For 2), we set x1 =· · ·=xr−1 = 0 in the formulas in Proposition 5.7 and have
∑∞ k=0
∑∞ n=0
B( z}|{r−1
0,...,0,−k) n
tn n!
xk
k! = (et−1)r−1 e−x+e−t−1
= (et−1)r−1 · et+x
1−(et−1)(ex−1)
=
∑∞ j=0
et(et−1)r+j−1ex(ex−1)j
=
∑∞ j=0
(r+j −1)!
∑∞ n=r+j−1
{n+ 1 r+j
}tn n!·j!
∑∞ k=j
{k+ 1 j+ 1
}xk k!
and
∑∞ k=0
∑∞ n=0
C( z}|{r−1
0,...,0,−k) n
tn n!
xk k! =
∑∞ j=0
(r+j−1)!
∑∞ n=r+j−1
{ n r+j−1
}tn n!· · ·j!
∑∞ k=j
{k+ 1 j+ 1
}xk k!. Comparing the coefficients of tn
n!
xk
k! on both sides and noting that {n
m }
= 0 if n < m, we obtain 2).
Proposition 5.11. For any multi-index (k1, . . . , kr), we have the following recursions:
Bn,⋆(k1,...,kr) = 1 n+ 1
(
Bn,⋆(k1−1,k2,...,kr)−
n−1
∑
j=1
( n j−1
)
Bj,⋆(k1,...,kr) )
,
Cn,⋆(k1,...,kr) = 1 n+ 1
(
Cn,⋆(k1−1,k2,...,kr)−
n−1
∑
j=0
(n+ 1 j
)
Cj,⋆(k1,...,kr) )
.
Proposition 5.12. For any multi-index (k1, . . . , kr), ki ∈Z, we have Bn,⋆(k1,...,kr) = ∑
n+1≥m1≥···≥mr≥1
(−1)m1+n−1(m1−1)!
mk11· · ·mkrr
{ n m1−1
}
and
Cn,⋆(k1,...,kr) = ∑
n+1≥m1≥···≥mr≥1
(−1)m1+n−1(m1 −1)!
mk11· · ·mkrr
{n+ 1 m1
} . Proposition 5.13. For any multi-index (k1, . . . , kr), we have
Cn,⋆(k1,...,kr) =B(kn,⋆1,...,kr)−Cn(k−11,⋆−1,k2,...,kr).
Proposition 5.14. For any multi-index (k1, . . . , kr) with ki ∈Z, we have Bn,⋆(k1,...,kr) =
∑n j=0
(n j
)
Cj,⋆(k1,...,kr) and
Cn,⋆(k1,...,kr) =
∑n j=0
(−1)n−j (n
j )
Bj,⋆(k1,...,kr).
We present the sum formulas and duality relation for multi-poly-Bernoulli-star num-bers. These results hold not for multi-poly-Bernoulli numbers but for multi-poly-Bernoulli-star numbers.
The former one is the following:
Theorem 5.15. We have
∑
k1+···+kr=k 1≤r≤k,ki≥1
(−1)rBn,⋆(k1,...,kr) = (−1)k k
( n k−1
)
Bn(1)−k+1,⋆
and ∑
k1+···+kr=k 1≤r≤k,ki≥1
(−1)rCn,⋆(k1,...,kr) = (−1)k k
( n k−1
)
Cn(1)−k+1,⋆. (5.3)
Proof. We consider the generating functions of both sides by multiplyingtn/n! and taking the summation onn:
(LHS) =
∑∞ n=0
∑
k1+···+kr=k 1≤r≤k,ki≥1
(−1)rBn,⋆(k1,...,kr)tn n!
= ∑
k1+···+kr=k 1≤r≤k,ki≥1
(−1)rLi⋆k1,...,kr(1−e−t) 1−e−t ,
(RHS) = (−1)k k
∑∞ n=0
( n k−1
)
Bn(1)−k+1,⋆tn n!
= (−1)k k!
∑∞ n=0
Bn(1)−k+1,⋆ tn (n−k+ 1)!
= (−1)k k!
∑∞ n=0
Bn,⋆(1)tn+k−1 n!
= (−1)k k!
tk 1−e−t.
Since both sides have the same denominator, it suffices to prove the following identity:
∑
k1+···+kr=k 1≤r≤k,ki≥1
(−1)rLi⋆k1,...,kr(1−e−t) = (−1)k
k! tk. (5.4)
This equality is proved by the induction on the weight. When k = 1, the left-hand side is −Li⋆1(1−e−t) = −t and is equal to the right-hand side. Next we assume the identity holds when the weight is k. Then by differentiating the left-hand side of the identity of weight k+ 1, we obtain
d dt
∑
k1+···+kr=k+1 1≤r≤k+1,ki≥1
(−1)rLi⋆k1,...,kr(1−e−t)
= ∑
k1+···+kr=k 1≤r≤k,ki≥1
(−1)r
( e−t
1−e−t − 1 1−e−t
)
Li⋆k1,...,kr(1−e−t)
= ∑
k1+···+kr=k 1≤r≤k,ki≥1
(−1)r+1Li⋆k
1,...,kr(1−e−t) (−1)k+1 k
We used the induction hypothesis in the last equality. Therefore we have
∑
k1+···+kr=k+1 1≤r≤k+1,ki≥1
(−1)rLi⋆k1,...,kr(1−e−t) = (−1)k+1
(k+ 1)!tk+1+C
with some constantC, which we find is 0 by puttingt= 0. The sum formula for Cn,⋆ can also be obtained from (5.4) since the generating function of Cn,⋆ differs only by the factor et to the one of Bn,⋆.
Next we present the duality relation for the multi-poly-Bernoulli-star numbers.
Theorem 5.16 (K. Imatomi [13]). For any multi-index (k1, . . . , kr) with ki ≥1(1 ≤i≤ r), we have
Cn,⋆(k1,...,kr)= (−1)nBn,⋆(k1′,...,k′l),
where (k1′, . . . , kl′) is the dual index of (k1, . . . , kr) in Hoffman’s sense.
Proof. We also consider the generating functions of both sides:
(LHS) =
∑∞ n=0
Cn,⋆(k1,...,kr)tn
n! = Li⋆k
1,...,kr(1−e−t) et−1 , (RHS) =
∑∞ n=0
(−1)nBn,⋆(k′1,...,k′l)tn
n! = Li⋆k′
1,...,k′l(1−et) 1−et . Hence we have to show the following identity:
Li⋆k
1,...,kr(1−e−t) +Li⋆k′
1,...,kl′(1−et) = 0. (5.5)
This identity also follows from the induction on the weight. First, it is trivial in the case k = 1. We assume the above identity holds when the weight is k. Since k1 = 1 is equivalent tok1′ ̸= 1, we may assumek1 = 1 by the symmetry of the identity. Then when the weight isk+ 1, the derivative of the left-hand side yields
d dt
( Li⋆1,k
2,...,kr(1−e−t) +Li⋆
k1′,...,k′l(1−et) )
= 1
1−e−tLi⋆k2,...,kr(1−e−t) + −et 1−etLi⋆k′
1−1,k′2,...,kl′(1−et)
= 1
1−e−t (
Li⋆k
2,...,kr(1−e−t) +Li⋆k′
1−1,k′2,...,kl′(1−et) )
= 0.
Therefore we obtain
Li⋆1,k2,...,kr(1−e−t) +Li⋆k′
1,...,k′l(1−et) =C with some constant C, and by puttingt = 0, we conclude C = 0.
5.5 Connection to finite multiple zeta(-star) values
First, we introduce the following congruence formula between multi-poly-Bernoulli(-star) numbers and finite multiple zeta(-star) values.
Proposition 5.17. We have
Hp(k1, . . . , kr) ≡ −Cp(k−12−1,k2,...,kr) modp., Hp⋆(k1, . . . , kr) ≡ −Cp(k−12,⋆−1,k2,...,kr) modp.
More generally, for (strict) multiple harmonic sums, we have for any r ≥1, j ≥0, and ki ∈Z
Hp(1, . . . ,1
| {z }
j
, k1, . . . , kr)≡ −Cp(k−1j−−1,k2 2,...,kr)modp.
Proof. By the explicit formula in Theorem 5.4, we have (we may assume pis odd) Cp(k−12−1,k2,...,kr) = − ∑
p−1≥m1>m2>···>mr>0
(−1)m1−1(m1−1)!
mk11−1mk22· · ·mkrr
{p−1 m1
}
= ∑
p−1≥m1>m2>···>mr>0
(−1)m1m1! mk11mk22· · ·mkrr
{p−1 m1
} . By the well-known formula (cf. [9,§6.1])
(−1)m1m1!
{p−1 m1
}
=
m1
∑
l=0
(−1)l (m1
l )
lp−1, we see
(−1)m1m1!
{p−1 m1
}
≡
m1
∑
l=1
(−1)l (m1
l )
modp,
the sum on the right being equal to (1−1)m1 −1 =−1. This proves that Cp(k−12−1,k2,...,kr)≡ −Hp(k1, . . . , kr) mod p.
Congruence formula for star version is proved in the same manner.
For the second identity, we proceed as follows. First note
∑
p−1≥i1>···>ij>m1>···>mr≥1
1
i1· · ·ijmk11· · ·mkrr
= ∑
p−j>m1>···>mr≥1
1 mk11· · ·mkrr
∑
p−1≥i1>···>ij>m1
1 i1· · ·ij
. By changing il →p−il, we see that
∑ 1
i · · ·i ≡(−1)j ∑ 1
i · · ·i modp.
Using the formula
∑
p−m1>ij>···i1≥1
1 i1· · ·ij
= 1
(p−m1−1)!
[p−m1 j+ 1
]
and the congruences [n
m
] ≡ {p−m
p−n
} mod p (1 ≤ m ≤ n ≤ p− 1) (see [11, §5]) and
1
(p−m1−1)! ≡(−1)m1+1m1! mod p, we obtain (for odd p) ζA(1, . . . ,1
| {z }
j
, k1, . . . , kr)(p) = ∑
p−j>m1>···>mr≥1
(−1)j+m1+1m1!{p−j−1
m1
}
mk11· · ·mkrr modp
= −Cp(k−1j−−1,k2 2,...,kr)modp.
This concludes the proof of the theorem.
Combining the proposition with Example 5.2, we see that
ζA(2,
k−2
z }| {
1, . . . ,1) = (
−C(
k−1
z}|{1,...,1)
p−2 modp)
p
= (
− 1 p−1
(p−1 k−1
)
Bp−k modp )
p
= (Bp−k modp)p. (We used (p−1
k−1
) ≡ (−1)k−1 modp and (−1)p−kBp−k = Bp−k for large enough p.) As is discussed in [18], the element (Bp−kmodp)p on the right is regarded as an analogue of kζ(k).
Corollary 5.18 (M. Hoffman [11]). For any multi-index (k1, . . . , kr) with ki ≥1(1≤i≤ r), let (k1′, . . . , kl′) be the dual index for (k1, . . . , kr) in Hoffman’s sense. Then we have
ζA⋆(k1, . . . , kr) =−ζA⋆(k′1, . . . , kl′).
Proof. It is sufficient to prove the case k1 = 1. By (5.6) and the duality relation for the multi-poly-Bernoulli-star numbers, we obtain
(LHS) =
(−Cp(0,k−2,⋆2,···,kr) modp )
p, (RHS) =
( C(k
′
1−1,k′2,···,k′l)
p−2,⋆ modp
)
p
= (
(−1)pBp(k−22,⋆,···,kr) modp )
p
.
Hence we complete the proof if we prove Cn,⋆(0,k2,···,kr) = Bn,⋆(k2,···,kr) for all n. We consider
the generating functions of these numbers:
∑∞ n=0
Cn,⋆(0,k2,···,kr)tn
n! = Li⋆0,k2,...,kr(1−e−t) et−1
= 1
et−1
∑
m2≥···≥mr≥1
1 mk22· · ·mkrr
∑∞ m1=m2
(1−e−t)m1
= 1
e−t(et−1)Li⋆k2,...,kr(1−e−t)
=
∑∞ n=0
Bn,⋆(k2,···,kr)tn n!. From this we have
−Cp(0,k−2,⋆2,···,kr)= (−1)pBp(k−22,⋆,···,kr) for any odd primep.
The following corollary is a weaker version of Theorem 4.10.
Corollary 5.19. We have
∑
k1+···+kr=k r≥1,k1≥2,ki≥1
(−1)rζA⋆(k1, . . . , kr) = (Bp−kmodp)p,
where Bn is the classical Bernoulli number.
Proof. Equations (5.3) and (5.6) yield
∑
k1+···+kr=k+1 r≥1,k1≥2,ki≥1
(−1)r+1ζA⋆(k1, . . . , kr) =
((−1)k k
(p−2 k−1
)
Cp(1)−k−1,⋆modp )
p
=
(−Cp−k−1,⋆(1) modp )
p
. So replacing k byk−1, we obtain the desired identity.
6 Multi-Poly-Bernoulli Polynomials
In this section, we give multi-poly-Bernoulli polynomials which generalize the classical Bernoulli polynomials, and generalized Arakawa-Kaneko zeta function whose values at non-positive integers are multi-poly-Bernoulli polynomials .
6.1 Definition of multi-poly-Bernoulli polynomials
Definition 6.1. For any multi-index(k1, . . . , kr), multi-poly-Bernoulli(-star) polynomials Cn(k1,...,kr)(x), Cn,⋆(k1,...,kr)(x) are defined by the following series:
∑∞ n=0
Cn(k1,...,kr)(x)tn
n! = Lik1,...,kr(1−e−t)
et−1 ext, (6.1)
∑∞ n=0
Cn,⋆(k1,...,kr)(x)tn
n! = Li⋆k1,...,kr(1−e−t) et−1 ext.
When r = 1 and k1 = 1, Cn(1)(x) and Cn,⋆(1)(x) are the classical Bernoulli polynomials whose generating function is text/(et−1), and when x= 0, Cn(k1,...,kr)(0) and Cn,⋆(k1,...,kr)(0) are the poly-Bernoulli(-star) numbers. We need not consider “B-version” of multi-poly-Bernoulli(-star) polynomials since we obtain “B-version” by replacing xby x+ 1 in Definition 6.1.
We present basic results for multi-poly-Bernoulli(-star) polynomials. We first give the explicit expression in terms of multi-poly-Bernoulli(-star) numbers.
Proposition 6.2. For any multi-index (k1, . . . , kr) and n ≥0, we have Cn(k1,...,kr)(x) =
∑n j=0
(n j
)
Cn(k−1j,...,kr)xj, Cn,⋆(k1,...,kr)(x) =
∑n j=0
(n j
)
Cn(k−1j,⋆,...,kr)xj. Proof. By using Definition 6.1, we obtain
∑∞ n=0
Cn(k1,...,kr)(x)tn
n! = Lik1,...,kr(1−e−t) et−1 ext
= ( ∞
∑
i=0
Ci(k1,...,kr)ti i!
) ( ∞
∑
j=0
(xt)j j!
)
=
∑∞ n=0
∑n j=0
(n j
)
Cn(k−1j,...,kr)xjtn n!. The proof for multi-poly-Bernoulli-star polynomials is identical.
Proposition 6.3. For any multi-index (k1, . . . , kr) and n ≥1, we have d
dxCn(k1,...,kr)(x) = nCn(k−11,...,kr)(x), d
dxCn,⋆(k1,...,kr)(x) = nCn(k−11,⋆,...,kr)(x).
Proof. We differentiate both hands of (6.1) with respect to x:
d dx
∑∞ n=0
Cn(k1,...,kr)(x)tn
n! = Lik1,...,kr(1−e−t) et−1 text
= d
dx
∑∞ n=0
Cn(k1,...,kr)(x)tn+1 n! .
We can prove the identity for multi-poly-Bernoulli-star polynomials in the same manner.
Next, we give the symmetric relation for multi-poly-Bernoulli-star polynomials. We note that the similar relation does not hold for multi-poly-Bernoulli polynomials.
Proposition 6.4. Let (k1, . . . , kr)∗ = (k1′, . . . , k′l). Then the relation Cn,⋆(k1,...,kr)(x) = (−1)nCn,⋆(k1′,...,k′l)(1−x) holds.
Proof. Using the identity (5.5), we obtain 0 = Li⋆k1,...,kr(1−e−t)
et−1 ext+ Li⋆k′
1,...,kl′(1−et) et−1 ext
= Li⋆k
1,...,kr(1−e−t)
1−e−t ext− Li⋆k′
1,...,k′l(1−et)
e−t−1 e(1−x)(−t). Hence we have the required identity by comparing the coefficients of tn/n!.
6.2 Generalized Arakawa-Kaneko zeta function
In this section, we introduce generalized Arakawa-Kaneko zeta function whose values at non-positive integers are multi-poly-Bernoulli(-star) polynomials.
Definition 6.5. For complex numbers sand xwith ℜ(s)>1−r andℜ(x)>0, we define Zk1,...,kr(s, x) by the following integral:
Zk1,...,kr(s, x) = 1 Γ(s)
∫ ∞
0
Lik1,...,kr(1−e−t)
et−1 e−xtts−1dt.
Proposition 6.6. The function Zk1,...,kr(s, x) can be analytically continued to whole s-plane as an entire function. Further, we have
Zk1,...,kr(−n, x) = (−1)nCn(k1,...,kr)(−x).
Proof. We decompose the interval of integration of defining integral:
Zk1,...,kr(s, x) = 1 Γ(s)
∫ 1
0
Lik1,...,kr(1−e−t)
et−1 e−xtts−1dt
+ 1
Γ(s)
∫ ∞
1
Lik1,...,kr(1−e−t)
et−1 e−xtts−1dt.
The latter integral converges for s∈C since ∫ ∞
1
Lik1,...,kr(1−e−t)
et−1 e−xtts−1dt
≤
∫ ∞
1
Li1,...,1(1−e−t)
et−1 e−xtts−1dt
≤ ∃M
∫ ∞
1
e−ℜ(x)ttℜ(s)+r−1dt.
Also we find that the latter one vanishes at non-positive integers. The former one can be rewritten as the following form:
1 Γ(s)
∫ 1 0
Lik1,...,kr(1−e−t)
et−1 e−xtts−1dt = 1 Γ(s)
∫ 1 0
∑∞ n=0
Cn(k1,...,kr)(−x)tn+s−1 n! dt
= 1
Γ(s)
∑∞ n=0
Cn(k1,...,kr)(−x)1 n!
1 n+s. Second equality holds from Lebesgue’s convergence theorem and the estimate
∑∞ n=r
∫ 1 0
Cn(k1,...,kr)(−x)tn+s−1 n!
dt =
∑∞ n=r
Cn(k1,...,kr)(−x) 1 n!
1 n+ℜ(s)
≤
∑∞ n=r
Cn(k1,...,kr)(−x) 1 n! <∞
forℜ(s)>1−r. Therefore, we can continueZk1,...,kr(s, x) tos-plane and moreover, when s=−n for n≥0, we have
Zk1,...,kr(−n, x) = lim
s→−n
1
Γ(s)(s+n)Cn(k1,...,kr)(−x)1 n!
= (−1)nCn(k1,...,kr)(−x).
Proposition 6.7. For any multi-index (k1, . . . , kr), we have Zk1,...,kr(s, x) = ∑
m1>···>mr>0
(−∆)m1
mk11· · ·mkrrζ(s, x+ 1),
=: Lik1,...,kr(−∆)ζ(s, x+ 1).
where ζ(s, x) is Hurwitz zeta function and∆ is the forward difference operator defined by
∆f(x) = f(x+ 1)−f(x).
Proof. We rewrite multiple polylogarithm as series in Definition 6.5:
Zk1,...,kr(s, x) = 1 Γ(s)
∫ ∞
0
∑
m1>···>mr>0
(1−e−t)m1 mk11· · ·mkrr
e−xtts−1 et−1 dt
= 1
Γ(s)
∑
m1>···>mr>0
1 mk11· · ·mkrr
m1
∑
j=0
(−1)j (m1
j
) ∫ ∞
0
e−(j+x)tts−1 et−1 dt
= ∑
m1>···>mr>0
1 mk11· · ·mkrr
m1
∑
j=0
(−1)j (m1
j )
ζ(s, x+j+ 1)
= ∑
m1>···>mr>0
(−∆)m1
mk11· · ·mkrrζ(s, x+ 1).
We derive a formula for Z1,...,1(s, x) using Proposition 6.7.
Example 6.8. For m≥0, we have Z{1}m(s, x) = (−1)m
m! s(s+ 1)· · ·(s+m−1)ζ(s+m, x+ 1).
Proof. By using Proposition 6.7, we obtain
Z{1}m(s, x) = Li{1}m(−∆)ζ(s, x+ 1)
= 1
m!(Li1(−∆))mζ(s, x+ 1)
= 1
m!(−Dx)mζ(s, x+ 1)
= 1
m!s(s+ 1)· · ·(s+m−1)ζ(s+m, x+ 1),
where Dx is the differential operator on x. Notice that 1 + ∆ = exp(Dx) holds since eDxf(x) =
∑∞ n=0
Dnx
n! f(x) =
∑∞ n=0
f(n)(x)
n! 1n =f(x+ 1).
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