(k)n . Theorem 2.2. Forn≥1 we have
c(k)n = (−1)n Xn
l=1
Xn
m=1
(−1)m m!
hn m
i hm l
iBl(k).
Proof. By (1.1) and (1.2), we have RHS= (−1)n
Xn
l=1
Xn
m=1
(−1)m m!
hn m
i hm l
i(−1)l Xl
i=0
l i
(−1)ii!
(i+ 1)k
= (−1)n Xn
m=1
(−1)m m!
hn m
iXn
l=0
hm l
i(−1)l Xl
i=0
l i
(−1)ii!
(i+ 1)k
= (−1)n Xn
m=1
(−1)m m!
hn m
iXn
i=0
(−1)ii!
(i+ 1)k Xn
l=i
(−1)lhm l
i l i
= (−1)n Xn
m=0
(−1)m m!
hn m
i(−1)mm!
(m+ 1)k(−1)m
= (−1)n Xn
m=0
hn m
i (−1)m
(m+ 1)k =LHS. Note thatm
0
= 0 (m≥1) andm l
= 0(l > m), and
Xm
l=i
(−1)m−lhm l
i l i
=
(1 (i=m);
0 (i6=m).
3. Poly-Cauchy numbers of the second kind
Poly-Cauchy numbers of the second kindcˆ(k)n are defined by
ˆ c(k)n =
Z1
0
. . . Z1
| {z }0 k
(−x1x2. . . xk)(−x1x2. . . xk−1)
. . .(−x1x2. . . xk−n+ 1)dx1dx2. . . dxk
(see in [7]). Ifk= 1, then ˆc(1)n = ˆcn is the classical Cauchy numbers of the second kind defined by
ˆ cn =
Zk
0
(−x)(−x−1). . .(−x−n+ 1)dx
(see e.g. [2, 8]). The generating function of poly-Cauchy numbers of the second kind ([7, Theorem 5]) is given by
Lifk(−ln(1 +x)) = X∞ n=0
ˆ c(k)n xn
n! . An explicit formula forˆc(k)n ([7, Theorem 4]) is given by
ˆ
c(k)n = (−1)n Xn
m=0
hn m
i 1
(m+ 1)k (n≥0, k≥1). (3.1) In a similar way, we have a relationship, expressing poly-Cauchy numbers of the second kind ˆc(k)n by using poly-Bernoulli numbers B(k)n . The proof is similar and omitted.
Theorem 3.1. Forn≥1 we have
ˆ
c(k)n = (−1)n Xn
l=1
Xn
m=1
1 m!
hn m
i hm l
iB(k)l .
In addition, we also obtain the corresponding relationship to Theorem 2.1.
Theorem 3.2. Forn≥1 we have
Bn(k)= (−1)n Xn
l=1
Xn
m=1
m!nn m
o nm l
oˆc(k)l .
Proof. By (1.2) and (3.1), we have RHS= (−1)n
Xn
l=1
Xn
m=1
m!nn m
o nm l
o(−1)l Xl
i=0
l i
1 (i+ 1)k
= (−1)n Xn
m=1
m!nn m
oXn
l=0
nm l
o(−1)l Xl
i=0
l i
1 (i+ 1)k
= (−1)n Xn
m=1
m!nn m
oXn
i=0
1 (i+ 1)k
Xn
l=i
(−1)lnm l
o l i
= (−1)n Xn
m=0
m!nn m
o 1
(m+ 1)k(−1)m
= (−1)n Xn
m=0
nn m
o(−1)mm!
(m+ 1)k =LHS. Note that
Xm
l=i
(−1)m−lnm l
o l i
=
(1 (i=m);
0 (i6=m).
4. Poly-Cauchy polynomials and poly-Bernoulli polynomials
Poly-Cauchy polynomials of the first kindc(k)n (z)are defined by
c(k)n (z) =n!
Z1
0
. . . Z1
| {z }0 k
(x1x2. . . xk−z)(x1x2. . . xk−1−z)
· · ·(x1x2. . . xk−(n−1)−z)dx1dx2. . . dxk, and are expressed explicitly in terms of Stirling numbers of the first kind ([5, Theorem 1])
c(k)n (z) = Xn
m=0
hn m
i(−1)n−m Xm
i=0
m i
(−z)i (m−i+ 1)k . Poly-Cauchy polynomials of the second kindˆc(k)n (z)are defined by
ˆ
c(k)n (z) =n!
Z1
0
. . . Z1
| {z }0 k
(−x1x2. . . xk+z)(−x1x2. . . xk−1 +z)
· · ·(−x1x2. . . xk−(n−1) +z)dx1dx2. . . dxk, and are expressed explicitly in terms of Stirling numbers of the first kind ([5, Theorem 4].
ˆ
c(k)n (z) = Xn
m=0
hn m
i(−1)n Xm
i=0
m i
(−z)i (m−i+ 1)k .
In 2010, Coppo and Candelpergher [3], 2011 Bayad and Hamahata [1, (1.5)] intro- duced the poly-Bernoulli polynomialsB(k)n (z)given by
Lik(1−e−x) 1−e−x e−xz=
X∞ n=0
Bn(k)(z)xn n!,
and
Lik(1−e−x) 1−e−x exz=
X∞ n=0
Bn(k)(z)xn n! , respectively, satisfyingB(k)n (0) =B(k)n .
If we define still different poly-Bernoulli polynomialsB(k)n by
Bn(k)(z) = (−1)n Xn
m=0
nn m
o(−1)mm!
Xm
i=0
m i
(−z)i (m−i+ 1)k,
satisfyingBn(k)(0) =Bn(k) (n≥0, k≥1), then we have relationships between the poly-Bernoulli polynomials and poly-Cauchy polynomials similar to those between the poly-Bernoulli numbers and the poly-Cauchy numbers.
Theorem 4.1. Forn≥1 we have
Bn(k)(z) = Xn
l=1
Xn
m=1
m!nn m
o m−1 l−1
c(k)l (z),
= (−1)n Xn
l=1
Xn
m=1
m!nn m
o nm l
oˆc(k)l (z),
c(k)n (z) = (−1)n Xn
l=1
Xn
m=1
(−1)m m!
hn m
i hm l
iBl(k)(z)
ˆ
c(k)n (z) = (−1)n Xn
l=1
Xn
m=1
1 m!
hn m
i hm l
iBl(k)(z).
References
[1] A. Bayad and Y. Hamahata, Polylogarithms and poly-Bernoulli polynomials, Kyushu J. Math., Vol. 65 (2011), 15–24.
[2] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.
[3] M.-A. Coppo and B. Candelpergher, The Arakawa-Kaneko zeta functions, Ra- manujan J., Vol. 22 (2010), 153–162.
[4] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Second Edition, Addison-Wesley, Reading, 1994.
[5] K. Kamano and T. Komatsu, Poly-Cauchy polynomials, (preprint).
[6] M. Kaneko, Poly-Bernoulli numbers,J. Th. Nombres Bordeaux, Vol. 9 (1997), 199–
206.
[7] T. Komatsu, Poly-Cauchy numbers,Kyushu J. Math., Vol. 67 (2013), (to appear).
[8] D. Merlini, R. Sprugnoli and M. C. Verri, The Cauchy numbers, Discrete Math., Vol. 306 (2006) 1906–1920.
