Bernoulli Numbers and Polynomials from a More General Point of View

ENTE PER LE NUOVE TECNOLOGIE, L’ENERGIA E L’AMBIENTE

Serie Innovazione

I BERNOULLI NUMBERS AND POLYNOMIALS FROM A MORE GENERAL POINT OF VIEW

GIUSEPPE. DATTOLI ENEA - Divisione Fisica Applicata Centro Ricerche Frascati, Roma

CLEMENTE CESARANO ULM University, Department of Mathematics, ULM, Germany

SILVERIA LORENZUTTA ENEA -P Divisione Fisica Applicata Centro Ricerche “Ezio Clementel”, Bologna

RT/1NN/2ooo/11 ,

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Riassunto Si applica il metodo dells 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 piti indici ed a piii 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 ciasses of partial sums involving generalized many index many variable polynomials. Analogous considerations are developed for the Euler numbers and polynomials.

Key words: numeri di Bernoulli, polinomi di Bernoulli, polinomi di Hermite, polinomi di AppEl, somme parziali

INDICE

1- ~TRODUCTION ...... 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 App&ll 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 Herrnite polynomials.

To this aim we remind that:

N–1 .ri_ (1) where Bn(x) are Bernoulli polynomials defined by the generating function:

(2) or in terms of Bernoulli numbers Bn as:

Bn(x)= ~ ~ n_,Xs . (3) ‘+ o Hermite polynomials with two variables and one parameter can be defined by means of the operational identity [2]:

32 e=axdy xmYn = hm,n(x,y I~), (4) ‘{ 1 8

and the hrm,n(x,yl~) are defined by the double sum:

min(m,n) s m–s n–s hm,n(x,yl~)=m!n! ~ ~ x y (5) s!(m–s)!(n– s)! SD()

The identity (4) can be used to state that:

#_ e axay(ax+ b)m(cy + d)n} = hm,n(ax + b,cy + d Iac~) , (6) { and to introduce the following two variable one parameter Bernoulli polynomials:

(7)

It is easy to note that from Eq. (4) we get:

dz ez5w{Barbs} ‘h ‘r,s(x, y I‘) (8)

This new class of Bernoulli polynomials can be used to derive the following partial sum:

(9) -hBr+ls+l(;>N+~*)+hBr+3} 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:

Uveux+vy+zuv co ~ m “n

“zzL– (hBm,n(x~Y ~~))r (10) (e” - l)(ev - 1) ~=~n=o ‘! ‘!

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 9

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:

‘n‘2] n!Bn_2~B~ (11) HBn = x SC() s!(n – 2s)! which are recognized as an Hermite convolution of Bernoulli numbsers on themselves. The generating functions of the ~Bn is provided by:

t3 (12) ~2 =~$(HBn) (et - l)(e – 1) n=O “ which suggests the following generalization ((a,b)#O):

t3 (13) = 2 ~(HBX(a,b)) (eat - l)(ebt2 – 1) n=O “ with:

~ [n/2] n!an–2sBn_2sBs ~B~(a,b)= ~ ~ (14) S=() s!(n–2s)! -

It is also evident that the generating function:

t3ext+yt2 = ~ ~(H&(%bl LY)) (15) (eat - l)(eb’2 – 1) n=() . can be exploited to define the polynomials:

HBx(%bIX,Y) = ~ ~ (HB~-s(a,b)Hs(xy)) (16) +) () as also results from Eqs. (13) and (15) which are also expressed as: 10

~f [n/2] ~!an-zsbs #:(a, f+@=; ~Bs; s=Oz s!(n _ 2s)! ‘n-2s ()(1a (l?)

yielding:

H~:(Ab10,0)‘H B~(a,b) (18)

s n–2s Hn(x, Y) = n!~[n/21 Y x where S=O S!(n–ZS)! are the Kamp& de F&iet polynomials. The use elf polynomials (16) is suggested by partial sums of the type:

M-IN–1

Z ~ Hr(x+rnY,z+nw)= 1 m=() n=() (r+ l)(r + 2)(r -t 3)

{HB;+3(Y>w lx+ My, z+ Nw)– H B;+3(Y>WIx + My,z) (19’) ‘HB;+3(y, wjx, z+ Nw)+ ‘ H ‘;+3(3’~W I X, z)} .

A further application of polynomials HB~ (a,b \ x, y) is relevant to the multiplication theorems. We find indeed:

HB.(mx>m2y)=(n+ ‘n-’l)(n + 2, 51ms(HB:+2(lllx+k=() h=() :1Y+$++)) (20)

HB;+2 l,llx+~,y+— -[ ( m :2 )) and

(21) and:

(22) can be proved by exploiting the procedure outlined in Appendix. 11

3- EULER POLYNOMIALS

Theexarnples we have provided yields anideaof theimplications 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]:

(23)

In analogy of Ref. [1] and the results of previous sections, we introduce the following classes of Euler polynomials:

(24)

and:

22eXU+JW+TUV m co urn ~~

‘z z–– (h Em,n(X, Y I ~)) (25) (e” + l)(ev + 1) ~=o ~=o m! n!

It is easily realized that:

‘n ‘2] ysEn Zs(x) ~En(x, y)= ~ sf(n - ~s)l (26) ~=o. —. and:

~Em.(xJ’JT)=~:(~](~)‘rn-sEn-rhs,r(xYl~) (27) where En(x) are the ordinary Euler polynomials and that the following theorems hold:

~En(x + z,Y+w) = ~ n (HEn_~(X>Y))HJZ,W) (28) s+ (1s and (see appendix): 12

@Jmx’’)=mn:;(-’)k(HE”(x+(29)

As to the polynomials @m,n(x,yl~) wa can also state that:

n #n(x+z,y+wl~)=~~ m (30) , )( hEm-s,n-r(x’y ‘))zswr Szor=()(s and:

p–l q–l k h ‘t ~ ~ (-l)k+hhEm “ K+— ,+y —— (31) hEm,n{Px~qY I ~) = Pmqn- , ‘=0 h=O P q ‘q )

4- CONCLUDING REMARKS

The introduction of the Hermite-Euler polynomials given by Eq. (24) offers the possibility olf speculating about alternative definitions as e.g. (t< A):

~ext+yt* = ~ ~@2)(x,y) . (32) (et + l)(etz +1) n=O ‘!

The polynomials E\2) (x, y) are defined as: [n’21E~(y)En-2s(x) E$2)(x, y) = n! x (33) S=() s!(n – 2s)! and are sb.own to satisfy the following differential equation:

a2 (2) ~13(2)(x,y) = ~En (’,Y) (34) dy n and the multiplication theorem:

m–1m2–1 E$2;)(mx,m2y) = m“ ~ ~ (-l)k+h E\2) x+ ~, y + — (35) :2 ‘=0 h=O [( ) 13

The more general theorem relevant to E~2)(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]:

Bn(x+ l)– Bn(x) = nxn-l (36) can be generalized as:

HBn(x + l,Y)–H Bn(x, y) = nHn_l(x, y) (37) which is a consequence ofi

~2 ~y~x2Bn(x) ‘H Bn(x>Y). (38)

For analogous reasons, we find:

hBm,n(x+ l,y+ll~)–h Bm,n(x, Y+ll~)–h Bm,n(x+l,Yl~) (39) +hBm,n (x, y I~) = rnnhm-l,n-l (x,y I~) and:

HB~(a, b Ix+a, y+ b)–H B~(a,b Ix,y) = n(n – l)(n –2)Hn_3(x, y) (40) and

HE$2)(X + 1,y) +H E\2)(x, y) = 2Hn(x, y). (41)

We can also define the further generalized form:

N ti+lexiti * tn (42) rI i = ~ :( HB:({ai}l{xi})) i=l (eait -) n=O “ thus getting (n>N 14

H13~(“{ail{xi})-H ‘~({ai}l{xi})= (n ‘~), ‘n-N({xi}) (43) with:

+c= n (441) Z ~Hn({xil)=eZg’xi’i < n=() “

The results if this paper show that the combination of operational rules and the properties c~f 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:

(Al)

The r.h.s. of the above relation can be more conveniently rewritten as:

~px+Vqy+UvPq* ~eUp _ l)(evq – 1) Uveupx+vqy+mlv _ (up)(vq) uve (A.2) (e” - l)(ev - 1) - pq (eup - I)(evq – 1) (e” – l)(ev - 1) by nothing that:

e“P – 1 p–1 ru = e x (A.3) e“–l r=() we can rearrange (A.2) as:

~veupx+vqy+mlv p–1 q–1

=zx~fW$~ (e” – l)(ev – 1) k=oh=() Pqm=on=() “ n! (A.4) k h T hBm,n X+; ,y+— — ( q Pq ) which once confronted with (A. 1) yields Eq. (22). A symilar procedure can be exploited to prove the multiplication theorems relevant to the Euler’s generalized forms.

We note indeed:

(A.5) and handling the r.h.s. of the above equations, we find: 16

(A.(i)

By noting that:

2e mxt emt .— ‘1 epyt’ =~l(-l)k e ‘L+l et+l k=O (A.7)

we obtain:

~ $-(HEn(wpy))=~trmn n=o “ n=() (A.8)

and using the (26) we finally state the theorem (29). 17

REFERENCES

[1 G. Dattoli, S. Lorenzutta and C. Cesarano, “Finite sums and generalization forms of Bernoulli polynomials”, Rend. Di Matematica (1999).

[2 G. Dattoli and A. Terre, “ Theory and applications of generalized Bessel functions”, ARACNE Rome (1996).

[3] L.M. Milne-Thomson, “Calculus of finite differences”, MacMillan & C. Ltd., London (1951).

[4] M. Abramovitz and I. Stegun, “Hand book of Mathematical Functions”, Dover Pub. Inc. New York (1970).