AN EXPLICIT FORMULA FOR THE BERNOULLI POLYNOMIALS IN TERMS OF THE r-STIRLING NUMBERS OF THE SECOND KIND ´ MEZO ˝ AND FENG QI BAI-NI GUO, ISTVAN
ABSTRACT. In the paper, the authors establish an explicit formula for computing the Bernoulli polynomials at non-negative integer points in terms of the r-Stirling numbers of the second kind.
1. Introduction. It is well known that the Bernoulli numbers Bk for k ≥ 0 can be generated by ∞
∞
k=0
k=1
X tk t X t2k t = Bk = 1 − + B2k , t e −1 k! 2 (2k)!
|t| < 2π
and that the Bernoulli polynomials Bn (x) for n ≥ 0 and x ∈ R can be generated by (1)
∞ X text tn = B (x) , n et − 1 n=0 n!
|t| < 2π.
In combinatorics, the Stirling numbers of the second kind S(n, k) are equal to the number of partitions of the set {1, 2, . . . , n} into k nonempty disjoint sets. The Stirling numbers of the second kind S(n, k) for n ≥ k ≥ 0 can be computed by k 1 X k−` k `n . (−1) S(n, k) = ` k! `=0
2010 AMS Mathematics subject classification. Primary 11B73; Secondary 05A18. Keywords and phrases. explicit formula; Bernoulli number; Bernoulli polynomial; Stirling number of the second kind; r-Stirling number of the second kind. Please cite this article as “Bai-Ni Guo, Istv´ an Mez˝ o, and Feng Qi, An explicit formula for the Bernoulli polynomials in terms of the r-Stirling numbers of the second kind, Rocky Mountain Journal of Mathematics 46 (2016), no. 6, 1919–1923; Available online at http://dx.doi.org/10.1216/RMJ-2016-46-6-1919.” Received by the editors on March 8, 2015; Accepted on May 27, 2015.
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˝ AND F. QI B.-N. GUO, I. MEZO
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In the paper [1], among other things, the Stirling numbers S(n, k) were combinatorially generalized as the r-Stirling numbers of the second kind, denoted by Sr (n, k) here, for r ∈ N, which can be alternatively defined as the number of partitions of the set {1, 2, . . . , n} into k nonempty disjoint subsets such that the numbers 1, 2, . . . , r are in distinct subsets. It is not difficult to see that S0 (n, k) = S(n, k). In [4, p. 536] and [5, p. 560], the simple formula (2)
Bn =
n X
(−1)k
k=0
k! S(n, k), k+1
n ∈ N ∪ {0}
for computing the Bernoulli numbers Bn in terms of the Stirling numbers of the second kind S(n, k) was incidentally obtained. Recently, four alternative proofs for the formula (2) were supplied in [6, 7, 16]. For more information on calculation of the Bernoulli numbers Bn , please refer to the papers [8, 9, 10, 11, 13, 15, 17], especially to the article [3], and plenty of references therein. The aim of this paper is to generalize the formula (2). Our main result can be formulated as the following theorem. Theorem 1. For all integers n, r ≥ 0, the Bernoulli polynomials Bn (r) can be computed in terms of the r-Stirling numbers of the second kind Sr (n, k) by (3)
Bn (r) =
n X
(−1)k
k=0
k! Sr (n, k). k+1
In the final section of this paper, several remarks are listed. 2. Proof of Theorem 1. We are now in a position to verify our main result. For n, r ≥ 0, let Fn,r (x) =
n X
k!Sr (n, k)xk .
k=0
By [1, p. 250, Theorem 16], we have ∞ X n=0
Sr (n, k)
∞ X tn tn = Sr (n, k) n! n! n=k
AN EXPLICIT FORMULA FOR BERNOULLI POLYNOMIALS
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1 rt t e (e − 1)k , k! where Sr (n, m) = 0 for m > n, see [1, p. 243, (10)]. Accordingly, we obtain ∞ ∞ X n X X tn tn Fn,r (x) = k!xk Sr (n + r, k + r) n! n=0 n! n=0 =
k=0
=
∞ X
k!xk
k=0
= ert
∞ X
Sr (n, k)
n=k
∞ X
tn n!
xk (et − 1)k
k=0 rt
=
e . 1 − x(et − 1)
Integrating with respect to x ∈ [0, s] for s ∈ R on both sides of the above equation yields n ∞ Z s X ln(1 + s − set ) t = −ert . (4) Fn,r (x) d x n! et − 1 0 n=0 On the other hand, Z s n X Fn,r (x) d x = 0
k=0
k! Sr (n, k)sk+1 . k+1
Substituting this into the equation (4) concludes that n ∞ X X n=0 k=0
k! tn ln(1 + s − set ) Sr (n, k)sk+1 = −ert . k+1 n! et − 1
Taking s = −1 in the above equation and making use of the generating function (1) result in " n # ∞ X ∞ X X k! tn tert tn (−1)k+1 Sr (n, k) =− t = [−Bn (r)] , k+1 n! e − 1 n=0 n! n=0 k=0
which implies the formula (3). The proof of Theorem 1 is complete.
3. Remarks. Finally we would like to give several remarks on Theorem 1 and its proof.
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˝ AND F. QI B.-N. GUO, I. MEZO
Remark 1. Since Bn (0) = Bn and S0 (n, k) = S(n, k), when r = 0, the formula (3) becomes (2). Therefore, our Theorem 1 generalizes the formula (2). Remark 2. It is easy to see that Fn,0 (1) =
n X
k!S(n, k),
k=0
which are just the classical ordered Bell numbers. For more information, please refer to the papers [2, 14] and closely related references therein. Remark 3. In the PhD thesis [12], the second author defined a variant of the polynomials Fn,r (x). Hence, a simple combinatorial study and interpretation of the polynomials Fn,r (x) is available therein. REFERENCES 1. A. Z. Broder, The r-Stirling numbers, Discrete Math. 49 (1984), no. 3, 241– 259; Available online at http://dx.doi.org/10.1016/0012-365X(84)90161-4. 2. M. B. Can and M. Joyce, Ordered Bell numbers, Hermite polynomials, skew Young tableaux, and Borel orbits, J. Combin. Theory Ser. A 119 (2012), no. 8, 1798–1810; Available online at http://dx.doi.org/10.1016/j.jcta.2012.06.002. 3. H. W. Gould, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly 79 (1972), 44–51; Available online at http://dx.doi.org/10.2307/2978125. 4. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics—A Foundation for Computer Science, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1989. , Concrete Mathematics—A Foundation for Computer Science, 2nd 5. ed., Addison-Wesley Publishing Company, Reading, MA, 1994. 6. B.-N. Guo and F. Qi, A new explicit formula for the Bernoulli and Genocchi numbers in terms of the Stirling numbers, Glob. J. Math. Anal. 3 (2015), no. 1, 33–36; Available online at http://dx.doi.org/10.14419/gjma.v3i1.4168. 7. , Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers, Analysis (Berlin) 34 (2014), no. 2, 187–193; Available online at http://dx.doi.org/10.1515/anly-2012-1238. , An explicit formula for Bernoulli numbers in terms of Stirling num8. bers of the second kind, J. Anal. Number Theory 3 (2015), no. 1, 27–30; Available online at http://dx.doi.org/10.12785/jant/030105. 9. , Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math. 272 (2014), 251–257; Available online at http://dx.doi.org/10.1016/j.cam.2014.05.018. 10. , Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math. 255 (2014), 568–579; Available online at http: //dx.doi.org/10.1016/j.cam.2013.06.020.
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[email protected],
[email protected] URL: https://www.researchgate.net/profile/Bai-Ni_Guo/ Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing City, 210044, China Email address:
[email protected] URL: http://www.inf.unideb.hu/valseg/dolgozok/mezoistvan/mezoistvan.html Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China Email address:
[email protected],
[email protected] URL: https://qifeng618.wordpress.com
