Special Multi-Poly-Bernoulli Numbers

Y. Hamahata and H. Masubuchi Department of Mathematics Tokyo University of Science Noda, Chiba, 278-8510 Japan

              hamahata yoshinori@ma.noda.tus.ac.jp

Abstract

In this paper we investigate generalized poly-Bernoulli numbers. We call them multi-poly-Bernoulli numbers, and we establish a closed formula and a duality property for them.

1 Introduction and background

Kaneko [5] introduced poly-Bernoulli numbers B_{n}^{(k)} (k ∈ ℤ, n = 0, 1, 2, …) which are generalizations of Bernoulli numbers. One knows that special values of certain zeta functions at non-positive integers can be described in terms of poly-Bernoulli numbers. Kaneko [6] suggests to study multi-poly-Bernoulli numbers, which are generalizations of poly-Bernoulli numbers, as an open problem. Kim and Kim [7] consider them and give a relationship with special values of certain zeta functions. We consider special multi-poly-Bernoulli numbers. It seems for the authors that they are more natural than multi-poly-Bernoulli numbers when one tries to generalize the results of poly-Bernoulli numbers. The purpose of the present paper is to establish some results for them. To be more precise, we prove the closed formula and the duality for them.

We briefly recall poly-Bernoulli numbers. For an integer k ∈ ℤ, put

Li_{k}(z) = ∑_{n=1}^{∞} \frac{z^{n}}{n^{k}} .

The formal power series Li_{k}(z) is the k-th polylogarithm if k ≥ 1, and a rational function if k ≤ 0. When k = 1, Li_{1}(z) = − log(1 − z). Using Li_{k}(z), one can introduce poly-Bernoulli numbers. The poly-Bernoulli numbers B_{n}^{(k)} (n = 0, 1, 2, …) are defined by the generating series

\frac{Li_{k}(1 − e^{−x})}{1 − e^{−x}} = ∑_{n=0}^{∞} B_{n}^{(k)} \frac{x^{n}}{n!} .

We find that for any n ≥ 0, B_{n}^{(1)} = B_{n}, the classical Bernoulli number.

For nonnegative integers n, m, put

\{ \begin{matrix}n \\ m \end{matrix} \} = \frac{(−1)^{m}}{m!} ∑_{l=0}^{m}(−1)^{l} ( \begin{matrix}m \\ l \end{matrix} ) l^{n}.

We call it the Stirling number of the second kind. Kaneko obtained in [5] an explicit formula for B_{n}^{(k)}: Theorem 1 ([5]). For a nonnegative integer n and an integer k, we have

B_{n}^{(k)} = (−1)^{n} ∑_{m=1}^{n+1} \frac{(−1)^{m−1}(m − 1)! \{ \begin{matrix}n \\ m − 1 \end{matrix} \}}{m^{k}}.

Using it, the following formula can be shown: Theorem 2 (Closed formula [1]). For any n, k ≥ 0, we have

B_{n}^{(−k)} = ∑_{j=0}^{min(n,k)}(j!)^{2} \{ \begin{matrix}n + 1 \\ j + 1 \end{matrix} \} \{ \begin{matrix}k + 1 \\ j + 1 \end{matrix} \} .

By this theorem, we get Theorem 3 (Duality [1], [5]). For n, k ≥ 0, B_{n}^{(−k)} = B_{k}^{(−n)} holds.

The last theorem can be proved in another way. Namely, using Theorem 1, we have Theorem 4 (Symmetric formula [5]).

∑_{n=0}^{∞} ∑_{k=0}^{∞} B_{n}^{(−k)} \frac{x^{n}}{n!} \frac{y^{k}}{k!} = \frac{e^{x+y}}{e^{x} + e^{y} − e^{x+y}} .

As corollary to this theorem, we have the duality theorem.

We would like to extend these results to our generalized poly-Bernoulli numbers.

2 Multi-poly-Bernoulli numbers

In this section, we investigate generalized poly-Bernoulli numbers. First, we define a generalization of Li_{k}(z).

Definition 5. For k_{1}, k_{2}, …, k_{r} ∈ ℤ, define

Li_{k_{1},k_{2},...,k_{r}}(z) = ∑_{{m_{1},...,mr∈ℤ \atop 0<m_{1}<m_{2}<···<mr}} \frac{z^{m_{r}}}{m_{1}^{k_{1}} ⋯ m_{r}^{k_{r}}} .

Now let us introduce a generalization of poly-Bernoulli numbers with the use of L_{k_{1},k_{2},...,k_{r}}(z). Definition 6. Multi-poly-Bernoulli numbers B_{n}^{(k_{1},k_{2},...,k_{r})} (n = 0, 1, 2, …) are defined for each integer k_{1}, k_{2}, …, k_{r} by the generating series

\frac{Li_{k_{1},k_{2},...,k_{r}}(1 − e^{−t})}{(1 − e^{−t})^{r}} = ∑_{n=0}^{∞} B_{n}^{(k_{1},k_{2},...,k_{r})} \frac{t^{n}}{n!}.

We generalize Theorem 1: Theorem 7. For a nonnegative integer n and integers k_{1}, …, k_{r}, we have

B_{n}^{(k_{1},k_{2},...,k_{r})} = (−1)^{n} ∑_{m_{r}=r}^{n+r} ∑_{0<m_{1}<···<m_{r}} \frac{(−1)^{m_{r}−r}(m_{r} − r)! \{ \begin{matrix}n \\ m_{r} − r \end{matrix} \}}{m_{1}^{k_{1}} ⋯ m_{r}^{k_{r}}}.

Proof. By definition,

∑_{n=0}^{∞} B_{n}^{(k_{1},k_{2},...,k_{r})} \frac{t^{n}}{n!} = ∑_{0<m_{1}<···<m_{r}} \frac{(−1)^{m_{r}−r}(e^{−t} − 1)^{m_{r}−r}}{m_{1}^{k_{1}} ⋯ m_{r}^{k_{r}}} .

Here we apply the formula

\frac{(e^{t} − 1)^{m}}{m!} = ∑_{n=m}^{∞} \{ \begin{matrix}n \\ m \end{matrix} \} \frac{t^{n}}{n!} (n ≥ m ≥ 0)

(see (4) in [5]) to the right hand side of the above equation.

R.H.S. = ∑_{0<m_{1}<···<m_{r}} \frac{(−1)^{m_{r}−r}(m_{r} − r)!}{m_{1}^{k_{1}} ⋯ m_{r}^{k_{r}}} ∑_{n=m_{r}−r}^{∞} \{ \begin{matrix}n \\ m_{r} − r \end{matrix} \} \frac{(−t)^{n}}{n!}
= ∑_{0<m_{1}<···<m_{r}} ∑_{n=m_{r}−r}^{∞} \frac{(−1)^{m_{r}−r}(m_{r} − r)! \{ \begin{matrix}n \\ m_{r}−r \end{matrix} \}}{m_{1}^{k_{1}} ⋯ m_{r}^{k_{r}}} · \frac{(−1)^{n}t^{n}}{n!}
= ∑_{n=0}^{∞} ∑_{m_{r}=r}^{n+r} ∑_{0<m_{1}<···<m_{r}} \frac{(−1)^{m_{r}−r}(m_{r} − r)! \{ \begin{matrix}n \\ m_{r}−r \end{matrix} \}}{m_{1}^{k_{1}} ⋯ m_{r}^{k_{r}}} · \frac{(−1)^{n}t^{n}}{n!}
= ∑_{n=0}^{∞} ((−1)^{n} ∑_{m_{r}=r}^{n+r} ∑_{0<m_{1}<···<m_{r}} \frac{(−1)^{m_{r}−r}(m_{r} − r)! \{ \begin{matrix}n \\ m_{r}−r \end{matrix} \}}{m_{1}^{k_{1}} ⋯ m_{r}^{k_{r}}}) \frac{t^{n}}{n!}.

This shows the claim.

By this theorem, for the smaller n, we can compute B_{n}^{(k_{1},k_{2},...,k_{r})} more specifically. For example,

B_{0}^{(k_{1},k_{2},...,k_{r})} = \frac{1}{1^{k_{1}}2^{k_{2}} ⋯ r^{k_{r}}} ,
B_{1}^{(k_{1},k_{2},...,k_{r})} = ∑_{0<m_{1}<···<m_{r}} \frac{1}{m_{1}^{k_{1}} ⋯ m_{r−1}^{k_{r−1}}(r + 1)^{k_{r}}} .

3 Closed formula and duality

Let n be a nonnegative integer, r a positive integer, and k ∈ ℤ. We define

B[r]_{n}^{(k)} = B_{n}^{(\overbrace{0, …, 0}^{r−1},k)}.

We say that B[r]_{n}^{(k)} is a special multi-Bernoulli number of order r. By definition, it is clear that B[1]_{n}^{(k)} = B_{n}^{(k)}. Hence the notion of special multi-poly-Bernoulli number is a generalization of that of poly-Bernoulli number. For B[r]_{n}^{(k)}, we establish closed formula:

Theorem 8 (Closed formula). For n, k ≥ 0, we have

B[r]_{n}^{(−k)}

= ∑_{{n=n_{1}+···+nr \atop n_{1},...,nr≥0}} ∑_{{k=k_{1}+···+kr \atop k_{1},...,kr≥0}} \frac{n!k!}{n_{1}! ⋯ n_{r}!k_{1}! ⋯ k_{r}!}

× ( ∑_{j_{1}=0}^{min(n_{1},k_{1})} ⋯ ∑_{j_{r}=0}^{min(n_{r},k_{r})}(j_{1}! ⋯ j_{r}!)^{2} \{ \begin{matrix}n_{1} + 1 \\ j_{1} + 1 \end{matrix} \} ⋯ \{ \begin{matrix}n_{r} + 1 \\ j_{r} + 1 \end{matrix} \} \{ \begin{matrix}k_{1} + 1 \\ j_{1} + 1 \end{matrix} \} ⋯ \{ \begin{matrix}k_{r} + 1 \\ j_{r} + 1 \end{matrix} \}) .

To prove the theorem, we need the following lemma:

Lemma 9.

∑_{n=m+1}^{∞}(e^{y} − e^{y−x})^{n} = \frac{e^{x+y}(1 − e^{−x})}{e^{x} + e^{y} − e^{x+y}} (e^{y} − e^{y−x})^{m}.

Proof.

L.H.S. = \frac{1}{1 − (e^{y} − e^{y−x})}(e^{y} − e^{y−x})^{m+1}
= \frac{e^{y} − e^{y−x}}{1 − (e^{y} − e^{y−x})}(e^{y} − e^{y−x})^{m}
= R.H.S..

Proof of Theorem 8.

∑_{n=0}^{∞} ∑_{k=0}^{∞} B[r]_{n}^{(−k)} \frac{x^{n}}{n!} \frac{y^{k}}{k!}

= ∑_{n=0}^{∞} ∑_{k=0}^{∞} ((−1)^{n} ∑_{m_{r}=r}^{n+r} ∑_{0<m_{1}<···<m_{r}}(−1)^{m_{r}−r}(m_{r} − r)! \{ \begin{matrix}n \\ m_{r} − r \end{matrix} \} m_{r}^{k}) \frac{x^{n}}{n!} \frac{y^{k}}{k!}

(by Theorem 7)

= ∑_{n=0}^{∞} ∑_{m_{r}=r}^{n+r} ∑_{m_{r−1}=r−1}^{m_{r}−1} ⋯ ∑_{m_{1}=1}^{m_{2}−1} \{ \begin{matrix}n \\ m_{r} − r \end{matrix} \} \frac{(−x)^{n}}{n!} (m_{r} − r)!(−1)^{m_{r}−r} ∑_{k=0}^{∞} \frac{(m_{r}y)^{k}}{k!}

= ∑_{n=0}^{∞} ∑_{m_{1}=1}^{n+1} ∑_{m_{2}=m_{1}+1}^{n+2} ⋯ ∑_{m_{r}=m_{r−1}+1}^{n+r} \{ \begin{matrix}n \\ m_{r} − r \end{matrix} \} \frac{(−x)^{n}}{n!} (m_{r} − r)!(−1)^{m_{r}−r} ∑_{k=0}^{∞} \frac{(m_{r}y)^{k}}{k!}

= ∑_{m_{1}=1}^{∞} ∑_{m_{2}=m_{1}+1}^{∞} ⋯ ∑_{m_{r}=m_{r−1}+1}^{∞} ∑_{n=m_{r}−r}^{∞} \{ \begin{matrix}n \\ m_{r} − r \end{matrix} \} \frac{(−x)^{n}}{n!} (m_{r} − r)!(−1)^{m_{r}−r} ∑_{k=0}^{∞} \frac{(m_{r}y)^{k}}{k!}

= ∑_{m_{1}=1}^{∞} ∑_{m_{2}=m_{1}+1}^{∞} ⋯ ∑_{m_{r}=m_{r−1}+1}^{∞}(1 − e^{−x})^{m_{r}−r}e^{m_{r}y}

= \frac{1}{(1 − e^{−x})^{r}} ∑_{m_{1}=1}^{∞} ∑_{m_{2}=m_{1}+1}^{∞} ⋯ ∑_{m_{r}=m_{r−1}+1}^{∞}(e^{y} − e^{y−x})^{m_{r}}.

We use Lemma 9 repeatedly to the right hand side of the last equation. Then

R.H.S = \frac{1}{(1 − e^{−x})^{r}} (\frac{e^{x+y}(1 − e^{−x})}{e^{x} + e^{y} − e^{x+y}} )^{r−1} ∑_{m_{1}=1}^{∞}(e^{y} − e^{y−x})^{m_{1}}

= \frac{1}{(1 − e^{−x})^{r}} (\frac{e^{x+y}(1 − e^{−x})}{e^{x} + e^{y} − e^{x+y}} )^{r}
= (\frac{e^{x+y}}{e^{x} + e^{y} − e^{x+y}} )^{r} .

By Theorem 2, the right hand side of the last equation is equal to

(∑_{n=0}^{∞} ∑_{k=0}^{∞} B_{n}^{(−k)} \frac{x^{n}}{n!} \frac{y^{k}}{k!} )^{r}

= (∑_{n=0}^{∞} ∑_{k=0}^{∞} ∑_{j=0}^{min(n,k)}(j!)^{2} \{ \begin{matrix}n + 1 \\ j + 1 \end{matrix} \} \{ \begin{matrix}k + 1 \\ j + 1 \end{matrix} \} \frac{x^{n}}{n!} \frac{y^{k}}{k!} )^{r} .

This implies the theorem. □ A generalization of Theorem 3 follows from the last theorem:

Corollary 10 (Duality). For n, k ≥ 0, we have

B[r]_{n}^{(−k)} = B[r]_{k}^{(−n)}.

We note that in the process of proof of the last theorem, we have obtained two formulae:

Proposition 11 (Symmetric formula). For n, k ≥ 0,

∑_{n=0}^{∞} ∑_{k=0}^{∞} B[r]_{n}^{(−k)} \frac{x^{n}}{n!} \frac{y^{k}}{k!} = (\frac{e^{x+y}}{e^{x} + e^{y} − e^{x+y}} )^{r} .

Proposition 12. For n, k ≥ 0,

B[r]_{n}^{(−k)} = ∑_{{n=n_{1}+···+nr \atop n_{1},...,nr≥0}} ∑_{{k=k_{1}+···+kr \atop k_{1},...,kr≥0}} \frac{n!k!}{n_{1}! ⋯ n_{r}!k_{1}! ⋯ k_{r}!}B_{n_{1}}^{(−k_{1})} ⋯ B_{n_{r}}^{(−k_{r})}.

Proposition 11 is a generalization of Theorem 4.

References

[1] T. Arakawa and M. Kaneko, On poly-Bernoulli numbers. Comment. Math. Univ. St. Pauli 48 (1999), 159–167.

[2] L. Carlitz, Some theorems on Bernoulli numbers of higher order. Pacific J. Math. 2 (1952), 127–139.

[3] R. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, 1989.

[4] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory. Second Edition, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.

[5] M. Kaneko, Poly-Bernoulli numbers. J. Théorie de Nombres 9 (1997), 221–228.

[6] M. Kaneko, Multiple Zeta Values and Poly-Bernoulli Numbers. Tokyo Metropolitan University Seminar Report, 1997.

[7] M.-S. Kim and T. Kim, An explicit formula on the generalized Bernoulli number with order n. Indian J. Pure Appl. Math. 31 (2000), 1455–1461.

[8] R. Sánchez-Peregrino, Closed formula for poly-Bernoulli numbers. Fibonacci Quart. 40 (2002), 362–364.

2000 Mathematics Subject Classification: Primary 11B68; Secondary 11B73.

Keywords: poly-Bernoulli numbers, Stirling numbers.

Received February 24 2007; revised version received April 13 2007. Published in Journal of Integer Sequences, April 13 2007.

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