Poly-Bernoulli number
In mathematics, poly-Bernoulli numbers, denoted as [math]\displaystyle{ B_{n}^{(k)} }[/math], were defined by M. Kaneko as
- [math]\displaystyle{ {Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!} }[/math]
where Li is the polylogarithm. The [math]\displaystyle{ B_{n}^{(1)} }[/math] are the usual Bernoulli numbers.
Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined as follows
- [math]\displaystyle{ {Li_{k}(1-(ab)^{-x})\over b^x-a^{-x}}c^{xt}=\sum_{n=0}^{\infty}B_{n}^{(k)}(t;a,b,c){x^{n}\over n!} }[/math]
where Li is the polylogarithm.
Kaneko also gave two combinatorial formulas:
- [math]\displaystyle{ B_{n}^{(-k)}=\sum_{m=0}^{n}(-1)^{m+n}m!S(n,m)(m+1)^{k}, }[/math]
- [math]\displaystyle{ B_{n}^{(-k)}=\sum_{j=0}^{\min(n,k)} (j!)^{2}S(n+1,j+1)S(k+1,j+1), }[/math]
where [math]\displaystyle{ S(n,k) }[/math] is the number of ways to partition a size [math]\displaystyle{ n }[/math] set into [math]\displaystyle{ k }[/math] non-empty subsets (the Stirling number of the second kind).
A combinatorial interpretation is that the poly-Bernoulli numbers of negative index enumerate the set of [math]\displaystyle{ n }[/math] by [math]\displaystyle{ k }[/math] (0,1)-matrices uniquely reconstructible from their row and column sums. Also it is the number of open tours by a biased rook on a board [math]\displaystyle{ \underbrace{1\cdots1}_{n}\underbrace{0\cdots0}_{k} }[/math] (see A329718 for definition).
The Poly-Bernoulli number [math]\displaystyle{ B_{k}^{(-k)} }[/math] satisfies the following asymptotic:[1]
[math]\displaystyle{ B_{k}^{(-k)} \sim (k!)^2 \sqrt{\frac{1}{k\pi(1-\log 2)}}\left( \frac{1}{\log 2} \right) ^{2k+1}, \quad \text{as } k \rightarrow \infty. }[/math]
For a positive integer n and a prime number p, the poly-Bernoulli numbers satisfy
- [math]\displaystyle{ B_n^{(-p)} \equiv 2^n \pmod p, }[/math]
which can be seen as an analog of Fermat's little theorem. Further, the equation
- [math]\displaystyle{ B_x^{(-n)} + B_y^{(-n)} = B_z^{(-n)} }[/math]
has no solution for integers x, y, z, n > 2; an analog of Fermat's Last Theorem. Moreover, there is an analogue of Poly-Bernoulli numbers (like Bernoulli numbers and Euler numbers) which is known as Poly-Euler numbers.
See also
- Bernoulli numbers
- Stirling numbers
- Gregory coefficients
- Bernoulli polynomials
- Bernoulli polynomials of the second kind
- Stirling polynomials
References
- ↑ Khera, J.; Lundberg, E.; Melczer, S. (2021), "Asymptotic Enumeration of Lonesum Matrices", Advances in Applied Mathematics 123 (4): 102118, doi:10.1016/j.aam.2020.102118, https://www.sciencedirect.com/science/article/abs/pii/S0196885820301214.
- Arakawa, Tsuneo; Kaneko, Masanobu (1999a), "Multiple zeta values, poly-Bernoulli numbers, and related zeta functions", Nagoya Mathematical Journal 153: 189–209, doi:10.1017/S0027763000006954, http://projecteuclid.org/euclid.nmj/1114630825.
- Arakawa, Tsuneo; Kaneko, Masanobu (1999b), "On poly-Bernoulli numbers", Commentarii Mathematici Universitatis Sancti Pauli 48 (2): 159–167
- Brewbaker, Chad (2008), "A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues", Integers 8: A02, 9, http://www.integers-ejcnt.org/vol8.html.
- Hamahata, Y.; Masubuchi, H. (2007), "Special multi-poly-Bernoulli numbers", Journal of Integer Sequences 10 (4): Article 07.4.1, Bibcode: 2007JIntS..10...41H.
- Kaneko, Masanobu (1997), "Poly-Bernoulli numbers", Journal de Théorie des Nombres de Bordeaux 9 (1): 221–228, doi:10.5802/jtnb.197, http://jtnb.cedram.org/item?id=JTNB_1997__9_1_221_0.
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