A Generalization of Poly-Cauchy Numbers and Their Properties

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 a q 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.

1. Introduction

Let 𝑛≥0, 𝑘≥1 be integers. Poly-Cauchy numbers of the first kind    𝑐_𝑛^(𝑘) are defined by 𝑐_𝑛^(𝑘)=¹₀⋯¹_0𝑘𝑥₁𝑥₂⋯𝑥_𝑘𝑥₁𝑥₂⋯𝑥_𝑘⋯𝑥−1₁𝑥₂⋯𝑥_𝑘−𝑛+1𝑑𝑥₁𝑑𝑥₂⋯𝑑𝑥_𝑘 (1) [1]. The concept of poly-Cauchy numbers is a generalization of that of the classical Cauchy numbers 𝑐_𝑛=𝑐_𝑛⁽¹⁾ defined by 𝑐_𝑛=¹₀𝑥(𝑥−1)⋯(𝑥−𝑛+1)𝑑𝑥 (2) (see, e.g., [2, 3]). The generating function of poly-Cauchy numbers ([1], Theorem 2) is given by Lif_𝑘(ln(1+𝑥))=^∞_𝑛=0𝑐_𝑛^(𝑘)𝑥^𝑛,𝑛! (3) where Lif_𝑘(𝑧)=^∞_𝑚=0𝑧^𝑚𝑚!(𝑚+1)^𝑘 (4) is the 𝑘th polylogarithm factorial function. An explicit formula for 𝑐_𝑛^(𝑘) ([1], Theorem 1) is given by 𝑐_𝑛^(𝑘)=^𝑛_𝑚=0𝑛𝑚(−1)^𝑛−𝑚(𝑚+1)^𝑘(𝑛≥0,𝑘≥1), (5) where [^𝑛_𝑚] are the (unsigned) Stirling numbers of the first kind, arising as coefficients of the rising factorial 𝑥(𝑥+1)⋯(𝑥+𝑛−1)=^𝑛_𝑚=0𝑛𝑚𝑥^𝑚 (6) (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 numbers 𝐵_𝑛^(𝑘) [6] defined by Li_𝑘1−𝑒^−𝑥1−𝑒^−𝑥=^∞_𝑛=0𝐵_𝑛^(𝑘)𝑥^𝑛,𝑛! (7) where Li_𝑘(𝑧)=^∞_𝑚=1𝑧^𝑚𝑚^𝑘 (8) is the 𝑘th polylogarithm function. When 𝑘=1, 𝐵_𝑛=𝐵_𝑛⁽¹⁾ is the classical Bernoulli number with 𝐵₁⁽¹⁾=1/2, defined by the generating function 𝑥𝑒^𝑥𝑒^𝑥=−1^∞_𝑛=0𝐵_𝑛𝑥^𝑛.𝑛! (9) An explicit formula for 𝐵_𝑛^(𝑘) ([6], Theorem 1) is given by 𝐵_𝑛^(𝑘)=(−1)^𝑛𝑛_𝑚=0𝑛𝑚(−1)^𝑚𝑚!(𝑚+1)^𝑘(𝑛≥0,𝑘≥1), (10) where {^𝑛_𝑚} are the Stirling numbers of the second kind, determined by 𝑛𝑚=1𝑚!^𝑚_𝑗=0(−1)^𝑗𝑚𝑗𝑚−𝑗^𝑛 (11) (see, e.g., [4]).

There are some kinds of generalizations of poly-Cauchy numbers. One is the poly-Cauchy number with a 𝑞 parameter   𝑐^(𝑘)_𝑛,𝑞 [7] defined by 𝑐^(𝑘)_𝑛,𝑞=¹₀…¹_0𝑘𝑥₁⋯𝑥_𝑘𝑥₁⋯𝑥_𝑘⋯𝑥−𝑞₁⋯𝑥_𝑘−(𝑛−1)𝑞𝑑𝑥₁⋯𝑑𝑥_𝑘. (12) Another is the shifted poly-Cauchy number   𝑐^(𝑘)_𝑛,𝑎 [8] defined by 𝑐^(𝑘)_𝑛,𝑎=¹₀…¹_0𝑘𝑥₁⋯𝑥_𝑘^𝑎𝑥₁⋯𝑥_𝑘⋯𝑥−1₁⋯𝑥_𝑘−(𝑛−1)𝑑𝑥₁⋯𝑑𝑥_𝑘. (13) Notice that 𝑐^(𝑘)_𝑛,𝑎 can be expressed as 𝑐^(𝑘)_𝑛,𝑎=^𝑛_𝑚=0𝑛𝑚(−1)^𝑛−𝑚(𝑚+𝑎)^𝑘. (14) For example, if 𝑛=5 and 𝑎=3, then 𝑐₅^(𝑘)=242^𝑘−503^𝑘+354^𝑘−105^𝑘+16^𝑘,𝑐^(𝑘)_5,3=244^𝑘−505^𝑘+356^𝑘−107^𝑘+18^𝑘. (15) Therefore, such numbers are shifted from the original poly-Cauchy numbers. Remember that the Hurwitz zeta function 𝜁(𝑠,𝑞)=∑^∞_𝑛=01/(𝑞+𝑛)^𝑠 is a generalization of the famous Riemann zeta function 𝜁(𝑠)=∑^∞_𝑛=11/𝑛^𝑠 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

Let 𝑛≥0, 𝑘≥1 be integers, and let 𝑎, 𝑞 and 𝑙₁,…,𝑙_𝑘 be nonzero real numbers. For simplicity, we write 𝐿=(𝑙₁,…,𝑙_𝑘) and ℓ=𝑙₁⋯𝑙_𝑘. Define 𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿} by 𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿}=^𝑙1₀…^𝑙𝑘_0𝑘𝑥₁⋯𝑥_𝑘^𝑎𝑥₁⋯𝑥_𝑘⋯𝑥−𝑞₁⋯𝑥_𝑘−(𝑛−1)𝑞𝑑𝑥₁⋯𝑑𝑥_𝑘. (16) Then, 𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿} can be expressed in terms of the Stirling numbers of the first kind [^𝑛_𝑚].

Theorem 1. Let 𝑎 be a positive real number. Then, 𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿}=^𝑛_𝑚=0𝑛𝑚−𝑞^𝑛−𝑚ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘(𝑛≥0,𝑘≥1). (17)

Remark 2. If 𝑎=ℓ=1, then 𝑐^(𝑘)_{𝑛,1,𝑞,(1,…,1)}=𝑐^(𝑘)_𝑛,𝑞 is the poly-Cauchy number with a 𝑞 parameter ([7], Theorem 1). If 𝑞=ℓ=1, then 𝑐^(𝑘)_{𝑛,𝑎,1,(1,…,1)}=𝑐^(𝑘)_𝑛,𝑎 is the shifted poly-Cauchy number ([8], Theorem 2).

Proof. By 𝑥(𝑥−1)⋯(𝑥−𝑛+1)=^𝑛_𝑚=0𝑛𝑚(−1)^𝑛−𝑚𝑥^𝑚, (18) we have 𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿}=^𝑙1₀…^𝑙𝑘₀^𝑛_𝑚=0𝑛𝑚(−1)^𝑛−𝑚×𝑥₁⋯𝑥_𝑘^{𝑚+𝑎−1}𝑞^𝑛−𝑚𝑑𝑥₁⋯𝑑𝑥_𝑘=^𝑛_𝑚=0𝑛𝑚−𝑞^𝑛−𝑚ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘. (19)

For an integer 𝑘 and a positive real number 𝑎, define the extended polylogarithm factorial function Lif_𝑘(𝑧;𝑎) by Lif_𝑘(𝑧;𝑎)=^∞_𝑚=0𝑧^𝑚𝑚!(𝑚+𝑎)^𝑘 (20) [8]. When 𝑎=1, Lif_𝑘(𝑧;1)=Lif_𝑘(𝑧) is the polylogarithm factorial function [1]. The generating function of the number 𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿} (𝑞≠0) is given by using the extended polylogarithm factorial function Lif_𝑘(𝑎;𝑧).

Theorem 3. One has ℓ^𝑎Lif_𝑘ℓln1+𝑞𝑥𝑞;𝑎=^∞_𝑛=0𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛.𝑛! (21)

Remark 4. If 𝑎=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 (ln(1+𝑥))^𝑚=()𝑚!−1^𝑚∞_𝑛=𝑚𝑛𝑚(−𝑥)^𝑛𝑛!, (22) by Theorem 1 we have ^∞_𝑛=0𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛=𝑛!^∞𝑛_𝑛=0𝑚=0𝑛𝑚−𝑞^𝑛−𝑚ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘𝑥^𝑛𝑛!=ℓ^𝑎∞_𝑚=0−𝑞^−𝑚ℓ^𝑚(𝑚+𝑎)^𝑘∞_𝑛=𝑚𝑛𝑚−𝑞𝑥^𝑛𝑛!=ℓ^𝑎∞_𝑚=0−𝑞^−𝑚ℓ^𝑚(𝑚+𝑎)^𝑘(−1)^𝑚ln(1+𝑞𝑥)^𝑚𝑚!=ℓ^𝑎∞_𝑚=01𝑚!(𝑚+𝑎)^𝑘ℓln(1+𝑞𝑥)𝑞^𝑚=ℓ^𝑎Lif_𝑘ℓln1+𝑞𝑥𝑞;𝑎. (23)

The generating function of the number 𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿} can be written in the form of iterated integrals.

Corollary 5. Let 𝑎 and 𝑞 be real numbers with 𝑎>0 and 𝑞≠0. For 𝑘=1, one has ℓ𝑞ln1+𝑞𝑥^𝑎𝑥₀ln(1+𝑞𝑥)𝑞^𝑎−11+𝑞𝑥^ℓ/𝑞−1=𝑑𝑥_∞𝑛=0𝑐⁽¹⁾_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛.𝑛! (24) For 𝑘>1, one has ℓ𝑞ln1+𝑞𝑥_𝑥0𝑞1+𝑞𝑥ln1+𝑞𝑥^𝑥₀⋯𝑞1+𝑞𝑥ln1+𝑞𝑥^𝑥_0𝑘ln1+𝑞𝑥𝑞^𝑎−11+𝑞𝑥^ℓ/𝑞−1𝑑𝑥⋯𝑑𝑥_𝑘=∑^∞_𝑛=0𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛.𝑛! (25)

Remark 6. If 𝑎=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. For 𝑘=1, Lif₁(𝑧;𝑎)=^∞_𝑚=0𝑧^𝑚()=1𝑚!𝑚+𝑎𝑧^𝑎∞_𝑚=0𝑧^𝑚+𝑎()=1𝑚!𝑚+𝑎𝑧^𝑎𝑧₀^∞_𝑚=0𝑧^{𝑚+𝑎−1}=1𝑚!𝑧^𝑎𝑧₀𝑧^𝑎−1𝑒^𝑧=1𝑑𝑧𝑧^𝑎(−1)^𝑎(𝑎−1)!+𝑒^𝑧𝑎−1_𝑖=0(−1)^𝑖(𝑎−1)!()!𝑧𝑎−𝑖−1^{𝑎−𝑖−1}. (26) Note that the last equation holds only if 𝑎 is an integer. For 𝑘>1, we have Lif_𝑘()=1𝑧;𝑎𝑧^𝑎∞_𝑚=0𝑧^𝑚+𝑎𝑚!(𝑚+𝑎)^𝑘=1𝑧^𝑎𝑧₀^∞_𝑚=0𝑧^{𝑚+𝑎−1}𝑚!(𝑚+𝑎)^𝑘−1=1𝑑𝑧𝑧^𝑎𝑧₀𝑧^𝑎−1Lif_𝑘−1(𝑧;𝑎)𝑑𝑧. (27) Hence, Lif_𝑘()=1𝑧;𝑎𝑧^𝑎𝑧₀1𝑧^𝑧₀⋯1𝑧^𝑧₀1^𝑧𝑧_0𝑘𝑧^𝑎−1𝑒^𝑧𝑑𝑧⋯𝑑𝑧_𝑘. (28) Putting 𝑧=ℓln(1+𝑞𝑥)/𝑞 and multiplying by ℓ^𝑎, 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 𝑐̂_𝑛^(𝑘) are defined by 𝑐̂_𝑛^(𝑘)=¹₀⋯¹_0𝑘−𝑥₁𝑥₂⋯𝑥_𝑘−𝑥₁𝑥₂⋯𝑥_𝑘−1⋯−𝑥₁𝑥₂⋯𝑥_𝑘−𝑛+1𝑑𝑥₁𝑑𝑥₂⋯𝑑𝑥_𝑘, (29) and the generating function is given by Lif_𝑘(−ln(1+𝑥))=^∞_𝑛=0𝑐̂_𝑛^(𝑘)𝑥^𝑛.𝑛! (30)

Then, the poly-Cauchy numbers of the second kind 𝑐̂_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 𝑐̂_𝑛^(𝑘)=(−1)^𝑛𝑛_𝑚=0𝑛𝑚1(𝑚+1)^𝑘. (31)

Let 𝑎 be a positive real number. Similar to generalized poly-Cauchy numbers of the first kind 𝑐^(𝑘)_{𝑛,𝑎,𝑞,𝐿}, define the poly-Cauchy numbers of the second kind 𝑐̂^(𝑘)_{𝑛,𝑎,𝑞,𝐿} (𝑛≥0, 𝑘≥1) by 𝑐̂^(𝑘)_{𝑛,𝑎,𝑞,𝐿}=(−1)^{𝑎−1𝑙1}₀⋯^𝑙𝑘₀−𝑥₁⋯𝑥_𝑘^𝑎−𝑥₁⋯𝑥_𝑘−𝑞⋯−𝑥₁⋯𝑥_𝑘−(𝑛−1)𝑞𝑑𝑥₁⋯𝑑𝑥_𝑘. (32) Then, similar to Theorem 1, 𝑐̂^(𝑘)_{𝑛,𝑎,𝑞,𝐿} can also be expressed in terms of the Stirling numbers of the first kind [^𝑛_𝑚].

Theorem 8. One has 𝑐̂^(𝑘)_{𝑛,𝑎,𝑞,𝐿}=(−1)^𝑛𝑛_𝑚=0𝑛𝑚𝑞^𝑛−𝑚ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘(𝑛≥0,𝑘≥1). (33)

Theorem 9. The generating function of the number 𝑐̂^(𝑘)_{𝑛,𝑎,𝑞,𝐿} is given by ℓ^𝑎Lif_𝑘−ℓln1+𝑞𝑥𝑞;𝑎=^∞_𝑚=0𝑐̂^(𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛,𝑛! (34) where Lif_𝑘(𝑧;𝑎)=^∞_𝑚=0𝑧^𝑚𝑚!(𝑚+𝑎)^𝑘. (35)

Remark 10. If 𝑎=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 number 𝑐̂^(𝑘)_{𝑛,𝑎,𝑞,𝐿} can be written in the form of iterated integrals.

Corollary 11. Let 𝑎 be a positive real number. For 𝑘=1, one has ℓ𝑞ln(1+𝑞𝑥)^𝑎𝑥₀ln(1+𝑞𝑥)𝑞^𝑎−11+𝑞𝑥^{−ℓ/𝑞−1}=𝑑𝑥_∞𝑛=0𝑐̂⁽¹⁾_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛.𝑛! (36) For 𝑘>1, one has ℓ𝑞ln1+𝑞𝑥_𝑥0𝑞1+𝑞𝑥ln1+𝑞𝑥^𝑥₀⋯𝑞1+𝑞𝑥ln1+𝑞𝑥^𝑥_0𝑘ln1+𝑞𝑥𝑞^𝑎−11+𝑞𝑥^{−ℓ/𝑞−1}𝑑𝑥⋯𝑑𝑥_𝑘=^∞_𝑛=0𝑐̂^(𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛.𝑛! (37)

Remark 12. When 𝑎=𝑞=𝑘=ℓ=1 in the first identity, we have the generating function of the classical Cauchy numbers of the second kind: 𝑥()()=1+𝑥ln1+𝑥^∞_𝑛=0𝑐_𝑛̂𝑥^𝑛.𝑛! (38)
In addition, there are relations between both kinds of poly-Cauchy numbers if 𝑞=1. For simplicity, we write 𝑐^(𝑘)_{𝑛,𝑎,𝐿}=𝑐^(𝑘)_{𝑛,𝑎,1,𝐿} and 𝑐̂^(𝑘)_{𝑛,𝑎,𝐿}=𝑐̂^(𝑘)_{𝑛,𝑎,1,𝐿}.

Theorem 13. Let 𝑘 be an integer and  𝑎 a positive real number. For 𝑛≥1, one has(−1)^𝑛𝑐^(𝑘)_{𝑛,𝑎,𝐿}=𝑛!^𝑛_𝑚=1𝑛−1𝑐̂𝑚−1^{(𝑘)𝑚,𝑎,𝐿},()𝑚!−1^𝑛𝑐̂^(𝑘)_{𝑛,𝑎,𝐿}=𝑛!^𝑛_𝑚=1𝑛−1𝑐𝑚−1^{(𝑘)𝑚,𝑎,𝐿}.𝑚! (39)

Remark 14. If 𝑎=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) (−1)^𝑖𝑛𝑖=𝑛!^𝑛_𝑚=1(−1)^𝑚𝑚𝑖𝑚!𝑛−1𝑚−1 (40) and Theorems 1 and 8, we have RHS=^𝑛_𝑚=1𝑛−11𝑚−1×𝑚!_𝑚𝑖=1𝑚𝑖(−1)^𝑚−𝑖ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘=^𝑛_𝑖=1(−1)^𝑖ℓ^𝑖(𝑖+𝑎)^𝑘𝑛_𝑚=𝑖(−1)^𝑚𝑚𝑖=𝑚!𝑛−1𝑚−1_𝑛𝑖=1(−1)^𝑖ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘(−1)^𝑖𝑛𝑖𝑛!=LHS. (41)

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

It is known that poly-Bernoulli numbers satisfy the duality theorem 𝐵_𝑛^(−𝑘)=𝐵_𝑘^(−𝑛) for 𝑛,𝑘≥0 ([6], Theorem 2) because of the symmetric formula ^∞∞_𝑛=0𝑘=0𝐵_𝑛^(−𝑘)𝑥^𝑛𝑦𝑛!^𝑘=𝑒𝑘!𝑒^𝑥+𝑒^𝑦−^𝑥+𝑦𝑒^𝑥+𝑦. (42) However, the corresponding duality theorem does not hold for poly-Cauchy numbers for any real number 𝑎, by the following results.

Proposition 15. Suppose that ℓ=1. Then, for nonnegative integers 𝑛 and 𝑘 and a real number 𝑎≠0, one has ^∞∞_𝑛=0𝑘=0𝑐^(−𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛𝑦𝑛!𝑘^𝑘!=𝑒^𝑎𝑦1+𝑞𝑥^{𝑒𝑦/𝑞},_{∞∞𝑛=0𝑘=0}𝑐̂^(−𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛𝑦𝑛!^𝑘=𝑒𝑘!1+𝑞𝑥^{𝑎𝑦𝑒𝑦/𝑞}. (43)

Remark 16. If 𝑎=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 Theorem 3, we have ^∞∞_𝑛=0𝑘=0𝑐^(−𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛𝑦𝑛!^𝑘=𝑘!_{∞𝑘=0∞𝑛=0}𝑐^(−𝑘)_{𝑛,𝑎,𝑞,𝐿}𝑥^𝑛𝑦𝑛!^𝑘=𝑘!_∞𝑘=0ℓ^𝑎∞_𝑚=0(𝑚+𝑎)^𝑘𝑚!ℓln(1+𝑞𝑥)𝑞^𝑚𝑦^𝑘=𝑘!_∞𝑚=01𝑚!ln(1+𝑞𝑥)𝑞^𝑚∞_𝑘=0(𝑚+𝑎)𝑦^𝑘=𝑘!_∞𝑚=01𝑚!ln(1+𝑞𝑥)𝑞^𝑚𝑒^{(𝑚+𝑎)𝑦}=𝑒^∞𝑎𝑦_𝑚=01𝑒𝑚!𝑞^𝑦ln1+𝑞𝑥^𝑚=𝑒^𝑎𝑦1+𝑞𝑥^{𝑒𝑦/𝑞}. (44)

By using Proposition 15, we have explicit expressions of poly-Cauchy numbers with negative indices. For simplicity, we write 𝑐^(−𝑘)_{𝑛,𝑎,𝑞}=𝑐^(−𝑘)_{𝑛,𝑎,𝑞,𝐿} and 𝑐̂^(−𝑘)_{𝑛,𝑎,𝑞}=𝑐̂^(−𝑘)_{𝑛,𝑎,𝑞,𝐿} if ℓ=1.

Theorem 17. For nonnegative integers 𝑛, 𝑘, and a real number 𝑎≠0, one has 𝑐^(−𝑘)_{𝑛,𝑎,𝑞}=^𝑘𝑖_𝑖=0^𝑛_𝑗=0^𝜆_𝜆=0𝜈=0𝑘𝑖𝑖𝑗𝑛𝜆𝑗×𝜆𝜈𝑎𝑗!𝑛−𝜆_𝑘−𝑖−𝑞^{𝑛−𝑗−𝜈},𝑐̂^(−𝑘)_{𝑛,𝑎,𝑞}=^𝑘𝑖_𝑖=0^𝑛_𝑗=0^𝜆_𝜆=0𝜈=0(−1)^𝑛𝑘𝑖𝑖𝑗𝑛𝜆×𝑗𝜆𝜈𝑎𝑗!𝑛−𝜆_𝑘−𝑖𝑞^{𝑛−𝑗−𝜈}. (45)

Remark 18. If 𝑎=𝑞=1, by ^𝑘_𝑖=0𝑘𝑖𝑖𝑗=𝑘+1𝑗+1 (46) [4], the above identities become 𝑐^(−𝑘)_𝑛,1,1=𝑐_𝑛^(−𝑘)=^𝑘_𝑗=0(−1)^𝑛+𝑗×𝑛𝑗𝑗,𝑐̂𝑗!−𝑛𝑛−1𝑘+1𝑗+1_{(−𝑘)𝑛,1,1}=𝑐̂_𝑛^(−𝑘)=^𝑘_𝑗=0(−1)^𝑛𝑗!.𝑛+1𝑗+1𝑘+1𝑗+1 (47)

Proof. By Proposition 15 together with 𝑒^𝑦−1^𝑗=𝑗!^∞_𝑘=𝑗𝑘𝑗𝑦^𝑘,((𝑘!−ln1+𝑥))^𝑗=𝑗!^∞_𝑛=𝑗𝑛𝑗(−𝑥)^𝑛𝑛! (48) [4], we have ^∞∞_𝑛=0𝑘=0𝑐^(−𝑘)_{𝑛,𝑎,𝑞}𝑥^𝑛𝑦𝑛!^𝑘=𝑘!1+𝑞𝑥^{1/𝑞𝑒𝑦−1}1+𝑞𝑥^1/𝑞𝑒^𝑎𝑦=exp𝑒^𝑦−1ln1+𝑞𝑥^1/𝑞×1+𝑞𝑥^1/𝑞𝑒^𝑎𝑦=^∞_𝑗=0𝑗!𝑞^𝑗𝑒^𝑦−1^𝑗𝑗!ln1+𝑞𝑥^𝑗×𝑗!1+𝑞𝑥^1/𝑞𝑒^𝑎𝑦=^∞_𝑗=0(−1)^𝑗𝑗!𝑞^𝑗𝑒^∞𝑎𝑦_𝑘=𝑗𝑘𝑗𝑦^𝑘𝑘!1+𝑞𝑥^1/𝑞×^∞_𝑛=𝑗𝑛𝑗−𝑞𝑥^𝑛.𝑛! (49) Since 𝑒^∞𝑎𝑦_𝑘=𝑗𝑘𝑗𝑦^𝑘=𝑘!^∞_𝑙=0𝑎𝑦^𝑙𝑙!^∞_𝑘=𝑗𝑘𝑗𝑦^𝑘=𝑘!_{∞𝑘=0𝑘𝑖=0}𝑎^𝑘−𝑖()!𝑖𝑗1𝑘−𝑖𝑦𝑖!^𝑘=^∞_𝑘=0^𝑘_𝑖=0𝑘𝑖𝑖𝑗𝑎^𝑘−𝑖𝑦^𝑘,𝑘!1+𝑞𝑥^∞1/𝑞_𝑛=𝑗𝑛𝑗−𝑞𝑥^𝑛=𝑛!_∞𝑙=01𝑞𝑙𝑞𝑥^𝑙∞_𝑛=𝑗𝑛𝑗−𝑞𝑥^𝑛=𝑛!_∞𝑙=0(−1)^𝑙𝑙_𝜈=0𝑙𝜈−1𝑞^𝜈𝑞𝑥^𝑙×𝑙!_∞𝑛=𝑗𝑛𝑗−𝑞𝑥^𝑛=𝑛!_{∞𝑙𝑙=0𝜈=0}𝑙𝜈−𝑞^𝑙−𝜈𝑥^𝑙𝑙!^∞_𝑛=0𝑛𝑗−𝑞𝑥^𝑛=𝑛!_{∞𝑛𝑛=0𝜆𝜆=0𝜈=0}𝜆𝜈−𝑞^𝜆−𝜈𝑗𝜆!𝑛−𝜆−𝑞^𝑛−𝜆()!𝑥𝑛−𝜆^𝑛=^∞𝑛_𝑛=0𝜆=0𝑛𝜆𝑗𝑛−𝜆^𝜆_𝜈=0𝜆𝜈−𝑞^𝑛−𝜈𝑥^𝑛,𝑛! (50) we obtain ^∞∞_𝑛=0𝑘=0𝑐^(−𝑘)_{𝑛,𝑎,𝑞}𝑥^𝑛𝑦𝑛!^𝑘=𝑘!_{∞∞𝑛=0𝑘𝑘=0𝑖𝑖=0𝑗=0}(−1)^𝑗𝑗!𝑞^𝑗𝑘𝑖𝑖𝑗𝑎^𝑘−𝑖×^𝑛_𝜆=0𝑛𝜆𝑗𝑛−𝜆^𝜆_𝜈=0𝜆𝜈−𝑞^𝑛−𝜈𝑥^𝑛𝑦𝑛!^𝑘=𝑘!_{∞∞𝑛=0𝑘𝑘=0𝑖𝑖=0𝑛𝑗=0𝜆𝜆=0𝜈=0}𝑘𝑖𝑖𝑗×𝑛𝜆𝑗𝜆𝜈𝑎𝑗!𝑛−𝜆_𝑘−𝑖×−𝑞^{𝑛−𝑗−𝜈}𝑥^𝑛𝑦𝑛!^𝑘.𝑘! (51) Similarly, by 1+𝑞𝑥^∞−1/𝑞_𝑛=𝑗𝑛𝑗−𝑞𝑥^𝑛=𝑛!_∞𝑙=0(−1)^𝑙𝑙_𝜈=0𝑙𝜈1𝑞^𝜈𝑞𝑥^𝑙×𝑙!_∞𝑛=𝑗𝑛𝑗−𝑞𝑥^𝑛=𝑛!_{∞𝑙𝑙=0𝜈=0}𝑙𝜈𝑞^𝑙−𝜈(−1)^𝑙𝑥^𝑙𝑙!^∞_𝑛=0𝑛𝑗−𝑞𝑥^𝑛=𝑛!_{∞𝑛𝑛=0𝜆𝜆=0𝜈=0}𝜆𝜈𝑞^𝜆−𝜈(−1)^𝜆×𝑗𝜆!𝑛−𝜆−𝑞^𝑛−𝜆()!𝑥𝑛−𝜆^𝑛=^∞_𝑛=0(−1)^𝑛𝑛_𝜆=0𝑛𝜆𝑗𝑛−𝜆^𝜆_𝜈=0𝜆𝜈𝑞^𝑛−𝜈𝑥^𝑛,𝑛! (52) we get 𝑐̂^∞∞_𝑛=0𝑘=0^(−𝑘)_{𝑛,𝑎,𝑞}𝑥^𝑛𝑦𝑛!^𝑘=𝑘!1+𝑞𝑥^{−𝑒𝑦/𝑞}𝑒^𝑎𝑦=exp𝑒^𝑦−1ln1+𝑞𝑥^−1/𝑞×1+𝑞𝑥^−1/𝑞𝑒^𝑎𝑦=^∞_𝑗=0𝑗!𝑞^𝑗𝑒^𝑦−1^𝑗𝑗!−ln(1+𝑞𝑥)^𝑗×𝑗!1+𝑞𝑥^−1/𝑞𝑒^𝑎𝑦=^∞_𝑗=0𝑗!𝑞^𝑗𝑒^∞𝑎𝑦_𝑘=𝑗𝑘𝑗𝑦^𝑘𝑘!1+𝑞𝑥^−1/𝑞×^∞_𝑛=𝑗𝑛𝑗−𝑞𝑥^𝑛=𝑛!_∞𝑗=0𝑗!𝑞^𝑗∞_𝑘=0^𝑘_𝑖=0𝑘𝑖𝑖𝑗𝑎^𝑘−𝑖𝑦^𝑘×𝑘!_∞𝑛=0(−1)^𝑛𝑛_𝜆=0𝑛𝜆𝑗𝑛−𝜆^𝜆_𝜈=0𝜆𝜈𝑞^𝑛−𝜈𝑥^𝑛=𝑛!_{∞∞𝑛=0𝑘𝑘=0𝑖𝑖=0𝑛𝑗=0𝜆𝜆=0𝜈=0}(−1)^𝑛×𝑘𝑖𝑖𝑗𝑛𝜆𝑗×𝜆𝜈𝑎𝑗!𝑛−𝜆_𝑘−𝑖𝑞^{𝑛−𝑗−𝜈}𝑥^𝑛𝑦𝑛!^𝑘.𝑘! (53)

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. Let 𝑘 be an integer and 𝑎 a positive real number. An explicit form of poly-Bernoulli number 𝐵_𝑛^(𝑘) is given by 𝐵_𝑛^(𝑘)=^𝑛_𝑚=0𝑛𝑚(−1)^𝑛−𝑚𝑚!(𝑚+1)^𝑘 (54) ([6], Theorem 1). In ([1], Theorem 8), one expression of 𝐵_𝑛^(𝑘) in terms of poly-Cauchy numbers 𝑐_𝑛^(𝑘) is given.

Proposition 19. One has 𝐵_𝑛^(𝑘)=^𝑛𝑛_𝑙=1𝑚=1𝑛𝑚𝑚!𝑐𝑚−1𝑙−1^𝑙(𝑘)(𝑛≥1). (55)

On the contrary, in ([9], Theorem 2.2), one expression of 𝑐_𝑛^(𝑘) in terms of 𝐵_𝑛^(𝑘) is given.

Proposition 20. One has 𝑐_𝑛^(𝑘)=^𝑛𝑛_𝑙=1𝑚=1(−1)^𝑛−𝑚𝑛𝑚𝑚𝑙𝐵𝑚!^𝑙(𝑘)(𝑛≥1). (56)

As a counterpart of a generalized poly-Cauchy number, we will define a generalized poly-Bernoulli number 𝐵^(𝑘)_{𝑛,𝑎,𝐿} by ℓ^𝑎−1Li_𝑘ℓ1−𝑒^−𝑡;𝑎−11−𝑒^−𝑡=^∞_𝑛=0𝐵^(𝑘)_{𝑛,𝑎,𝐿}𝑡^𝑛,𝑛! (57) where Li_𝑘(𝑧;𝑎) is the generalized polylogarithm function defined by Li_𝑘(𝑧;𝑎)=^∞_𝑚=0𝑧^𝑚(𝑚+𝑎)^𝑘, (58) so that Li_𝑘(𝑧;0)=Li_𝑘(𝑧).

Then, 𝐵^(𝑘)_{𝑛,𝑎,𝐿} can be expressed explicitly in terms of the Stirling numbers of the second kind. Note that 𝐵^(𝑘)_{𝑛,1,(1,…,1)}=𝐵_𝑛^(𝑘).

Proposition 21. One has 𝐵^(𝑘)_{𝑛,𝑎,𝐿}=^𝑛_𝑚=0𝑛𝑚(−1)^𝑛−𝑚𝑚!ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘(𝑛≥0). (59)

Proof. By 𝑒^𝑡−1^𝑚=𝑚!^∞_𝑛=𝑚𝑛𝑚𝑡^𝑛𝑛!, (60) we have ^∞_𝑛=0𝐵^(𝑘)_{𝑛,𝑎,𝐿}𝑡^𝑛=ℓ𝑛!ℓ(1−^𝑎𝑒^−𝑡)^∞_𝑚=1ℓ(1−𝑒^−𝑡)^𝑚(𝑚+𝑎−1)^𝑘=ℓ^𝑎∞_𝑚=0ℓ(1−𝑒^−𝑡)^𝑚(𝑚+𝑎)^𝑘=ℓ^𝑎∞_𝑚=0(−ℓ)^𝑚𝑚!(𝑚+𝑎)^𝑘∞_𝑛=𝑚𝑛𝑚(−𝑡)^𝑛=𝑛!_{∞𝑛=0𝑛𝑚=0}𝑛𝑚(−1)^𝑛−𝑚𝑚!ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘𝑡^𝑛.𝑛! (61) Comparing the coefficients on both sides, we get the result.

For simplicity, we write 𝑐^(𝑘)_{𝑛,𝑎,𝐿}=𝑐^(𝑘)_{𝑛,𝑎,1,𝐿} and 𝑐̂^(𝑘)_{𝑛,𝑎,𝐿}=𝑐̂^(𝑘)_{𝑛,𝑎,1,𝐿}. If 𝑎=ℓ=1, then our results below are reduced to those previous ones.

Theorem 22. For 𝑛≥0, one has 𝐵^(𝑘)_{𝑛,𝑎,𝐿}=^𝑛𝑛_𝑗=1𝑚=1𝑛𝑚𝑚!𝑐𝑚−1𝑗−1^{(𝑘)𝑗,𝑎,𝐿},𝑐^(𝑘)_{𝑛,𝑎,𝐿}=^𝑛𝑛_𝑗=1𝑚=1(−1)^𝑛−𝑚𝑛𝑚𝑚𝑗𝐵𝑚!^{(𝑘)𝑗,𝑎,𝐿}. (62)

Proof. For the first identity, RHS=^𝑛𝑛_{𝑗=1𝑚=𝑗}𝑛𝑚×𝑚!𝑚−1𝑗−1_𝑗𝑖=0𝑗𝑖(−1)^𝑗−𝑖ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘=^𝑛_𝑖=1(−1)^𝑖ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘×^𝑛𝑛_{𝑗=𝑖𝑚=𝑗}𝑛𝑚𝑚!()𝑚−1𝑗−1−1^𝑗𝑗𝑖=^𝑛_𝑖=1(−1)^𝑖ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘𝑛_𝑚=𝑖𝑛𝑚×𝑚!_{𝑚𝑗=𝑖}(−1)^𝑗𝑗𝑖=𝑚−1𝑗−1_𝑛𝑖=1(−1)^𝑖ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘𝑛_𝑚=𝑖𝑛𝑚𝑚!(−1)^𝑚=𝑚−1𝑖−1_𝑛𝑖=1(−1)^𝑖ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘(−1)^𝑛𝑛𝑖𝑖!=LHS. (63) For the second identity, RHS=^𝑛𝑛_𝑗=1𝑚=1(−1)^𝑛−𝑚𝑛𝑚𝑚𝑗×𝑚!_𝑗𝑖=0𝑗𝑖(−1)^𝑗−𝑖𝑖!ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘=^𝑛_𝑚=1(−1)^𝑛−𝑚𝑛𝑚×𝑚!_𝑛𝑗=0𝑚𝑗^𝑗𝑖=0𝑗𝑖(−1)^𝑗−𝑖𝑖!ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘=^𝑛_𝑚=1(−1)^𝑛−𝑚𝑛𝑚×𝑚!_𝑛𝑖=0(−1)^𝑖𝑖!ℓ^𝑖+𝑎(𝑖+𝑎)^𝑘𝑛_𝑗=𝑖(−1)^𝑗𝑚𝑗𝑗𝑖=^𝑛_𝑚=0(−1)^𝑛−𝑚𝑛𝑚×()𝑚!−1^𝑚𝑚!ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘(−1)^𝑚=^𝑛_𝑚=0𝑛𝑚(−1)^𝑛−𝑚ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘=LHS. (64) Note that [^𝑚₀]=0   (𝑚≥1) and ^𝑚_𝑗=0 (𝑗>𝑚), and ^𝑚_𝑗=𝑖(−1)^𝑚−𝑗𝑚𝑗𝑗𝑖=1();0().𝑖=𝑚𝑖≠𝑚 (65)

Similarly, concerning 𝑐̂^(𝑘)_{𝑛,𝑎,𝐿}=(−1)^𝑛𝑛_𝑚=0𝑛𝑚ℓ^𝑚+𝑎(𝑚+𝑎)^𝑘(𝑛≥0) (66) as a generalization of poly-Cauchy numbers of the second kind 𝑐̂_𝑛^(𝑘), we have the following.

Theorem 23. One has 𝐵^(𝑘)_{𝑛,𝑎,𝐿}=(−1)^𝑛𝑛𝑛_𝑗=1𝑚=1𝑛𝑚𝑚𝑗𝑚!𝑐̂^(𝑘)_{𝑗,𝑎,𝐿},𝑐̂^(𝑘)_{𝑛,𝑎,𝐿}=(−1)^𝑛𝑛𝑛_𝑗=1𝑚=11𝑛𝑚𝑚𝑗𝐵𝑚!^{(𝑘)𝑗,𝑎,𝐿}. (67)

Remark 24. If 𝑎=ℓ=1, these results are reduced to the identities in Theorems 3.2 and 3.1 in [9], respectively.

Acknowledgments

This work was partly done when the first author visited Eszterházy Károly College in September 2012 and Kasetsart University in October 2012. He thanks both institute and the college for their hospitality. This work was completed when the third author visited Hirosaki University in November 2012. This work was supported in part by the Grant-in-Aid for Scientific research (C) (no. 22540005), the Japan Society for the Promotion of Science.