# A Generalization of Poly-Cauchy Numbers and Their Properties

Authors: Takao Komatsu; Vichian Laohakosol; Kálmán Liptai
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In Komatsu's work (2013), the concept of poly-Cauchy numbers is introduced as an analogue of that of poly-Bernoulli numbers. Both numbers are extensions of classical Cauchy numbers and Bernoulli numbers, respectively. There are several generalizations of poly-Cauchy numbers, including poly-Cauchy numbers with aq parameter and shifted poly-Cauchy numbers. In this paper, we give a further generalization of poly-Cauchy numbers and investigate several arithmetical properties. We also give the corresponding generalized poly-Bernoulli numbers so that both numbers have some relations.


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## Body

## 1. Introduction

Letn≥0, k≥1 be integers. Poly-Cauchy numbers of the first kind    cn(k) are defined by
(1)cn(k)=∫01‍⋯∫01‍︸k(x1x2⋯xk)(x1x2⋯xk-1)⋯(x1x2⋯xk-n+1)dx1dx2⋯dxk
[1]. The concept of poly-Cauchy numbers is a generalization of that of the classical Cauchy numbers cn=cn(1) defined by
(2)cn=∫01x(x-1)⋯(x-n+1)dx
(see, e.g., [2, 3]). The generating function of poly-Cauchy numbers ([1], Theorem 2) is given by
(3)Lifk(ln(1+x))=∑n=0∞cn(k)xnn!,
where
(4)Lifk(z)=∑m=0∞zmm!(m+1)k
is the kth polylogarithm factorial function. An explicit formula for cn(k) ([1], Theorem 1) is given by
(5)cn(k)=∑m=0n[nm](-1)n-m(m+1)k(n≥0,k≥1),
where [nm] are the (unsigned) Stirling numbers of the first kind, arising as coefficients of the rising factorial
(6)x(x+1)⋯(x+n-1)=∑m=0n[nm]xm
(see, e.g., [4]). See ([5], A224094–A224101) for the sequences arising from poly-Cauchy numbers.The concept of poly-Cauchy numbers is an analogue of that of poly-Bernoulli numbersBn(k) [6] defined by
(7)Lik(1-e-x)1-e-x=∑n=0∞Bn(k)xnn!,
where
(8)Lik(z)=∑m=1∞zmmk
is the kth polylogarithm function. When k=1, Bn=Bn(1) is the classical Bernoulli number with B1(1)=1/2, defined by the generating function
(9)xexex-1=∑n=0∞Bnxnn!.
An explicit formula for Bn(k) ([6], Theorem 1) is given by
(10)Bn(k)=(-1)n∑m=0n{nm}(-1)mm!(m+1)k(n≥0,k≥1),
where {nm} are the Stirling numbers of the second kind, determined by
(11){nm}=1m!∑j=0m(-1)j(mj)(m-j)n
(see, e.g., [4]).There are some kinds of generalizations of poly-Cauchy numbers. One is thepoly-Cauchy number with aqparameter   cn,q(k) [7] defined by
(12)cn,q(k)=∫01‍…∫01‍︸k(x1⋯xk)(x1⋯xk-q)⋯(x1⋯xk-(n-1)q)dx1⋯dxk.
Another is the shifted poly-Cauchy number   cn,a(k) [8] defined by
(13)cn,a(k)=∫01‍…∫01‍︸k(x1⋯xk)a(x1⋯xk-1)⋯(x1⋯xk-(n-1))dx1⋯dxk.
Notice that cn,a(k) can be expressed as
(14)cn,a(k)=∑m=0n[nm](-1)n-m(m+a)k.
For example, if n=5 and a=3, then
(15)c5(k)=242k-503k+354k-105k+16k,c5,3(k)=244k-505k+356k-107k+18k.
Therefore, such numbers are shifted from the original poly-Cauchy numbers. Remember that the Hurwitz zeta function ζ(s,q)=∑n=0∞‍1/(q+n)s is a generalization of the famous Riemann zeta function ζ(s)=∑n=1∞‍1/ns since ζ(s)=ζ(s,1).In this paper, we give a further generalization of poly-Cauchy numbers, including both kinds of generalizations, and show several combinatorial and characteristic properties. We also give the corresponding poly-Bernoulli numbers so that both numbers have some relations.


## 2. Definitions and Basic Properties

Letn≥0, k≥1 be integers, and let a, q and l1,…,lk be nonzero real numbers. For simplicity, we write L=(l1,…,lk) and ℓ=l1⋯lk. Define cn,a,q,L(k) by
(16)cn,a,q,L(k)=∫0l1…∫0lk︸k(x1⋯xk)a(x1⋯xk-q)⋯(x1⋯xk-(n-1)q)dx1⋯dxk.
Then, cn,a,q,L(k) can be expressed in terms of the Stirling numbers of the first kind [nm].Theorem 1.
Leta be a positive real number. Then,
(17)cn,a,q,L(k)=∑m=0n[nm](-q)n-mℓm+a(m+a)k(n≥0,k≥1).Remark 2.
Ifa=ℓ=1, then cn,1,q,(1,…,1)(k)=cn,q(k) is the poly-Cauchy number with a q parameter ([7], Theorem 1). If q=ℓ=1, then cn,a,1,(1,…,1)(k)=cn,a(k) is the shifted poly-Cauchy number ([8], Theorem 2).Proof.
By(18)x(x-1)⋯(x-n+1)=∑m=0n[nm](-1)n-mxm,
we have
(19)cn,a,q,L(k)=∫0l1…∫0lk∑m=0n[nm](-1)n-m×(x1⋯xk)m+a-1qn-mdx1⋯dxk=∑m=0n[nm](-q)n-mℓm+a(m+a)k.For an integerk and a positive real number a, define the extended polylogarithm factorial function Lifk(z;a) by
(20)Lifk(z;a)=∑m=0∞zmm!(m+a)k
[8]. When a=1, Lifk(z;1)=Lifk(z) is the polylogarithm factorial function [1]. The generating function of the number cn,a,q,L(k) (q≠0) is given by using the extended polylogarithm factorial function Lifk(a;z).Theorem 3.
One has(21)ℓaLifk(ℓln(1+qx)q;a)=∑n=0∞cn,a,q,L(k)xnn!.Remark 4.
Ifa=ℓ=1, then Theorem 3 is reduced to Theorem 2 in [7]. If q=ℓ=1, then Theorem 3 is reduced to Theorem 3 in [8].Proof.
Since(22)(ln(1+x))mm!=(-1)m∑n=m∞[nm](-x)nn!,
by Theorem 1 we have
(23)∑n=0∞cn,a,q,L(k)xnn!=∑n=0∞∑m=0n[nm](-q)n-mℓm+a(m+a)kxnn!=ℓa∑m=0∞(-q)-mℓm(m+a)k∑n=m∞[nm](-qx)nn!=ℓa∑m=0∞(-q)-mℓm(m+a)k(-1)m(ln(1+qx))mm!=ℓa∑m=0∞1m!(m+a)k(ℓln(1+qx)q)m=ℓaLifk(ℓln(1+qx)q;a).The generating function of the numbercn,a,q,L(k) can be written in the form of iterated integrals.Corollary 5.
Leta and q be real numbers with a>0 and q≠0. For k=1, one has
(24)(ℓqln(1+qx))a∫0x‍(ln(1+qx)q)a-1(1+qx)ℓ/q-1dx=∑n=0∞cn,a,q,L(1)xnn!.
For k>1, one has
(25)(ℓqln(1+qx))∫0x‍q(1+qx)ln(1+qx)∫0x‍⋯q(1+qx)ln(1+qx)∫0x‍︸k(ln(1+qx)q)a-1(1+qx)ℓ/q-1dx⋯dx︸k=∑n=0∞‍cn,a,q,L(k)xnn!.Remark 6.
Ifa=ℓ=1, then Corollary 5 is reduced to Corollary 1 in [7]. If q=ℓ=1, then Corollary 5 is reduced to Corollary 1 in [8].Proof.
Fork=1,
(26)Lif1(z;a)=∑m=0∞zmm!(m+a)=1za∑m=0∞zm+am!(m+a)=1za∫0z∑m=0∞zm+a-1m!=1za∫0zza-1ezdz=1za(ez∑i=0a-1(-1)i(a-1)!(a-i-1)!za-i-1(-1)a(a-1)!+ez∑i=0a-1(-1)i(a-1)!(a-i-1)!za-i-1).
Note that the last equation holds only if a is an integer. For k>1, we have
(27)Lifk(z;a)=1za∑m=0∞zm+am!(m+a)k=1za∫0z∑m=0∞zm+a-1m!(m+a)k-1dz=1za∫0zza-1Lifk-1(z;a)dz.
Hence,
(28)Lifk(z;a)=1za∫0z1z∫0z⋯1z∫0z1z∫0z︸kza-1ezdz⋯dz︸k.
Putting z=ℓln(1+qx)/q and multiplying by ℓa, we get the result.


## 3. Poly-Cauchy Numbers of the Second Kind

In [1], the concept of poly-Cauchy numbers of the second kind is also introduced. The poly-Cauchy numbers of the second kind c^n(k) are defined by
(29)c^n(k)=∫01‍⋯∫01‍︸k(-x1x2⋯xk)(-x1x2⋯xk-1)⋯(-x1x2⋯xk-n+1)dx1dx2⋯dxk,
and the generating function is given by
(30)Lifk(-ln(1+x))=∑n=0∞c^n(k)xnn!.Then, the poly-Cauchy numbers of the second kindc^n(k) can also be expressed in terms of the Stirling numbers of the first kind ([1], Theorem 4). See ([5], A219247, A224102–A224107, A224109) for the sequences arising from poly-Cauchy numbers of the second kind.Proposition 7.
One has(31)c^n(k)=(-1)n∑m=0n[nm]1(m+1)k.Leta be a positive real number. Similar to generalized poly-Cauchy numbers of the first kind cn,a,q,L(k), define the poly-Cauchy numbers of the second kind c^n,a,q,L(k) (n≥0, k≥1) by
(32)c^n,a,q,L(k)=(-1)a-1∫0l1⋯∫0lk(-x1⋯xk)a(-x1⋯xk-q)⋯(-x1⋯xk-(n-1)q)dx1⋯dxk.
Then, similar to Theorem 1, c^n,a,q,L(k) can also be expressed in terms of the Stirling numbers of the first kind [nm].Theorem 8.
One has(33)c^n,a,q,L(k)=(-1)n∑m=0n[nm]qn-mℓm+a(m+a)k(n≥0,k≥1).Theorem 9.
The generating function of the numberc^n,a,q,L(k) is given by
(34)ℓaLifk(-ℓln(1+qx)q;a)=∑m=0∞c^n,a,q,L(k)xnn!,
where
(35)Lifk(z;a)=∑m=0∞zmm!(m+a)k.Remark 10.
Ifa=ℓ=1, then Theorem 8 is reduced to Theorem 3 in [7] and Theorem 9 is reduced to Theorem 4 in [7]. If q=ℓ=1, then Theorem 8 is reduced to Theorem 5 in [8] and Theorem 9 is reduced to Theorem 6 in [8].
The generating function of the numberc^n,a,q,L(k) can be written in the form of iterated integrals.Corollary 11.
Leta be a positive real number. For k=1, one has
(36)(ℓqln(1+qx))a∫0x(ln(1+qx)q)a-1(1+qx)-ℓ/q-1dx=∑n=0∞c^n,a,q,L(1)xnn!.
For k>1, one has
(37)(ℓqln(1+qx))∫0xq(1+qx)ln(1+qx)∫0x⋯q(1+qx)ln(1+qx)∫0x︸k(ln(1+qx)q)a-1(1+qx)-ℓ/q-1dx⋯dx︸k=∑n=0∞c^n,a,q,L(k)xnn!.Remark 12.
Whena=q=k=ℓ=1 in the first identity, we have the generating function of the classical Cauchy numbers of the second kind:
(38)x(1+x)ln(1+x)=∑n=0∞c^nxnn!.
In addition, there are relations between both kinds of poly-Cauchy numbers ifq=1. For simplicity, we write cn,a,L(k)=cn,a,1,L(k) and c^n,a,L(k)=c^n,a,1,L(k).Theorem 13.
Letk be an integer and  a a positive real number. For n≥1, one has(39)(-1)ncn,a,L(k)n!=∑m=1n(n-1m-1)c^m,a,L(k)m!,(-1)nc^n,a,L(k)n!=∑m=1n(n-1m-1)cm,a,L(k)m!.Remark 14.
Ifa=ℓ=1, then Theorem 13 is reduced to Theorem 7 in [1].Proof.
We will prove the second identity. The first one is proved similarly and omitted. By using the identity (see, e.g., [4], Chapter 6)
(40)(-1)in![ni]=∑m=1n(-1)mm!(n-1m-1)[mi]
and Theorems 1 and 8, we have
(41)RHS=∑m=1n(n-1m-1)1m!×∑i=1m[mi](-1)m-iℓi+a(i+a)k=∑i=1n(-1)iℓi(i+a)k∑m=in(-1)mm!(n-1m-1)[mi]=∑i=1n(-1)iℓi+a(i+a)k(-1)in![ni]=LHS.


## 4. Some Expressions of Poly-Cauchy Numbers with Negative Indices

It is known that poly-Bernoulli numbers satisfy the duality theoremBn(-k)=Bk(-n) for n,k≥0 ([6], Theorem 2) because of the symmetric formula
(42)∑n=0∞∑k=0∞Bn(-k)xnn!ykk!=ex+yex+ey-ex+y.
However, the corresponding duality theorem does not hold for poly-Cauchy numbers for any real number a, by the following results.Proposition 15.
Suppose thatℓ=1. Then, for nonnegative integers n and k and a real number a≠0, one has
(43)∑n=0∞∑k=0∞cn,a,q,L(-k)xnn!ykk!=eay(1+qx)ey/q,∑n=0∞∑k=0∞c^n,a,q,L(-k)xnn!ykk!=eay(1+qx)ey/q.Remark 16.
Ifa=ℓ=1, then Proposition 15 is reduced to Proposition 1 in [7]. If q=ℓ=1, then Proposition 15 is reduced to Proposition 3 in [8].Proof.
We will prove the first identity. The second identity is proved similarly. By Theorem3, we have
(44)∑n=0∞∑k=0∞cn,a,q,L(-k)xnn!ykk!=∑k=0∞(∑n=0∞cn,a,q,L(-k)xnn!)ykk!=∑k=0∞ℓa∑m=0∞(m+a)km!(ℓln(1+qx)q)mykk!=∑m=0∞1m!(ln(1+qx)q)m∑k=0∞((m+a)y)kk!=∑m=0∞1m!(ln(1+qx)q)me(m+a)y=eay∑m=0∞1m!(eyqln(1+qx))m=eay(1+qx)ey/q.By using Proposition15, we have explicit expressions of poly-Cauchy numbers with negative indices. For simplicity, we write cn,a,q(-k)=cn,a,q,L(-k) and c^n,a,q(-k)=c^n,a,q,L(-k) if ℓ=1.Theorem 17.
For nonnegative integersn, k, and a real number a≠0, one has
(45)cn,a,q(-k)=∑i=0k∑j=0i∑λ=0n∑ν=0λj!(ki){ij}(nλ)[n-λj]×[λν]ak-i(-q)n-j-ν,c^n,a,q(-k)=∑i=0k∑j=0i∑λ=0n∑ν=0λ(-1)nj!(ki){ij}(nλ)×[n-λj][λν]ak-iqn-j-ν.Remark 18.
Ifa=q=1, by
(46)∑i=0k(ki){ij}={k+1j+1}
[4], the above identities become
(47)cn,1,1(-k)=cn(-k)=∑j=0k(-1)n+jj!×([nj]-n[n-1j]){k+1j+1},c^n,1,1(-k)=c^n(-k)=∑j=0k(-1)nj![n+1j+1]{k+1j+1}.Proof.
By Proposition15 together with
(48)(ey-1)jj!=∑k=j∞{kj}ykk!,(-ln(1+x))jj!=∑n=j∞[nj](-x)nn!
[4], we have
(49)∑n=0∞∑k=0∞cn,a,q(-k)xnn!ykk!=((1+qx)1/q)ey-1(1+qx)1/qeay=exp((ey-1)(ln(1+qx))1/q)×(1+qx)1/qeay=∑j=0∞j!qj(ey-1)jj!(ln(1+qx))jj!×(1+qx)1/qeay=∑j=0∞(-1)jj!qjeay∑k=j∞{kj}ykk!(1+qx)1/q×∑n=j∞[nj](-qx)nn!.
Since
(50)eay∑k=j∞{kj}ykk!=∑l=0∞(ay)ll!∑k=j∞{kj}ykk!=∑k=0∞(∑i=0kak-i(k-i)!{ij}1i!)yk=∑k=0∞(∑i=0k(ki){ij}ak-i)ykk!,(1+qx)1/q∑n=j∞[nj](-qx)nn!=∑l=0∞(1ql)(qx)l∑n=j∞[nj](-qx)nn!=∑l=0∞(-1)l∑ν=0l[lν](-1q)ν(qx)ll!×∑n=j∞[nj](-qx)nn!=∑l=0∞∑ν=0l[lν](-q)l-νxll!∑n=0∞[nj](-qx)nn!=∑n=0∞∑λ=0n∑ν=0λ[λν](-q)λ-νλ![n-λj](-q)n-λ(n-λ)!xn=∑n=0∞∑λ=0n(nλ)[n-λj]∑ν=0λ[λν](-q)n-νxnn!,
we obtain
(51)∑n=0∞∑k=0∞cn,a,q(-k)xnn!ykk!=∑n=0∞∑k=0∞∑i=0k∑j=0i(-1)jj!qj(ki){ij}ak-i×∑λ=0n(nλ)[n-λj]∑ν=0λ‍[λν](-q)n-νxnn!ykk!=∑n=0∞‍∑k=0∞‍∑i=0k‍∑j=0i‍∑λ=0n‍∑ν=0λ‍j!(ki){ij}×(nλ)[n-λj][λν]ak-i×(-q)n-j-νxnn!ykk!.
Similarly, by
(52)(1+qx)-1/q∑n=j∞‍[nj](-qx)nn!=∑l=0∞‍(-1)l∑ν=0l‍[lν](1q)ν(qx)ll!×∑n=j∞‍[nj](-qx)nn!=∑l=0∞‍∑ν=0l‍[lν]ql-ν(-1)lxll!∑n=0∞‍[nj](-qx)nn!=∑n=0∞‍∑λ=0n‍∑ν=0λ‍[λν]qλ-ν(-1)λλ!×[n-λj](-q)n-λ(n-λ)!xn=∑n=0∞‍(-1)n∑λ=0n‍(nλ)[n-λj]∑ν=0λ‍[λν]qn-νxnn!,
we get
(53)∑n=0∞∑k=0∞c^n,a,q(-k)xnn!ykk!=(1+qx)-ey/qeay=exp((ey-1)(ln(1+qx))-1/q)×(1+qx)-1/qeay=∑j=0∞j!qj(ey-1)jj!(-ln(1+qx))jj!×(1+qx)-1/qeay=∑j=0∞j!qjeay∑k=j∞{kj}ykk!(1+qx)-1/q×∑n=j∞[nj](-qx)nn!=∑j=0∞j!qj∑k=0∞(∑i=0k(ki){ij}ak-i)ykk!×∑n=0∞(-1)n∑λ=0n(nλ)[n-λj]∑ν=0λ[λν]qn-νxnn!=∑n=0∞∑k=0∞∑i=0k∑j=0i∑λ=0n∑ν=0λ(-1)nj!×(ki){ij}(nλ)[n-λj]×[λν]ak-iqn-j-νxnn!ykk!.


## 5. Poly-Bernoulli Numbers Corresponding to Poly-Cauchy Numbers

In this section, we will consider the corresponding generalized poly-Bernoulli numbers to the generalized poly-Cauchy numbers discussed in the previous sections. Letk be an integer and a a positive real number. An explicit form of poly-Bernoulli number Bn(k) is given by
(54)Bn(k)=∑m=0n{nm}(-1)n-mm!(m+1)k
([6], Theorem 1). In ([1], Theorem 8), one expression of Bn(k) in terms of poly-Cauchy numbers cn(k) is given.Proposition 19.
One has(55)Bn(k)=∑l=1n∑m=1nm!{nm}{m-1l-1}cl(k)(n≥1).On the contrary, in ([9], Theorem 2.2), one expression of cn(k) in terms of Bn(k) is given.Proposition 20.
One has(56)cn(k)=∑l=1n∑m=1n(-1)n-mm![nm][ml]Bl(k)(n≥1).As a counterpart of a generalized poly-Cauchy number, we will define a generalized poly-Bernoulli numberBn,a,L(k) by
(57)ℓa-1Lik(ℓ(1-e-t);a-1)1-e-t=∑n=0∞Bn,a,L(k)tnn!,
where Lik(z;a) is the generalized polylogarithm function defined by
(58)Lik(z;a)=∑m=0∞zm(m+a)k,
so that Lik(z;0)=Lik(z).Then,Bn,a,L(k) can be expressed explicitly in terms of the Stirling numbers of the second kind. Note that Bn,1,(1,…,1)(k)=Bn(k).Proposition 21.
One has(59)Bn,a,L(k)=∑m=0n{nm}(-1)n-mm!ℓm+a(m+a)k(n≥0).Proof.
By(60)(et-1)mm!=∑n=m∞{nm}tnn!,
we have
(61)∑n=0∞Bn,a,L(k)tnn!=ℓaℓ(1-e-t)∑m=1∞(ℓ(1-e-t))m(m+a-1)k=ℓa∑m=0∞‍(ℓ(1-e-t))m(m+a)k=ℓa∑m=0∞(-ℓ)mm!(m+a)k∑n=m∞{nm}(-t)nn!=∑n=0∞(∑m=0n{nm}(-1)n-mm!ℓm+a(m+a)k)tnn!.
Comparing the coefficients on both sides, we get the result.For simplicity, we writecn,a,L(k)=cn,a,1,L(k) and c^n,a,L(k)=c^n,a,1,L(k). If a=ℓ=1, then our results below are reduced to those previous ones.Theorem 22.
Forn≥0, one has
(62)Bn,a,L(k)=∑j=1n∑m=1nm!{nm}{m-1j-1}cj,a,L(k),cn,a,L(k)=∑j=1n∑m=1n(-1)n-mm![nm][mj]Bj,a,L(k).Proof.
For the first identity,(63)RHS=∑j=1n∑m=jnm!{nm}{m-1j-1}×∑i=0j[ji](-1)j-iℓi+a(i+a)k=∑i=1n(-1)iℓi+a(i+a)k×∑j=in∑m=jnm!{nm}{m-1j-1}(-1)j[ji]=∑i=1n(-1)iℓi+a(i+a)k∑m=inm!{nm}×∑j=im(-1)j{m-1j-1}[ji]=∑i=1n(-1)iℓi+a(i+a)k∑m=inm!{nm}(-1)m(m-1i-1)=∑i=1n(-1)iℓi+a(i+a)k(-1)ni!{ni}=LHS.
For the second identity,
(64)RHS=∑j=1n∑m=1n(-1)n-mm![nm][mj]×∑i=0j{ji}(-1)j-ii!ℓi+a(i+a)k=∑m=1n(-1)n-mm![nm]×∑j=0n[mj]∑i=0j{ji}(-1)j-ii!ℓi+a(i+a)k=∑m=1n(-1)n-mm![nm]×∑i=0n(-1)ii!ℓi+a(i+a)k∑j=in(-1)j[mj]{ji}=∑m=0n(-1)n-mm![nm]×(-1)mm!ℓm+a(m+a)k(-1)m=∑m=0n[nm](-1)n-mℓm+a(m+a)k=LHS.
Note that [m0]=0   (m≥1) and [mj]=0 (j>m), and
(65)∑j=im(-1)m-j[mj]{ji}={1(i=m);0(i≠m).Similarly, concerning(66)c^n,a,L(k)=(-1)n∑m=0n[nm]ℓm+a(m+a)k(n≥0)
as a generalization of poly-Cauchy numbers of the second kind c^n(k), we have the following.Theorem 23.
One has(67)Bn,a,L(k)=(-1)n∑j=1n∑m=1nm!{nm}{mj}c^j,a,L(k),c^n,a,L(k)=(-1)n∑j=1n∑m=1n1m![nm][mj]Bj,a,L(k).Remark 24.
Ifa=ℓ=1, these results are reduced to the identities in Theorems 3.2 and 3.1 in [9], respectively.



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*Source: 179841-2013-12-04.xml*