TWO CLOSED FORMS FOR THE BERNOULLI POLYNOMIALS

Transcription

1 TWO CLOSED FORMS FOR THE BERNOULLI POLYNOMIALS FENG QI AND ROBIN J. CHAPMAN Abstract. In the paper, the authors find two closed forms involving the Stirling numbers of the second kind and in terms of a determinant of combinatorial numbers for the Bernoulli polynomials and numbers.. Introduction It is common knowledge that the Bernoulli numbers and polynomials B k and B k u for k satisfy B k B k and can be generated respectively by and z e z z k B k k! z 2 + k ze uz e z k k B 2k z 2k 2k!, B k u zk, z < 2π. k! z < 2π x Because the function e x + x 2 is odd in x R, all of the Bernoulli numbers B 2k+ for k N equal. It is clear that B and B 2. The first few Bernoulli numbers B 2k are B 2 6, B 4 3, B 6 42, B 8 3, B 5 66, B , B 4 7 6, B The first five Bernoulli polynomials are B u, B u u 2, B 2u u 2 u + 6, B 3 u u u2 + 2 u, B 4u u 4 2u 3 + u 2 3. In combinatorics, the Stirling numbers Sn, k of the second kind for n k can be computed and generated by Sn, k k! respectively. See [7, p. 26]. k k k l l n l l and e x k k! nk Sn, k xn 2 Mathematics Subject Classification. Primary B68, Secondary B73, 26A6, 26A9, 26C5. Key words and phrases. closed form; Bernoulli polynomial; Bernoulli number; Stirling numbers of the second kind; determinant. This paper was typeset using AMS-LATEX.

2 2 F. QI AND R. J. CHAPMAN It is easy to see that the generating function of B k u can be ze uz [ e uz e z e uz ] z u u ezt d t ezt u d t.. This expression will play important role in this paper. For related information on the integral expression., please refer to [2, 5, 7, 3, 32] and plenty of references cited in the survey and expository article [3]. In mathematics, a closed form is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, four arithmetic operations, and elementary functions, but usually no limit. The main aim of this paper is to find two closed forms for the Bernoulli polynomials and numbers B k u and B k for k N. The main results can be summarized as the following theorems. Theorem.. The Bernoulli polynomials B n u for n N can be expressed as n [ r n l! B n u k! m m! s l + r i+j l l + r! m + s! r i k r+sk l+mn i j ] m + s Sl + i, ism + j, j u m+s u l+r..2 s j Consequently, the Bernoulli numbers B k for k N can be represented as n n+ B n i i+ Sn + i, i..3 i n+i i Theorem.2. Under the conventions that and p q for q > p, the Bernoulli polynomials B k u for k N can be expressed as B k u k l + l + [ u l m+ u l m+] m,.4 l k, m k where l k, m k denotes a k k determinant. Consequently, the Bernoulli numbers B k for k N can be represented as B k k l +..5 l + m l k, m k 2. Lemmas For proving the main results, we need the following notation and lemmas. In combinatorial mathematics, the Bell polynomials of the second kind B n,k are defined by B n,k x, x 2,..., x n k+ for n k. See [7, p. 34, Theorem A]. l i {} N n i ilin n i lik n k+ i l i! n k+ i xi i! li

3 TWO CLOSED FORM FOR THE BERNOULLI POLYNOMIALS 3 Lemma 2. [, Example 2.6] and [7, p. 36, Eq. [3n]]. The Bell polynomials of the second kind B n,k meets B n,k x + y, x 2 + y 2,..., x n k+ + y n k+ n l r+sk l+mn Lemma 2.2 [7, p. 35]. For n k, we have B l,r x, x 2,..., x l r+ B m,s y, y 2,..., y m s+. 2. B n,k abx, ab 2 x 2,..., ab n k+ x n k+ a k b n B n,k x, x n,..., x n k+, 2.2 where a and b are any complex numbers. Lemma 2.3 [3, 39]. For n k, we have B n,k 2, 3,..., n k + 2 n + k! k n + k k i Sn + i, i. 2.3 k i Lemma 2.4. Let ft + k a kt k and gt + k b kt k be formal power series such that ftgt. Then a a 2 a b n n a 3 a 2 a a n a n 2 a n 3 a n 4 a n a n a n 2 a n 3 a Proof. The identity ftgt entails the matrix identity b a b 2 b a 2 a b n b n b n 2 a n a n a n 2 where stands for the inverse of an invertible matrix. Applying Cramer s rule for a system of linear equations proves Lemma 2.4. i 3. Proofs of Theorems. and.2 We are now in a position to prove our main results. Proof of Theorem.. In terms of the Bell polynomials of the second kind B n,k, the Faà di Bruno formula for computing higher order derivatives of composite functions is described in [7, p. 39, Theorem C] by d n d x n f gx n k f k gxb n,k g x, g x,..., g n k+ x. 3.,

4 4 F. QI AND R. J. CHAPMAN By the integral expression., applying the formula 3. to the functions fy y and y gx ext u d t results in d n xe ux d x n e x n k dn d x n ext u d t t ue xt u d t, k k! ext u d t k+ B n,k t u 2 e xt u d t,..., n k k!b n,k t u d t, k k t u n k+ e xt u d t t u 2 d t,..., t u n k+ d t n u k 2 u 2 k!b n,k, u3 u 3,..., un k+2 u n k n k + 2 as x. Further employing 2., 2.2, and 2.3 acquires d n xe ux d x n e x x k n k k! k r+sk l+mn u 2 B l,r, 2 B m,s r+sk l+mn n l u3,..., 3 ul r+2 l r + 2 u2, u3,..., um s m s + 2 n n k k! l u l+r B l,r 2, 3,..., u s u m B m,s 2, 3,..., m s + 2 n n l! k! m m! l l + r! m + s! k r+sk l+mn [ r s l + r m + s i+j r i s j i j Sl + i, ism + j, j l r + 2 ] u m+s u l+r. As a result, the formula.2 follows immediately. Letting u in.2, simplifying, and interchanging the order of sums lead to the formula.3. The proof of Theorem. is complete.

5 TWO CLOSED FORM FOR THE BERNOULLI POLYNOMIALS 5 The first proof of Theorem.2. Let u uz and v vz be differentiable functions. In [3, p. 4], the formula u v... d k u v v... u d z k k u v 2v... v v k u k v k k v k 2... v u k v k k v k... v for the kth derivative of the ratio uz vz was listed. For easy understanding and convenient availability, we now reformulate the formula 3.2 as where the matrices and satisfy d k d z k u v k k k v k+ A k+ B k+ k k+ k+, 3.3 A k+ a l, l k B k+ k b l,m l k, m k a l, u l z and b l,m l v l m z m under the conventions that v z vz and that p q and v p q z for p < q. See also [26, Section 2.2] and [38, Lemma 2.]. By the integral expression., applying the formula 3.3 to uz and vz ezt u d t yields a,, a l, for l >, l b l,m t u l m e zt u d t m l t u l m d t, z m u l m+ u l m+ l m l m + for l k and m k with l m, and d k ze uz d z k e z k b k+ b l,m l k, m k, l k t u l m d t m, z l k, m k l u k l m+ u l m+ m l m +. l k, m k The formula.4 is proved. The formula.5 follows readily from taking u in.4. The first proof of Theorem.2 is complete.

6 6 F. QI AND R. J. CHAPMAN The second proof of Theorem.2. Applying Lemma 2.4 to gt teut e t reveals that b n Bnu and a n un+ u n+. Hence, by virtue of Lemma 2.4, B n u n ul m+ u l m+. l n, m n l m+! and ft e ut e ut t Multiplying the row l of this determinant by l! and dividing the row m by m! gives B n u n l + [ u l m+ u l m+] l + m. l n, m n The formula.4 is thus proved. The second proof of Theorem.2 is complete. 4. Remarks and comparisons In this final section, we will remark on our main results and compare them with some known conclusions. Remark 4.. The formula.3 recovers the one appeared in [9, p. 48, ], [3, 6], [9, p. 59], and [35, p. 4]. For detailed infirmation, please refer to [3, Remark 4]. There are also some other formulas and inequalities for the Bernoulli numbers and polynomials in [,, 6, 8, 24, 25, 27, 29, 33] and references cited therein. Hence, Theorems. and.2 generalize those corresponding results obtained in these references. Remark 4.2. Motivated by the idea in [3, 32], Guo and Qi generalized in [5] the Bernoulli polynomials and numbers. Hereafter, some papers such as [2, 22] were published. Remark 4.3. The special values of the Bell polynomials of the second kind B n,k are important in combinatorics and number theory. Recently, some special values for B n,k were discovered and applied in [4, 26, 34, 39]. Remark 4.4. In [4, 2], several different approaches to the theory of Bernoulli polynomials B k u were surveyed. However, there is no any conclusion directly related to Theorems. and.2. Let {a n } n be a sequence of complex numbers and let {D n a k } n be a sequence of determinants such that D a k and a a a 2 a a D n a k , n N. a n a n 2 a n 3 a a n a n a n 2 a In [37], two identities and B 2n n 2 2 for n N were established. { n l l 2l! D n l 2k +! + D n B 2n n n D n 2k +! } 4. 2k +! 4.2

7 TWO CLOSED FORM FOR THE BERNOULLI POLYNOMIALS 7 In [8], six approaches to the theory of Bernoulli polynomials were mentioned. Mainly, a determinantal approach was introduced in [8] by defining B x and B n x n n! As a result, the Bernoulli numbers B n n n! x x 2 x 3 x n x n n n+ 2 3 n n 3 2 n n, n N n n n 2 n n n+ 2 3 n n 3 2 n n, n N n In [6, Theorem.], it was obtained that, if Az n a nz n and Bz n b nz n are the ordinary generating functions of {a n } n and {b n } n such that AxBx, then a and b n n D na k. Therefore, Lemma 2.4 is a special case of [6, Theorem.]. In the paper [6], a n+ as applications of [6, Theorem.], some properties of D n a k were discovered and applied to give an elegant proof of 4. and 4.2, and to express the Genocchi numbers, the tangent numbers, higher order Bernoulli numbers, the Stirling numbers of the first and second kinds, the harmonic numbers, higher order Euler numbers, higher order Bernoulli numbers of the second kind, and so on, in terms of D n a k. Especially, the formulas which recovers [5, Eq.4], and were derived. In [2], it was mentioned that the formula B n n n +! n n 2 n 2 B n n D n, k +! k B n D n k +! n+ n+ n+ 2 n+ 3 n+ n

8 8 F. QI AND R. J. CHAPMAN was traced back to the book [36]. The Bernoulli polynomials B n x were represented in [2] as x! 2! x B n x n 2 2! 3! 2! x 3 3! 4! 3! 2! x n n+! n! n 2! n 3! x 2! n n k k! x 2 2! 3! 2! x 3 3! 3! 4! 2! 3! x n n+! n! n 2! n 3! n x 2! n k! x 2 2! 3! 2! k! k x 3 3! 3! 3! 4! 2! 2!. x n n+! n! n 2!2! n 3!3! 2!n 2! Similar to the above representations for the Bernoulli polynomials B n x, some determinantal expressions for the hypergeometric Bernoulli polynomials were further presented in [2]. Remark 4.5. The idea of Lemma 2.4 was used in [23, pp ] to express determinants of complete symmetric functions in terms of determinants of elementary symmetric functions. Remark 4.6. This manuscript is a revision and extension of the first two versions of the preprint [28]. Acknowledgements. This manuscript was drafted while the first author visited Kyungpook National University and Hannam University between 25 3 May 25 to attend the 25 Conference and Special Seminar of the Jangjeon Mathematical Society. The first author appreciates Professors Dmitry V. Dolgy and Taekyun Kim at Kwangwoon University, Byung-Mun Kim at Dongguk University, Dae San Kim at Sogang University, Seog-Hoon Rim at Kyungpook National University, Jong Jin Seo at Pukyong National University, and many others for their invitation, support, and hospitality. The first author thanks Professor James F. Peters at University of Manitoba for his recommendation of the reference [] on 26 May 25 through the ResearchGate website and thanks Professor Hieu D. Nguyen at Rowan University for his providing the reference [2] on June 25. The authors appreciate the anonymous referee for his/her careful corrections to and valuable comments on the original version of this paper. References [] A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, O. Mallet, Bell polynomials in combinatorial Hopf algebras, available online at [2] R. Booth and H. D. Nguyen, Bernoulli polynomials and Pascal s square, Fibonacci Quart. 46/47 28/29, no.,

9 TWO CLOSED FORM FOR THE BERNOULLI POLYNOMIALS 9 [3] N. Bourbaki, Functions of a Real Variable, Elementary Theory, Translated from the 976 French original by Philip Spain. Elements of Mathematics Berlin. Springer-Verlag, Berlin, 24; Available online at http: //dx.doi.org/.7/ [4] D. Callan, Letter to the editor: A new approach to Bernoulli polynomials [Amer. Math. Monthly , no., 95 9] by D. H. Lehmer, Amer. Math. Monthly , no. 6, 5 5; Available online at [5] H.-W. Chen, Bernoulli numbers via determinants, Internat. J. Math. Ed. Sci. Tech , no. 2, ; Available online at [6] K.-W. Chen, Inversion of generating functions using determinants, J. Integer Seq. 27, no., Article 7..5, pages. [7] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 974. [8] F. Costabile, F. Dell Accio, M. I. Gualtieri, A new approach to Bernoulli polynomials Rend. Mat. Appl , no., 2. [9] H. W. Gould, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly , 44 5; Available online at [] B.-N. Guo, I. Mező, and F. Qi, An explicit formula for the Bernoulli polynomials in terms of the r-stirling numbers of the second kind, available online at Accepted by Rocky Mountain J. Math. on May 27, 25. [] 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 25, no., 33 36; Available online at v3i.468. [2] B.-N. Guo and F. Qi, A simple proof of logarithmic convexity of extended mean values, Numer. Algorithms 52 29, no., 89 92; Available online at [3] B.-N. Guo and F. Qi, An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind, J. Anal. Number Theory 3 25, no., 27 3; Available online at [4] B.-N. Guo and F. Qi, Explicit formulas for special values of the Bell polynomials of the second kind and the Euler numbers, ResearchGate Technical Report, available online at [5] B.-N. Guo and F. Qi, Generalization of Bernoulli polynomials, Internat. J. Math. Ed. Sci. Tech , no. 3, ; Available online at [6] B.-N. Guo and F. Qi, Some identities and an explicit formula for Bernoulli and Stirling numbers, J. Comput. Appl. Math , ; Available online at [7] B.-N. Guo and F. Qi, The function b x a x /x: Logarithmic convexity and applications to extended mean values, Filomat 25 2, no. 4, 63 73; Available online at [8] S.-L. Guo and F. Qi, Recursion formulae for n m mk, Z. Anal. Anwendungen 8 999, no. 4, 23 3; Available online at [9] S. Jeong, M.-S. Kim, and J.-W. Son, On explicit formulae for Bernoulli numbers and their counterparts in positive characteristic, J. Number Theory 3 25, no., 53 68; Available online at 6/j.jnt [2] D. H. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly , no., 95 9; Available online at [2] Q.-M. Luo, B.-N. Guo, F. Qi, and L. Debnath, Generalizations of Bernoulli numbers and polynomials, Int. J. Math. Math. Sci , no. 59, ; Available online at S [22] Q.-M. Luo and F. Qi, Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials, Adv. Stud. Contemp. Math. Kyungshang 7 23, no., 8. [23] I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., With contributions by A. Zelevinsky, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 995. [24] F. Qi, A double inequality for ratios of the Bernoulli numbers, ResearchGate Dataset, available online at [25] F. Qi, A new formula for the Bernoulli numbers of the second kind in terms of the Stirling numbers of the first kind, available online at Accepted by Publ. Inst. Math. Beograd N.S. on June, 25.

10 F. QI AND R. J. CHAPMAN [26] F. Qi, Derivatives of tangent function and tangent numbers, Appl. Math. Comput. 25, in press; Available online at [27] F. Qi, Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind, Filomat 28 24, no. 2, ; Available online at [28] F. Qi, Two closed forms for the Bernoulli polynomials, available online at [29] F. Qi and B.-N. Guo, Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers, Analysis Berlin 34 24, no. 3, 3 37; Available online at [3] F. Qi, Q.-M. Luo, and B.-N. Guo, The function b x a x /x: Ratio s properties, In: Analytic Number Theory, Approximation Theory, and Special Functions, G. V. Milovanović and M. Th. Rassias Eds, Springer, 24, pp ; Available online at [3] F. Qi and S.-L. Xu, Refinements and extensions of an inequality, II, J. Math. Anal. Appl , no. 2, 66 62; Available online at [32] F. Qi and S.-L. Xu, The function b x a x /x: Inequalities and properties, Proc. Amer. Math. Soc , no., ; Available online at [33] F. Qi and X.-J. Zhang, An integral representation, some inequalities, and complete monotonicity of the Bernoulli numbers of the second kind, Bull. Korean Math. Soc , no. 3, ; Available online at http: //dx.doi.org/.434/bkms [34] F. Qi and M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput , ; Available online at [35] S. Shirai and K.-I. Sato, Some identities involving Bernoulli and Stirling numbers, J. Number Theory 9 2, no., 3 42; Available online at [36] H. W. Turnbull, The Theory of Determinants, Matrices, and Invariants, 3rd ed., Dover Publications, New York, 96. [37] R. Van Malderen, Non-recursive expressions for even-index Bernoulli numbers: A remarkable sequence of determinants, available online at [38] C.-F. Wei and F. Qi, Several closed expressions for the Euler numbers, J. Inequal. Appl. 25, 25:29, 8 pages; Available online at [39] Z.-Z. Zhang and J.-Z. Yang, Notes on some identities related to the partial Bell polynomials, Tamsui Oxf. J. Inf. Math. Sci , no., Qi Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454, China; College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 2843, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 3387, China address: qifeng68@gmail.com, qifeng68@hotmail.com, qifeng68@qq.com URL: Chapman Mathematics Research Institute, University of Exeter, United Kingdom address: R.J.Chapman@exeter.ac.uk, rjc@maths.ex.ac.uk URL: