Guo, Bai-Ni

Luo, Qiu-Ming

Qi, Feng

English

Victoria University Institutional Repository (VUIR)

Generalizations of Bernoulli's Numbers and Polynomials

Name not available

Victoria University Eprints Repository

GENERALIZATIONS OF BERNOULLI’S NUMBERS ANDPOLYNOMIALSQIU-MING LUO, BAI-NI GUO, AND FENG QIAbstract. In this paper, the concepts of Bernoulli numbers Bn, Bernoullipolynomials Bn(x), the generalized Bernoulli numbers Bn(a, b) are furthergeneralized to one which is called as the generalized Bernoulli polynomials,and many basic properties and some relationships between Bn, Bn(x) andBn(a, b) are established.1. IntroductionIt is well known that Bernoulli’s numbers and polynomials play important rolesin mathematics, they are main objects in the Theory of Special Functions [4]. Theirdefinitions can be given as follows.Definition 1. The numbers Bn, 0 ≤ n ≤ ∞, are called Bernoulli numbers ifφ(t) =tet − 1 =∞∑n=0Bnn!tn, |t| < 2pi. (1)Definition 2. The functions Bn(x), 0 ≤ n ≤ ∞, are called Bernoulli polynomialsif they satisfyφ(x; t) =textet − 1 =∞∑n=0Bn(x)n!tn, |t| < 2pi, x ∈ R. (2)The usual definition of Bernoulli polynomials istσeut(et − 1)σ =∞∑n=0Bσn(u)n!tn, |t| < 2pi. (3)2000 Mathematics Subject Classification. Primary 11B68, 33E20; Secondary 26A48, 40A30.Key words and phrases. Bernoulli numbers, Bernoulli polynomials, generalized Bernoulli num-bers, generalized Bernoulli polynomials.The authors were supported in part by NNSF (#10001016) of China, SF for the ProminentYouth of Henan Province, SF of Henan Innovation Talents at Universities, NSF of Henan Province(#004051800), SF for Pure Research of Natural Science of the Education Department of HenanProvince (#1999110004), Doctor Fund of Jiaozuo Institute of Technology, China.This paper was typeset using AMS-LATEX.12 Q.-M. LUO, B.-N. GUO, AND F. QIIn [2], the last two authors generalized the concept of Bernoulli numbers asfollows.Definition 3. Let a, b > 0 and a 6= b. The generalized Bernoulli numbers Bn(a, b)are defined byφ(t; a, b) =tbt − at =∞∑n=0Bn(a, b)n!tn, |t| < 2pi|ln b− ln a| . (4)Among other things, some basic properties and relationships between Bn, Bn(x)and Bn(a, b) were also studied in [2] initially and originally.In this article, we first give the following definition of the generalized Bernoullipolynomials, which generalizes the concepts stated above, and then research theirbasic properties and relationships to Bernoulli numbers Bn, Bernoulli polynomialsBn(x) and the generalized Bernoulli numbers Bn(a, b).2. Definitions and Properties of Generalized Bernoulli PolynomialsIt is easy to see that the following definition is natural and essential generaliza-tions of the concepts of Bernoulli numbers Bn, Bernoulli polynomials Bn(x) andthe generalized Bernoulli numbers Bn(a, b).Definition 4. Let a, b, c > 0 and a 6= b. The generalized Bernoulli polynomialsBn(x; a, b, c) for nonnegative integer n are defined byφ(x; t; a, b, c) =tcxtbt − at =∞∑n=0Bn(x; a, b, c)n!tn, |t| < 2pi|ln b− ln a| , x ∈ R. (5)The generalized Bernoulli polynomials Bn(x; a, b, c) have the following proper-ties, which are stated as theorems below.Theorem 1. Let a, b, c > 0 and a 6= b. For x ∈ R and n ≥ 0, we haveBn(x; 1, e, e) = Bn(x), Bn(0; a, b, c) = Bn(a, b), Bn(0; 1, e, e) = Bn,Bn(x; a, b, 1) = Bn(a, b), Bn(x; 1, e, 1) = Bn,(6)Bn(x; a, b, c) =n∑k=0(nk)[ln c]n−kBk(a, b)xx−k, (7)Bn(x; a, b, c) =n∑k=0(nk)[ln c]n−k[ln b− ln a]k−1Bk(ln aln a− ln b)xx−k, (8)Bn(x; a, b, c) =n∑k=0k∑j=0(−1)k−j(nk)(kj)[ln c]n−k[ln a]k−j[lnba]j−1Bjxx−k. (9)GENERALIZATIONS OF BERNOULLI’S NUMBERS AND POLYNOMIALS 3Proof. Applying Definition 3 to the term tbt−at and expanding the exponentialfunction cxt at t = 0 yieldstcxtbt − at =( ∞∑k=0Bk(a, b)k!tk)( ∞∑i=0xi(ln c)ii!ti)=∞∑k=0k∑i=0(ln c)k−ii!(k − i)!Bi(a, b)xk−itk=∞∑n=0(n∑k=0(nk)(ln c)n−kBk(a, b)xn−k)tnn!.(10)Combining (10) and (5) and equating their coefficients of tn produces formula (7).The following two formulae were provided in [2]:Bn(a, b) = (ln b− ln a)n−1Bn(ln aln a− ln b), (11)Bn(a, b) =n∑i=0(−1)n−i(ln b− ln a)i−1(ln a)n−i(ni)Bi. (12)Substituting (11) and (12) into (7) leads to (8) and (9).Formulae in (6) are obvious. Now we give some results about derivatives and integrals of the generalizedBernoulli polynomials Bn(x; a, b, c) as follows.Theorem 2. Let a, b, c > 0, a 6= b, n ≥ 0, and x ∈ R. For any nonnegative integer` and real numbers α and β, we have∂`Bn(x; a, b, c)∂x`=n!(n− `)! (ln c)`Bn−`(x; a, b, c), (13)∫ βαBn(t; a, b, c) dt =1(n+ 1) ln c[Bn+1(β; a, b, c)−Bn+1(α; a, b, c)], (14)where B0(x; a, b, c) = 1ln b−ln a .Proof. Formula (13) follows from standard arguments and induction.Integrating on both sides of (13) with respect to variable x for ` = 1 turns outformula (14). Theorem 3. Let a, b, c > 0, a 6= b, n ≥ 0 and x ∈ R. ThenBn(x+ 1; a, b, c) =n∑k=0(nk)(ln c)n−kBk(x; a, b, c), (15)Bn(x+ 1; a, b, c) = Bn(x;ac,bc, c), (16)4 Q.-M. LUO, B.-N. GUO, AND F. QIand, for m ≥ 2,Bm(x+ 1; a, b, c) = Bm(x; a, b, c) +m(ln c)m−1xm−1+m−1∑k=0(mk)[(ln a)m−k − (ln b)m−k + (ln c)m−k]Bk(x; a, b, c). (17)Proof. By definition of the generalized Bernoulli polynomials, we havetc(x+1)tbt − at =∞∑n=0Bn(x+ 1; a, b, c)n!tn (18)andtc(x+1)tbt − at =tcxtbt − at · ct=( ∞∑n=0Bn(x; a, b, c)n!tn)( ∞∑k=0(ln c)kk!tk)=∞∑n=0∑nk=0(nk)(ln c)n−kBk(x; a, b, c)n!tn.(19)Combining (18) and (19) and equating their coefficients of tn leads to formula (15).Similarly, sincetc(x+1)tbt − at =tcxt(bc)t − (ac )t =∞∑n=0Bn(x; ac ,bc , c)n!tn, (20)equating the coefficients of tn in (18) and (20) leads to formula (16).Straightforward computation gives us thattc(x+1)tbt − at = tcxt +tcxt(at − bt + ct)bt − at=∞∑n=0(ln c)nxnn!tn+1+( ∞∑n=0Bn(x; a, b, c)n!tn)( ∞∑`=0[(ln a)` − (ln b)` + (ln c)`]`!t`)=∞∑n=0(ln c)nxnn!tn+1+∞∑n=0[n∑`=0(n`)[(ln a)n−` − (ln b)n−`+(ln c)n−`]B`(x; a, b, c)] tnn!(21)GENERALIZATIONS OF BERNOULLI’S NUMBERS AND POLYNOMIALS 5= B0(x; a, b, c) +[1 +B1(x; a, b, c) +B0(x; a, b, c)(ln a− ln b+ ln c)]t+∞∑n=2[n(ln c)n−1xn−1 +Bn(x; a, b, c)] tnn!+∞∑n=2{n−1∑`=0(n`)[(ln a)n−` − (ln b)n−` + (ln c)n−`]B`(x; a, b, c)} tnn!.Equating (5) and (21) yields (17). Corollary 1. For n ≥ 1, b > 0 and x ∈ R, we haveBn(x+ 1; 1, b, b) = Bn(x; 1, b, b) + n(ln b)n−1xn−1. (22)Remark 1. If taking b = e in (22), the following well known result is deducedBn(x+ 1) = Bn(x) + nxn−1, n ≥ 1. (23)Similarly, from (13) it follows thatB′i(t) = iBi−1(t), B0(t) = 1. (24)Actually, the Bernoulli polynomials Bi(t), i ∈ N, are uniquely determined by for-mulae (23) and (24), see [1, 23.1.5 and 23.1.6] or [4].Theorem 4. Let a, b, c > 0, a 6= b, n ≥ 0, and x ∈ R. ThenBn(1− x; a, b, c) = (−1)nBn(x;cb,ca, c)(25)= Bn(−x; ac,bc,1c), (26)Bn(x+ y; a, b, c) =n∑k=0(nk)(ln c)n−kBk(x; a, b, c)yn−k (27)=n∑k=0(nk)(ln c)n−kBk(y; a, b, c)xn−k. (28)Proof. From Definition 5, it follows thattc(1−x)tbt − at =∞∑n=0Bn(1− x; a, b, c)n!tn. (29)Meanwhile, we havetc(1−x)tbt − at =tc−xt(bc)t − (ac )t =∞∑n=0Bn(−x; ac , bc , c)n!tn, (30)tc(1−x)tbt − at =−tcx(−t)(ca)−t − ( cb)−t =∞∑n=0(−1)nBn(x; cb ,ca , c)n!tn. (31)6 Q.-M. LUO, B.-N. GUO, AND F. QITherefore, formulae (25) and (26) follow from equating series-expansions in (29),(30) and (31).Similarly, we havetc(x+y)tbt − at =∞∑n=0Bn(x+ y; a, b, c)n!tn, (32)tc(x+y)tbt − at =tcxtbt − at · cyt=( ∞∑n=0Bn(x; a, b, c)n!tn)( ∞∑i=0yi(ln c)ii!ti)=∞∑n=0(n∑k=0(nk)yn−k(ln c)n−kBk(x; a, b, c))tnn!,(33)tc(x+y)tbt − at =tcytbt − at · cxt=∞∑n=0(n∑k=0(nk)xn−k(ln c)n−kBk(y; a, b, c))tnn!.(34)Hence, formulae (27) and (28) follow from equating series-expansions in (32), (33)and (34). The proof is complete. Theorem 5. Let m,n be natural numbers. Then, for any positive number b, thefollowing identity holds:m∑j=1jn =1(n+ 1)(ln b)n[Bn+1(m+ 1; 1, b, b)−Bn+1(0; 1, b, b)]=1(n+ 1)(ln b)n[Bn+1(m+ 1; 1, b, b)−Bn+1(1; 1, b, b)].(35)Proof. Rewritting formula (22) yieldsxn−1 =1n(ln b)n−1[Bn(x+ 1; 1, b, b)−Bn(x; 1, b, b)], (36)which impliesjn =1(n+ 1)(ln b)n[Bn+1(j + 1; 1, b, b)−Bn+1(j; 1, b, b)]. (37)Summing up on both sides of (37) from 0 to m or from 1 to m with respect to jleads to formula (35) easily. Remark 2. The calculation of values of∑mj=1 jn is an interesting problem, andthere has been a rich literature, for example [3].GENERALIZATIONS OF BERNOULLI’S NUMBERS AND POLYNOMIALS 7Remark 3. At last, it is pointed out that the Bernoulli’s and Euler’s numbers andthe Bernoulli’s and Euler’s polynomoals can be further generalized to more generalresults in this manner. These conclusions will be published in some subsequentpapers.Acknowledgements. This paper was finalized while the third author’s visit to theRGMIA between November 1, 2001 and January 31, 2002, as a Visiting Professorwith grants from the Victoria University of Technology and Jiaozuo Institute ofTechnology.References[1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tebles, National Bureau of Standards, Applied Mathematics Series55, 4th printing, Washington, 1965.[2] B.-N. Guo and F. Qi, Generalisation of Bernoulli polynomials, Internat. J. Math. Ed. Sci.Tech. (2002), in the press. RGMIA Res. Rep. Coll. 4 (2001), no. 4, Art. 10, 591–595. Availableonline at http://rgmia.vu.edu.au/v4n4.html.[3] S.-L. Guo and F. Qi, Recursion formulae for∑nm=0mk, Z. Anal. Anwendungen 18 (1999),no. 4, 1123–1130.[4] Zh.-X. Wang and D.-R. Guo, Te`shu¯ Ha´nshu` Ga`ilu`n (Introduction to Special Function), TheSeries of Advanced Physics of Peking University, Peking University Press, Beijing, China,2000. (Chinese)(Luo) Department of Broadcast-Television-Teaching, Jiaozuo University, JiaozuoCity, Henan 454002, ChinaE-mail address: luoqm236@sohu.com(Guo) Department of Applied Mathematics and Informatics, Jiaozuo Institute ofTechnology, Jiaozuo City, Henan 454000, ChinaE-mail address: guobaini@jzit.edu.cn(Qi) Department of Applied Mathematics and Informatics, Jiaozuo Institute of Tech-nology, Jiaozuo City, Henan 454000, ChinaE-mail address: qifeng@jzit.edu.cn or qifeng618@hotmail.comURL: http://rgmia.vu.edu.au/qi.html

Generalisation of Bernoulli polynomials,

Recursion formulae for Pn m=0 mk,

T` esh¯ u H´ ansh` u G` ail` un (Introduction to Special Function), The Series of Advanced Physics

Generalizations of Bernoulli's Numbers and Polynomials

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