IT0100106 ENTE PER LE NUOVE TECNOLOGIE, L'ENERGIA E L'AMBIENTE Serie Innovazione BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW GIUSEPPE. DATTOLI ENEA - Divisione Fisica Applicata Centra Ricerche Frascati, Roma CLEMENTE CESARANO ULM University, Department of Matematics, ULM, Germany SILVERIA LORENZUTTA ENEA -P Divisione Fisica Applicata Centra Ricerche "Ezio Clementel", Bologna RT/INN/2000/11 Questo rapporto ё stato preparato da: Servizio Edizioni Scientifiche - ENEA, Centro Ricerche Frascati, C.P. 65 - 00044 Frascati, Roma, Italia I contenuti tecnico-scientifici dei rapporti tecnici dell'ENEA rispecchiano I'opinione degli autori e non necessariamente quella dell'Ente. ISSN/1120-5571 BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW Riassunto Si applica il metodo della funzione generatrice per introdurre nuove forme di numeri e polinomi di Bernoulli che vengono utilizzati per sviluppare e per calcolare somme parziali che coinvolgono polinomi a piu indici ed a piu variabili. Si sviluppano considerazioni analoghe per i polinomi ed i numeri di Eulero. BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW Abstract We apply the method of generating function, to introduce new forms of Bernoulli numbers and polynomials, which are exploited to derive further classes of partial sums involving generalized many index many variable polynomials. Analogous considerations are develo- ped for the Euler numbers and polynomials. Key words: numeri di Bernoulli, polinomi di Bernoulli, polinomi di Hermite, polinomi di AppEl, somme parziali INDICE 1 -INTRODUCTION p. 7 2 - FINITE SUMS AND NEW CLASSES OF BERNOULLI NUMBERS p. 9 3 - EULER POLYNOMIALS p. 11 4 - CONCLUDING REMARKS p. 12 APPENDIX p. 15 REFERENCES p. 17 BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW 1 - INTRODUCTION In a previous paper [1] we have derived partial sums involving Hermite, Laguerre and Appell polynomials in terms of generalized Bernoulli polynomials. The new and interesting possibilities offered by this class of polynomials are better illustrated by an example relevant to the derivations of a partial sum involving two-index Hermite polynomials. To this aim we remind that: N-l (1) n=0 where Bn(x) are Bernoulli polynomials defined by the generating function: TBnOO (2) or in terms of Bernoulli numbers Bn as: Hermite polynomials with two variables and one parameter can be defined by means of the operational identity [2]: (4) and the hm n(x,y|x) are defined by the double sum: min(m,n) s m-s n-s l hmn(x,y|x) = m!n! У — (5) m ' stj s!(m-s)!(n-s)! The identity (4) can be used to state that: (6) and to introduce the following two variable one parameter Bernoulli polynomials: B hBr,s(x,y|x)= J r-qBs-khq,k(x,y|x) . (7) It is easy to note that from Eq. (4) we get: ЭхЭ e У{Вг(х)В5(у)}=ьВГ)8(х,у|х). (8) This new class of Bernoulli polynomials can be used to derive the following partial sum: M-1N-1 yrws (I m=0n=0 X z r> . , X Z В (9) w — - r M + -,— — • ywj ? . у w ywj which is a consequence of Eqs. (1,6,7,8). The generating function of this last class of Bernoulli polynomials can be shown to be provided by: ux+vy+xuv о» m n uve (10) This introductory examples shows that a wealth of implications is offered by the use of generalized forms of Bernoulli polynomials. In the forthcoming sections we will develop a more systematic analysis which yields a deeper insight into the effectiveness of this type of generalizations. 2 - FINITE SUMS AND NEW CLASSES OF BERNOULLI NUMBERS In Ref. [1] we have touched on the following new class of numbers: 21 '".f n!Bn_2sBs which are recognized as an Hermite convolution of Bernoulli numbsers on themselves. The generating functions of the HBn is provided by: 15 -z.-VH-n; (12) •1)(е1 -: which suggests the following generalization t? ~ (eat-l)(ebt2-l) n=0xi: with: 1 B"(a,b" ) = — У n ab ^ s!(n-2s)! b S—\J It is also evident that the generating function: 2 3 xt+yt tn / . 2 (eat _ 1)(ebt _ 1} " —(нВп(а,Ь|х,у)) (15) can be exploited to define the polynomials: HB*n(a,b|x,y)= ip(HB;_s(a,b)Hs(X)y)) (16) as also results from Eqs. (13) and (15) which are also expressed as: 10 [n 2] n 2s s * Пn! ^ n!a - b нцВ пn(а,v Ь i х,у,yj) = — (17) ab % s!(n-2s)! S—KJ yielding: HBn(a,b|0,0)=HBn(a,b) (18) Гп /91 V,s.,n-2 X s are the Kam de where Hn(x,y) = п!1^=0 g!(n-2s)! P^ Feriet polynomials. The use of polynomials (16) is suggested by partial sums of the type: M-lN-l 1 (r + l)(r + 2)(r + 3) m=0n=0 B B w x z Nw B ( W X ~HH r+r+33 (У' I ' + ) +H r+r+33 У' I ' A further application of polynomials HBn(a,b|x,y) is relevant to the multiplication theorems. We find indeed: n 2 h 1 B (mx, 2 N m ~ VVY „• H n m в vl = -X X н m TIT nr (20) m m^ and m-lm —1, т,-1 u i ak bh HBn a,b|x + — ,y + -j (21) k=l h=l m and: p-1 q-1, к h — 1 V 1 '% 1 (22) k=0 h=C pq can be proved by exploiting the procedure outlined in Appendix. 11 3 - EULER POLYNOMIALS The examples we have provided yields an idea of the implications offered by this type of generalization. It is also evident that the considerations we have developed for Bernoulli polynomials can be extended to Euler polynomials [3]: In analogy of Ref. [1] and the results of previous sections, we introduce the following classes of Euler polynomials: 2 2 xt+yt ~ tn ХМЕ^» (24) and: m n It is easily realized that: ySE2(X) (26) s=0 s!(n-2s)! and: m n /'щЛ/'^Л E A t c l:: I1 A l А1 h 'm>nv >.yi / ~ Zu 2-Л m-s 'n-r s,rv >.y I / \ ) s r s=o r=o^ A y where En(x) are the ordinary Euler polynomials and that the following theorems hold: : Z w) 28 _ s ...... , s^ ' ( > and (see appendix): 12 m-1 / / , \\ n k HEn(mx,py) = m X (~l) l HEn^ + ->^TJJ • (29) As to the polynomials hEm^C^yl't) wa can also state that: s r Em_sn_r(x,y|t))z w (30) and: Цx k=Oh=O pq 4 - CONCLUDING REMARKS The introduction of the Hermite-Euler polynomials given by Eq. (24) offers the possibility of speculating about alternative definitions as e.g. (t < л/тг): (32) (е1+1)(ег +1) n=0n- The polynomials Е„ (х,у) are defined as: -2s(X) (33) s! n 2s ! s=o ( - > - and are shown to satisfy the following differential equation: ( 2) ( 2) |-E n (x,y) = ^TE n (x,y) (34) Эу Эх and the multiplication theorem: m-lm -1 ( f л и \Л 2 Р "]Г J (-1)^^(2)^1^^ (35) 13 The more general theorem relevant to Е„ '(mx,py) requires the introduction of a class of Euler polynomials analogous to those provided by Eq. (13) for the Bernoulli case. Before concluding this paper we want to emphasize that most of the identities holding for the ordinary Bernoulli or Euler polynomials can be extended to the generalized case. For example the identity [4]: n 1 Bn(x + 1)-Bn(x) = nx - (36) can be generalized as: HBn(x + l,y)-HBn(x,y) = nHn_1(x,y) (37) which is a consequence of: y 2 e 3x Bn(x)=HBn(x,y). (38) For analogous reasons, we find: hBm,n(x + l,y + l|T)-hBm>n(x,y + l|T)-hBm>n(x + l,y|T) (39) B +h m,n<Xy IT) = mnhm_1>n_1(x,y I T) and: HB* (a,b | x + а,у + b) -H B^(a,b | x,y) = n(n - l)(n - 2)Hn_3(x,y) (40) and ( 2) ( 2) HE n (x + l,y)+HE n (x,y) = 2Hn(x,y). (41) We can also define the further generalized form: thus getting (n>N): 14 ({}|{j)({}|{}) ^J({}) (43) with: 2: Xit ^({}) e i=i . (44) n=0 П' The results if this paper show that the combination of operational rules and the properties of ordinary and generalized polynomials offer a wealth of possibilities to introduce new familes of Euler and Bernoulli polynomials which provides a powerful tool in applications. 15 APPENDIX The multiplication theorems are easily stated by exploiting the method of the generating functions and suitable manipulations. We note indeed that: m n (h-^m,n(Px'qy|'r) = • (A.I) m=Un=O m" П' (e ~*№ -1) The r.h.s. of the above relation can be more conveniently rewritten as: (eu-l)(ev-l) pq (euP - by nothing that: up , p-l e -1 r=0 we can rearrange (A.2) as: uveuPx+vqy+Tuv P-l Я-1 } ~ - (up)m (vq)n (A.4) X P^ pq which once confronted with (A.I) yields Eq. (22). A symilar procedure can be exploited to prove the multiplication theorems relevant to the Euler's generalized forms. We note indeed: 00 tn \ 2emxt+pyt2 X ~7 (н En(mx' РУ))= 1 (A-5) n=0n! e +1 and handling the r.h.s.