On multi-poly-Bernoulli–Carlitz numbers

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On multi-poly-Bernoulli–Carlitz numbers Ryotaro Harada Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan

a r t i c l e

i n f o

Article history: Received 4 April 2018 Received in revised form 25 October 2018 Accepted 18 November 2018 Available online xxxx Communicated by F. Pellarin Keywords: Multiple zeta values Bernoulli numbers Function fields

a b s t r a c t We introduce multi-poly-Bernoulli–Carlitz numbers, function field analogues of multi-poly-Bernoulli numbers of Imatomi– Kaneko–Takeda. We explicitly describe multi-poly-Bernoulli Carlitz numbers in terms of the Carlitz factorial and the Stirling–Carlitz numbers of the second kind and also show their relationships with function field analogues of finite multiple zeta values. © 2018 Elsevier Inc. All rights reserved.

Contents 0. 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Definition of finite multiple zeta values . . . . . . . . . . 1.3. Definition of (finite Carlitz) multiple polylogarithms . 2. Multi-poly-Bernoulli(-Carlitz) numbers . . . . . . . . . . . . . . . 2.1. Characteristic 0 case . . . . . . . . . . . . . . . . . . . . . . . 2.2. Characteristic p case . . . . . . . . . . . . . . . . . . . . . . . 3. Several properties of multi-poly-Bernoulli(-Carlitz) numbers 3.1. Characteristic 0 case . . . . . . . . . . . . . . . . . . . . . . . 3.2. Characteristic p case . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

E-mail address: [email protected]. https://doi.org/10.1016/j.jnt.2018.11.003 0022-314X/© 2018 Elsevier Inc. All rights reserved.

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0. Introduction In this paper, we introduce and study function field analogues of the Bernoulli numbers. In 1997, M. Kaneko introduced and investigated generalizations of the Bernoulli numbers, poly-Bernoulli numbers in [15]. He obtained explicit formulae for poly-Bernoulli numbers which includes the Stirling numbers of the second kind. Moreover, He and T. Arakawa found that they are also related to the Arakawa–Kaneko zeta functions at non-positive integers in [2]. From 2000, several multi-poly-Bernoulli numbers, generalizations of poly-Bernoulli numbers, were posted by Hamahata–Masubuchi [13], Imatomi– Kaneko–Takeda [14] and M.-S. Kim-T. Kim [18] in different ways each other. In [14], K. Imatomi, M. Kaneko and E. Takeda established relationships between multi-polyBernoulli numbers and finite multiple zeta values by obtaining some fundamental formulae. In 1935, L. Carlitz [3] introduced and investigated function field analogues of the Bernoulli numbers, the Bernoulli–Carlitz numbers BCn . By using them, he obtained m an analogue of Euler’s famous formula ζ(m) = − (2πi) 2(m!) Bm (for even m) in [3] and the von Staudt–Clausen theorem in [4,5]. The latter result was revisited and put in a more conceptual setting by D. Goss in [12].1 E. Gekeler proved several identities for the Bernoulli–Carlitz numbers in [11]. Furthermore, H. Kaneko and T. Komatsu obtained explicit formulae for them by using function field analogues of the Stirling numbers in [16]. In this paper, we introduce in §2.2 multi-poly-Bernoulli–Carlitz numbers as function field analogues of multi-poly-Bernoulli numbers and also discuss generalizations of the vanishing condition BCn = 0 (q − 1  n) shown in [4] and explicit formulae BCn = ∞ (−1)j Dj  n  j=0 q j −1 C shown in [16]. In §3.2 we show that multi-poly-Bernoulli–Carlitz L2j numbers with special indices are expressed by Bernoulli–Carlitz numbers. We also show their connection to finite multiple zeta values in function field which were introduced by C.-Y. Chang and Y. Mishiba [8] as finite variants of Thakur’s multiple zeta values in [19]. 1. Notations and definitions 1.1. Notations We recall the following notation. q Fq θ, t 1

a power of a prime number p. a finite field with q elements. independent variables.

An analogue of von Staudt–Clausen theorem stated in [4,5,12] was corrected by L. Carlitz [6] for q = 2.

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A A+ k k∞ Di Li Γn+1 Π(n)

3

the polynomial ring Fq [θ]. the set of monic polynomials in A. the rational function field Fq (θ). Fq ((1/θ)), the completion of k at ∞. i−1 qi qj j=0 (θ − θ ) ∈ A+ where D0 := 1. i qj where L0 := 1. j=1 (θ − θ ) ∈ A+   the Carlitz gamma, i Dini (n = i ni q i ∈ Z≥0 (0 ≤ ni ≤ q − 1)). the Carlitz factorial, Γn+1

1.2. Definition of finite multiple zeta values In this subsection, we recall the definition of finite multiple zeta values and its function field analogues which were introduced in [8]. 1.2.1. Characteristic 0 case We begin this subsection by recalling the finite multiple zeta values those were introduced by M. Kaneko and D. Zagier in [17]. Definition 1 ([17]). We set a Q-algebra as follows:     A := Z pZ Z pZ p

p

where p runs over all prime numbers. For s := (s1 , . . . , sr ) ∈ Zr , the finite multiple zeta values are defined as follows: ζA (s) := (ζA (s)(p) ) ∈ A where ζA (s)(p) :=

p>m1 >···>mr

 1 s1 sr ∈ Z pZ. m · · · m r 1 >0

1.2.2. Characteristic p case Next, let us turn into function field situation. In 1935, L. Carlitz [3] considered an analogue of the Riemann zeta values in function field which we call the Carlitz zeta values. For s ∈ N, they are defined by ζA (s) :=

1 ∈ k∞ . as

a∈A+

D.S. Thakur [19] generalized this definition to that of multiple zeta values in A, which are defined for s = (s1 , . . . , sr ) ∈ Nr ,

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ζA (s) :=

as 1 deg a1 >···>deg ar ≥0 1 a1 ,...,ar ∈A+

1 ∈ k∞ . · · · asrr

Also, Chang–Mishiba and D.S. Thakur concerned v-adic variant ([9], [10] [19]) and finite variant ([8], [20]). In this paper, we consider Chang and Mishiba’s finite variant ([8]). Definition 2 ([8], (2.1)). We set a k-algebra as follows: Ak :=

    A ℘A A ℘A ℘

℘

where ℘ runs over all monic irreducible polynomials in A. For s := (s1 , . . . , sr ) ∈ Nr and a monic irreducible polynomial ℘ ∈ A, finite multiple zeta values are defined as follows: ζAk (s) := (ζAk (s)℘ ) ∈ Ak where ζAk (s)℘ :=

as 1 deg ℘>deg a1 >···>deg ar ≥0 1 a1 ,...,ar ∈A+

 1 sr ∈ A ℘A. · · · ar

1.3. Definition of (finite Carlitz) multiple polylogarithms In this subsection, we recall the definition of multiple polylogarithms in characteristic 0 and p. 1.3.1. Characteristic 0 case Definition 3. For s := (s1 , . . . , sr ) ∈ Zr , the multiple polylogarithm series Lis (z1 , . . . , zr ) are defined as the following series of r-variables z1 , . . . , zr :

Lis (z1 , . . . , zr ) :=

m1 >···>mr

z1m1 · · · zrmr ∈ Q[[z1 , . . . , zr ]]. ms11 · · · msrr >0

1.3.2. Characteristic p case In 2014, C.-Y. Chang [7] introduced the Carlitz multiple polylogarithms as function field analogues of the multiple polylogarithms. Definition 4 ([7], Definition 5.1.1). For s = (s1 , . . . , sr ) ∈ Nr , the Carlitz multiple polylogarithms are defined as the following series of r-variables z1 , . . . , zr : Lis (z1 , . . . , zr ) :=

i1 >···>ir ≥0

i1

ir

z1q · · · zrq ∈ k[[z1 , . . . , zr ]]. Lsi11 · · · Lsirr

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Remark 5. We recover the Carlitz logarithms in the case of r = 1 and s1 = 1 logC (z) :=

z qi i≥0

Li

∈ k[[z]].

In [8], C.-Y. Chang and Y. Mishiba introduced finite Carlitz multiple polylogarithms, a finite variant of the Carlitz multiple polylogarithms. Definition 6 ([8], (3.1)). For s = (s1 , . . . , sr ) ∈ Nr and r-tuple of variables z = (z1 , . . . , zr ), finite Carlitz multiple polylogarithms are defined as follows: LiAk ,s (z) := (LiAk ,s (z1 , . . . , zr )℘ ) ∈ Ak,z where

LiAk ,s (z1 , . . . , zr )℘ :=

deg ℘>i1 >···>ir ≥0

i1

ir

z1q · · · zrq mod ℘ ∈ A[z1 , . . . , zr ]/℘A. Lsi11 · · · Lsirr

Here Ak,z is the following quotient ring Ak,z :=



 A[z]/℘A[z] A[z]/℘A[z]

℘

℘

(we put A[z] := A[z1 , . . . , zr ]). Remark 7. In the above definition, we remark that ℘ does not divide Li for i < deg ℘. In [8], they established an explicit formula expressing ζAk (s) as a k-linear combination of LiAk ,s (z1 , . . . , zr )℘ evaluated at some integral points. Before we recall it, let us prepare the Anderson–Thakur polynomial. Definition 8 ([1], (3.7.1)). Let θ, t, x be independent variables. For n ∈ Z≥0 , Anderson– Thakur polynomial Hn ∈ A[t] is defined by  1−

∞

i=0

i j=1

qi j t − θq

Di |θ=t

x

qi

−1 =

∞

Hn xn . Γ | n=0 n+1 θ=t

Remark 9. We note that Hn = 1 for 0 ≤ n ≤ q − 1. mi Notation 10. For r-tuple s = (s1 , . . . , sr ) ∈ Nr , let Hsi −1 = j=0 uij tj i ≤ r) and then, we set following symbols which are introduced in [8]: Js := {0, 1, . . . , m1 } × · · · × {0, 1, . . . , mr }.

(uij ∈ A, 1 ≤

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For each j = (j1 , . . . , jr ) ∈ Js , we set uj := (u1j1 , . . . , urjr ) ∈ Ar , and aj := aj (t) := tj1 +···+jr . Example 11. We note that when s = (s1 , . . . , sr ) = (1, . . . , 1), by Remark 9, we have Js = {0} × · · · × {0} and uj = (1, . . . , 1) for j ∈ Js . The following equation was obtained by C.-Y. Chang and Y. Mishiba in [8]. Proposition 12 ([8], p. 1056). For s = (s1 , . . . , sr ) ∈ Nr , let ℘ ∈ A be a monic irreducible polynomial which satisfy ℘  Γs1 · · · Γsr . Then we have ζAk (s)℘ =

1 aj (θ)LiAk ,s (uj )℘ . Γs1 · · · Γsr j∈Js

2. Multi-poly-Bernoulli(-Carlitz) numbers In this section, we define multi-poly-Bernoulli–Carlitz numbers which are function field analogues of multi-poly-Bernnoulli numbers. 2.1. Characteristic 0 case The Bernoulli numbers Bn (n = 0, 1, . . .) are rational numbers defined by the following generating function ∞

Bn

n=0

zez zn := z . n! e −1

(1)

It is known that we have the following equation Bn = 0 (for n ≥ 3 so that 2  n). Moreover, we know that the Bernoulli numbers are expressed as follows:   (−1)m−1 (m − 1)! n Bn = (−1) , m m−1 m=1 n

n+1

(2)

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where

n m

7

∈ Z are the Stirling numbers of the second kind defined by ∞   n

(ez − 1)m n z = . m! n! m n=0

(3)

In 2014, K. Imatomi, M. Kaneko and E. Takeda [14] concerned two types of the multi-poly-Bernoulli numbers which generalize the Bernoulli numbers. Definition 13 ([14], (1) and [2], (8)). For s := (s1 , . . . , sr ) ∈ Zr , the multi-poly-Bernoulli numbers (MPBNs for short) of B-type, C-type are the rational numbers which are defined by following generating functions respectively r−1

   Lis (1 − e , 1, . . . , 1) sz := Bn , n! 1 − e−z n=0 ∞

−z

n

r−1

   Lis (1 − e , 1, . . . , 1) sz := . Cn n! ez − 1 n=0 ∞

−z

n

Remark 14. When r = 1, Bns and Cns are the poly-Bernoulli numbers of B-type, C-type (1) (cf. [2,15]). When r = 1 and s1 = 1, Lis (z1 , . . . , zr ) = − log(1 − z) and then Bn agrees (1) (1) with (1) of the Bernoulli numbers. We note that B1 = 1/2 and C1 = −1/2 and (1) (1) Bn = Cn = Bn for n = 1. 2.2. Characteristic p case In 1935, L. Carlitz [3] introduced the Bernoulli–Carlitz numbers, function field analogues of the Bernoulli numbers by using the Carlitz factorials Π(n) and the Carlitz exponentials

eC (z) :=

z qi i≥0

Di

as follows. Definition 15 ([3]). For n ∈ Z≥0 , the Bernoulli–Carlitz numbers BCn are the elements of k defined by ∞

n=0

BCn

z zn := . Π(n) eC (z)

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In [4], L. Carlitz obtained the following: BCn = 0

for (q − 1)  n.

In 2016, H. Kaneko and T. Komatsu [16] introduced the Stirling–Carlitz numbers of the first and second kind as an analogue of the Stirling numbers which were introduced in (3). We recall below those of the second kind. Definition 16 ([16], (15)). For m ∈ Z≥0 , the Stirling–Carlitz numbers of the second kind n m C ∈ k are defined by ∞  

n n=0

(eC (z))m zn := . Π(m) m C Π(n)

In addition, they [16] showed that   n = 0 (n ≥ 1), 0 C

  n = 0 (n < m), m C

  n = 1 (n ≥ 0) n C

(4)

and the following property. Proposition 17 ([16], Proposition 8). For n, m ∈ Z>0 with λ(n) > λ(m),   n =0 m C here we noted λ(n) :=

 i

ni where ni are the digits of q-adic expansion n =

 i

ni q i .

By using the Stirling–Carlitz numbers of the second kind, they obtained the following proposition as a function field analogue of (2). Proposition 18 ([16], Theorem 2). For n ∈ Z≥0 , we have BCn =

 ∞

(−1)j Dj j=0



n qj − 1

L2j

. C

j i Remark 19. In [16], they put Lj by i=1 (θq − θ). But the above equation is same to their equation (cf. [16], (20)) due to the appearance of L2j . Remark 20. By the definition of Dm , we have q Dm =

m−1  i=0

m

i

(θq − θq )q =

m−1  i=0

(θq

m+1

− θq

i+1

)=

m 

(θq

i =1

m+1

i

− θq ) = −

Dm+1 . (θ − θqm+1 )

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Thus we obtain q−1 Dm =−

Dm+1 . Dm (θ − θqm+1 )

By the definition of Carlitz factorial, Lj and the above equation, we have the following: Π(q j − 1) =

j−1 

q−1 Dm =

m=0

j−1  m=0

−

Dm+1 Dj = (−1)j m+1 q Lj Dm (θ − θ )

(j ∈ Z≥0 ).

(5)

Thus we may write the formula in Theorem 18 as follows: BCn =

  ∞

n Π(q j − 1) . Lj qj − 1 C j=0

Next we introduce multi-poly-Bernoulli–Carlitz numbers (MPBCNs) as function field analogues of MPBNs (Definition 13). It is defined by the following generating function. Definition 21. For s = (s1 , . . . , sr ) ∈ Nr , j = (j1 , . . . , jr ) ∈ Js (for Js , see Notation 10), we define multi-poly-Bernoulli–Carlitz numbers (MPBCNs for short) BCns,j to be elements of k as follows:

BCns,j

n≥0

Lis (eC (z)u1j1 , u2j2 , . . . , urjr ) zn := . Π(n) eC (z)

(6)

Remark 22. In the case when r = 1 and s1 = 1 in the above definition, we have Lis (z1 , . . . , zr ) = logC (z) and Js = {0}, u1j1 = u10 = 1 since Hs1 −1 = H0 = 1. Hence we recover the Definition 15 by

n≥0

BCn(1),(0)

logC (eC (z)) z zn = = . Π(n) eC (z) eC (z)

This is the one we have seen in Definition 15 so we have BCn(1),(0) = BCn .

(7)

Remark 23. Let g be a generator of F× q then we have g n = 1 ⇔ (q − 1)|n.

(8)

By the definition, it follows that eC (gz) = geC (z). Then by (6) and Definition 4, we have

n≥0

BCns,j

Lis (eC (gz)u1j1 , u2j2 , . . . , urjr ) (gz)n = = Π(n) eC (gz)

i1 >···>ir ≥0

i1

eC (gz)q

i1 −1

ir

uq1j1 · · · uqrjr Lsi11 · · · Lsirr

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by using eC (gz) = geC (z) and (8),

=

i1



eC (z)

q i1 −1

i1 >···>ir ≥0

=



BCns,j

n≥0

ir

uq1j1 · · · uqrjr Lis (eC (z)u1j1 , u2j2 , . . . , urjr ) s1 sr = Li1 · · · Lir eC (z)

zn . Π(n)

By comparing the coefficients of z n , we have g n BCns,j = BCns,j . Therefore we obtain the following by (8): BCns,j = 0 for (q − 1)  n. The MPBNs are defined for si ∈ Z, on the other hand our MPBCNs are defined for si ∈ N. It is because in Definition 21, we use uiji , the coefficients of Hsi −1 which are defined for si ∈ N. We remark that we do not have two kinds of MPBCNs as we do in Definition 13. 3. Several properties of multi-poly-Bernoulli(-Carlitz) numbers In this section, we obtain function field analogues of some results in [14]. In subsection 3.1, we recall their results in characteristic 0 case. In subsection 3.2, we prove their analogue in characteristic p case. 3.1. Characteristic 0 case In [14], K. Imatomi, M. Kaneko, and E. Takeda obtained explicit formulae for MPBNs. They are the following finite sums involving the Stirling numbers of the second kind. Proposition 24 ([14], Theorem 3). For s = (s1 , . . . , sr ) ∈ Zr and n ≥ 0, we have Bns

n

= (−1)



m1 −1

(−1)

n+1≥m1 >m2 >···>mr >0

  1 n (m1 − 1)! s1 m1 − 1 m1 · · · msrr

and Cns

n

= (−1)

n+1≥m1 >m2 >···>mr >0

m1 −1

(−1)

  1 n+1 (m1 − 1)! . m1 ms11 · · · msrr

In [14], they derived the following relations between the MPBNs and the Bernoulli numbers for the special case (s1 , . . . , sr ) = (1, . . . , 1).

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Proposition 25 ([14], Proposition 4). For r ≥ 1 and n ≥ r − 1, we have    Bn(1, . . . , 1) = r

   r

Cn(1, . . . , 1)

  1 n+1 (1) Bn−r+1 , n+1 r

  n+1 1 (1) Cn−r+1 . = n+1 r

In [14], they obtained the following relations which connect the MPBNs and finite multiple zeta values. Proposition 26 ([14], Theorem 8). For s = (s1 , . . . , sr ) ∈ Zr , we have (s −1,s2 ,...,sr )

1 ζA (s)(p) = −Cp−2

mod p

and for d ≥ 0, (s −1,s2 ,...,sr )

1 ζA (1, . . . , 1, s1 , . . . , sr )(p) = −Cp−d−2   

mod p.

d

Here we note that the second relation generalizes the first relation. 3.2. Characteristic p case We prove function field analogues of Proposition 24-26. The following theorem is a function field analogue of Proposition 24. Theorem 27. For r ∈ N, s = (s1 , . . . , sr ) ∈ Nr , j = (j1 , . . . , jr ) ∈ Js and n ∈ Z≥0 , BCns,j



=

ir  qi1  u1j1 · · · uqrjr n Π(q − 1) i1 . q − 1 C Lsi11 · · · Lsirr

i1

logq (n+1)≥i1 >···>ir ≥0

Proof. By Definition 4, the right hand side of (6) is translated as follows. Lis (eC (z)u1j1 , u2j2 , . . . , urjr ) = eC (z)



i1

eC (z)

i1 >···>ir ≥0

q i1 −1

by Definition 16 for m = q i1 − 1,

=





i1 >···>ir ≥0 n≥0

 Π(q − 1) i1



n i 1 q −1

i1

ir

uq1j1 · · · uqrjr Lsi11 · · · Lsirr

ir

q q z n u1j1 · · · urjr s1 sr C Π(n) Li1 · · · Lir

(9)

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=



n q i1 − 1

i1

n≥0 i1 >···>ir ≥0

=







 Π(q − 1)



n≥0 logq (n+1)≥i1 >···>ir ≥0

i1

ir

uq1j1 · · · uqrjr z n s1 sr C Li1 · · · Lir Π(n)

ir  qi1  u1j1 · · · uqrjr z n n . Π(q i1 − 1) i1 q − 1 C Lsi11 · · · Lsirr Π(n)

Then by Definition 21, we have

BCns,j

n≥0



zn = Π(n)

ir  qi1  u1j1 · · · uqrjr z n n . Π(q − 1) i1 q − 1 C Lsi11 · · · Lsirr Π(n)

i1

n≥0 logq (n+1)≥i1 >···>ir ≥0

By comparing the coefficients of z n , we obtain the formula (9).

2

From (4) and (9), we easily deduce that BCns,j = 0 if n < q r−1 − 1. We note that for x, y ∈ N, all digits of the q-adic expansion of q x − 1 and q y − 1 are q − 1. Therefore we have   x  0 if x = y, q −1 = (10) y q −1 C 1 if x = y, by Proposition 17 and (4). Thus we have the following from Theorem 27. Corollary 28. For m ∈ N, we have BCqs,j m −1

=

m



Π(q

m

m>i2 >···>ir ≥0

i2

ir

uq1j uq2j · · · uqrjr − 1) s11 s22 . Lm Li2 · · · Lsirr

(11)

Remark 29. When r = 1 and s1 = 1, we have Hs1 −1 = H0 = 1. Then Js = {0}, u1j1 = u10 = 1 hence we have  

1 n BCn(1),(0) = Π(q i1 − 1) i1 q − 1 C Li1 logq (n+1)≥i1 ≥0

by using (5), =



(−1)i1

logq (n+1)≥i1 ≥0

  n Di1 . L2i1 q i1 − 1 C

Therefore by Remark 22 our Theorem 27 includes H. Kaneko and T. Komatsu’s result (Proposition 18) in the case of r = 1 and s1 = 1. We obtain the following relation between the MPBCNs and the Bernoulli–Carlitz numbers for the tuple (1, . . . , 1) as a function field analogue of Proposition 25.

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Theorem 30. For r ≥ 1 and n ≥ q r−1 − 1, we have       BC (1, . . . , 1),(0, . . . , 0) r

r

n



=





n q i1 − 1

logq (n+1)≥i1 >···>ir ≥0

BCqi1 −1 C

BCqi2 −1 BCqir −1 ··· . Π(q i2 − 1) Π(q ir − 1)

(12)

Proof. Let us first prove an equation BCqi −1 1 = i Π(q − 1) Li

(13)

for each i ≥ 0. It follows from Proposition 18 that we have BCqi −1 =

  ∞

(−1)j Dj q i − 1 qj − 1

L2j

j=0

.

C

The right hand side is translated as follows:   ∞

(−1)j Dj q i − 1 j=0

L2j

−1

qj

=

C

(−1)i Di Π(q i − 1) = . 2 Li Li

The first equality follows from Proposition 17, the second one follows from (5). Then we have the equation (13). It follows from Theorem 27 that we have       (1, . . . , 1),(0, . . . , 0) BCn = r

r



logq (n+1)≥i1 >···>ir ≥0



n q i1 − 1

Π(q i1 − 1) . C Li1 · · · Lir

By using the equation (13) to the right hand side,       BC (1, . . . , 1),(0, . . . , 0) r

r

n

=

logq (n+1)≥i1 >···>ir ≥0





n q i1 − 1

Π(q i1 − 1) C

BCqi1 −1 BCqi2 −1 BCqir −1 ··· . Π(q i1 − 1) Π(q i2 − 1) Π(q ir − 1)

Therefore we obtain the desired equation (12). 2 Next, before we see a function field analogue of Proposition 26, we prepare the following lemma.

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Lemma 31. When r ≥ 2, we have the following equation for s = (s1 , . . . , sr ) ∈ Nr , j ∈ Js and m ≥ r − 1. BCqs,j m −1

(s ),(j ) BCqm1−1 1

=

m−1

1 (s2 ,...,sr ),(j2 ,...,jr ) BCqi −1 . i − 1) Π(q i=r−2

(14)

Proof. By using Theorem 27, we have BCqs,j m −1



=

m≥i1 >···>ir ≥0

i1 ir  m  uq1j1 · · · uqrjr q −1 i1 Π(q − 1) s1 . Li1 · · · Lsirr q i1 − 1 C

Then by using (10), we have BCqs,j m −1

m



=

Π(q

m

m>i2 >···>ir ≥0 m

= Π(q

m

uq1j − 1) s11 Lm

i2

ir

uq1j uq2j · · · uqrjr − 1) s11 s22 Lm Li2 · · · Lsirr i2

m>i2 >···>ir ≥0

ir

uq2j2 · · · uqrjr . Lsi22 · · · Lsirr

By using Theorem 27, we have BCqs,j m −1

=



(s ),(j ) BCqm1−1 1

m>i2 >···>ir ≥0

=

(s ),(j ) BCqm1−1 1

i2

ir

uq2j2 · · · uqrjr Lsi22 · · · Lsirr

m−1

1 i Π(q − 1) i=r−2

i>i3 >···>ir ≥0

(15) i

i3

ir

uq2j uq3j · · · uqrjr Π(q − 1) s22 s33 Li Li3 · · · Lsirr i

again by using Theorem 27, (s ),(j )

= BCqm1−1 1

m−1

1 (s2 ,...,sr ),(j2 ,...,jr ) BCqi −1 . i − 1) Π(q i=r−2

Then we obtain the desired equation (14). 2 The following result is an analogue of Proposition 26 which provides the connection between MPBCNs and finite multiple zeta values in the function field case. Theorem 32. For s = (s1 , . . . , sr ) ∈ Nr and a monic irreducible polynomial ℘ ∈ A so that ℘  Γs1 · · · Γsr , we have the following: ζAk (s)℘ =

s,j deg ℘−1



1 1 BCqi −1 aj (θ) Γs1 · · · Γsr Li BCqi −1 i=r−1 j∈Js

mod ℘.

(16)

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For s = (1, . . . , 1, s1 , . . . , sr ) ∈ Nr+d (d ≥ 0) and a monic irreducible polynomial ℘ ∈ A    d

so that ℘  Γs1 · · · Γsr , we have the following:

1 ζAk (s)℘ = aj (θ) Γs1 · · · Γsr  j ∈Js



deg ℘>i0 >···>id ≥r−1



BCqsid,j−1 1 Li0 · · · Lid BCqid −1

mod ℘. (17)

Here we put s = (s1 , . . . , sr ). We remark that both sides of the equation (16) become 0 for deg ℘ < r (resp. (17), deg ℘ < d + r). Proof. We first prove that the equation (16). From Definition 4, the equation (11) and (13) we have i1



LiAk ,s (uj )℘ =

deg ℘>i1 >···>ir ≥0

=

deg ℘−1

i=r−1

=

deg ℘−1

i=r−1

=

deg ℘−1

i=r−1

ir

uq1j1 · · · uqrjr Lsi11 · · · Lsirr

1 i Π(q − 1)

i1



i2

ir

uq1j uq2j · · · uqrjr Π(q − 1) s11 s22 Li1 Li2 · · · Lsirr i

i>i2 >···>ir ≥0

  1 ,j BCqsi −1 − 1)

Π(q i





s ,j 1 BCqi −1 . Li BCqi −1

By our assumption ℘  Γs1 · · · Γsr we may apply Proposition 12 and obtain the desired formula (16). Next we prove the equation (17). By using (16) for s = (1, . . . , 1, s1 , . . . , sr ), the ℘-part    d

of ζ(s)℘ is computed as

ζ(s)℘ =

s,j deg ℘−1



1 1 BCqi −1 a (θ) j Li BCqi −1 Γd1 Γs1 · · · Γsr j∈J i=d+r−1 s

=

deg ℘−1



BCqs,j i0 −1 1 . a (θ) j Π(q i0 − 1) Γd1 Γs1 · · · Γsr j∈J i =d+r−1 s

0

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Then by using Lemma 31, BCqs,j i0 −1 Π(q i0 − 1)

BCqi0 −1

Π(q i0 − 1)

Π(q i0 − 1)

id−2 −1



···

id−1 =r

1 = Li0

(1,...,1,s1 ,...,sr ),(0,...,0,j1 ,...,jr )

Π(q i1 − 1)

i

1 −1

(1),(0)

i1 =d+r−2

Π(q i1 − 1)

(1),(0)

BCqi2 −1

i2 =d+r−3

Π(q i2 − 1)

,...,sr ),(j1 ,...,jr )

BCq(sid1−1

Π(q id−1 − 1) i

i1 =d+r−2

Π(q i2 − 1)

id−1 −1

BCqid−1 −1

d =r−1

1 Li1

(1,...,1,s1 ,...,sr ),(0,...,0,j1 ,...,jr )

BCqi2 −1

i2 =d+r−3

BCqi1 −1

i

1 −1 i2 =d+r−3

Π(q id − 1) id−2 −1

1 1 ··· Li2 L i =r id−1 d−1

id−1 −1



id

(s ,...,s ),(j ,...,jr )

1 r 1 1 BCqid −1 L BCqid −1 =r−1 id

(s ,...,s ),(j ,...,jr )

i0 >i1 >···>id il ≥d+r−l−1 for each l

BCqid1−1 r 1 1 Li0 Li1 · · · Lid BCqid −1

(s ,...,s ),(j ,...,jr )



=

i

1 −1

(1),(0)

i

0 −1

i

0 −1

=

Π(q i1 − 1) BCqi1 −1

i1 =d+r−2

BCqi0 −1

Π(q i0 − 1)

is computed as

BCqi1 −1

i

0 −1

BCqi0 −1

(1),(0)

=

−1

i1 =d+r−2

(1),(0)

=

q

i

0 −1

(1),(0)

=

BC s,j i0

Π(q i0 −1)

i0 >i1 >···>id ≥r−1

BCqid1−1 r 1 1 Li0 Li1 · · · Lid BCqid −1

.

Thus by Γ1 = 1, we have

1 aj (θ) Γs1 · · · Γsr

(s ,...,s ),(j ,...,jr )



deg ℘>i0 >···>id ≥r−1

j∈Js

BCqid1−1 r 1 1 Li0 · · · Lid BCqid −1

= ζAk (s)℘ mod ℘.

For s = (1, . . . , 1, s1 , . . . , sr ), we have Js = {0} × · · · × {0} × {0, 1, . . . , m1 } × · · · × {0, 1, . . . , mr } so aj (θ) = θj1 +···+jr for j = (0, . . . , 0, j1 , . . . , jr ) ∈ Js and thus aj (θ) depends only on j = (j1 , . . . jr ) ∈ Js . Therefore the above equation is rewritten as follows:

1 aj (θ) Γs1 · · · Γsr  j ∈Js

deg ℘>i0 >···>id ≥r−1





BCqsid,j−1 1 = ζAk (s)℘ mod ℘. Li0 · · · Lid BCqid −1

Thus we obtain the equation (17). 2 We remark that the relation (17) is a generalization of (16).

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17

Acknowledgments The author is deeply grateful to Professor H. Furusho for guiding him towards this topic. This paper could not have been written without his continuous encouragements. He gratefully acknowledges Professor Y. Mishiba for indicating him towards generalizations of Remark 23 and the relation (17) which improved this paper. He would also like to thank NCTS for their kind support during his stay at NTHU and Daiko Foundation for financial support. References [1] G.W. Anderson, D.S. Thakur, Tensor powers of Carlitz module and zeta values, Ann. of Math. 132 (1) (1990) 159–191. [2] T. Arakawa, M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999) 189–209. [3] L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935) 137–168. [4] L. Carlitz, An analogue of the von Staudt–Clausen theorem, Duke Math. J. 3 (3) (1937) 503–517. [5] L. Carlitz, An analogue of the Staudt–Clausen theorem, Duke Math. J. 7 (1940) 62–67. [6] L. Carlitz, An analogue of the Bernoulli polynomials, Duke Math. J. 8 (1941) 405–412. [7] C.-Y. Chang, Linear indepandence of monomials of multizeta values in positive characteristic, Compos. Math. 150 (11) (2014) 1789–1808. [8] C.-Y. Chang, Y. Mishiba, On finite multiple Carlitz polylogarithms, J. Théor. Nombres Bordeaux 29 (3) (2017) 1049–1058. [9] C.-Y. Chang, Y. Mishiba, On multiple polylogarithms in characteristic p: v-adic vanishing versus ∞-adic Eulerianness, Int. Math. Res. Not., in press, https://doi.org/10.1093/imrn/rnx151. [10] C.-Y. Chang, Y. Mishiba, Logarithmic interpretation of multiple zeta values in positive characteristic, arXiv:1710.10849. [11] E.-U. Gekeler, Some new identities for Bernoulli–Carlitz numbers, J. Number Theory 33 (2) (1989) 209–219. [12] D. Goss, Von Staudt for Fq [T ], Duke Math. J. 45 (4) (1978) 885–910. [13] Y. Hamahata, H. Masubuchi, Special multi-poly-Bernoulli numbers, J. Integer Seq. 10 (2007). [14] K. Imatomi, M. Kaneko, E. Takeda, Multi-poly-Bernoulli numbers and finite multiple zeta values, J. Integer Seq. 17 (2014). [15] M. Kaneko, Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux 9 (1) (1997) 199–206. [16] H. Kaneko, T. Komatsu, Cauchy–Carlitz numbers, J. Number Theory 163 (2016) 238–254. [17] M. Kaneko, D. Zagier, Finite multiple zeta values, in preparation. [18] M.-S. Kim, T. Kim, An explicit formula on the generalized Bernoulli number with order n, Indian J. Pure Appl. Math. 31 (2000) 1455–1461. [19] D.S. Thakur, Function Field Arithmetic, World Sci., NJ, 2004. [20] D.S. Thakur, Multizeta values for function fields: a survey, J. Théor. Nombres Bordeaux 29 (3) (2017) 997–1023.