MULTIDIMENSIONAL EXTENSIONS OF THE BERNOULLI AND APPELL POLYNOMIALS. Gabriella Bretti and Paolo E. Ricci

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1 TAIWANESE JOURNAL OF MATHEMATICS Vol. 8, No. 3, pp , September 004 This paper is available olie at MULTIDIMENSIONAL EXTENSIONS OF THE BERNOULLI AND APPELL POLYNOMIALS Gabriella Bretti ad Paolo E. Ricci Abstract. Multidimesioal etesios of the Beroulli ad Appell polyomials are defied geeralizig the correspodig geeratig fuctios, ad usig the Hermite-Kampé de Fériet or Gould-Hopper) polyomials. Furthermore the differetial equatios satisfied by the correspodig D polyomials are derived eploitig the factorizatio method, itroduced i [ INTRODUCTION The Hermite-Kampé de Fériet or Gould-Hopper) polyomials [, 11, 18, have bee recetly used i order to costruct additio formulas for differet classes of geeralized Gegebauer polyomials [6. They are defied by the geeratig fuctio: 1.1) e t+ytj = H j), y) t! or by the eplicit form 1.) H j) [ j, y)=! s=0 js y s js)!s!, where j is a iteger. The case whe j =1is ot cosidered, sice the correspodig D polyomials are simply epressed by the Newto biomial formula. It is worth recallig that the polyomials H j), y) are a atural solutio of the geeralized heat equatio: Received March 0, 00; revised October 30, 00. Commuicated by H. M. Srivastava. 000 Mathematics Subject Classificatio: 33C99, 34A35. Key words ad phrases: Beroulli polyomials, Appell polyomials, Multidimesioal polyomials, Hermite-Kampé de Fériet or Gould-Hopper) polyomials, Differetial equatios. 415

2 416 Gabriella Bretti ad Paolo E. Ricci y j F, y) = F, y), j F, 0) =. The case whe j =is the particularly importat see [0), ad it was recetly used i order to defie D etesios of the Beroulli ad Euler polyomials [8. Further geeralizatios icludig the H j), y) polyomials as a particular case, are give by 1.3) 1.4) e 1t+ t + + rt r = H 1,,..., r ) t!. Note that the geeratig fuctio of eq. 1.3) ca be writte i the form: e 1t+ t + + rt r = = = 1 k! 1 t + t + + r t r ) k k 1 +k + +k r=k π k r) k! k! k 1!k! k r! k 1 1 k kr r tk 1+k + +rk r! k 1 1 k kr r k 1!k! k r! t!, where k := k 1 + k + + k r, := k 1 +k + + rk r, ad the sum rus over all the restricted partitios π k r) cotaiig at most r sizes) of the iteger, k deotig the umber of parts of the partitio ad k i the umber of parts of size i. Note that, usig the ordiary otatio for the partitios of, i.e. = k 1 +k + + k,wehavetoassumek r+1 = k r+ = = k =0. From eq. 1.4) the followig eplicit form of the multidimesioal Hermite- Kampé defériet polyomials follows H 1,,..., r )= π k r)! k 1 1 k kr r k 1!k! k r!. Furthermore, they satisfy for all the isobaric property of weight ): 1.5) H t 1,t,...,t r r )=t H 1,,..., r ) ad cosequetly, they are solutios of the first order partial differetial equatio: 1.6) 1 H 1 + H + + r r H r = H.

3 Multidimesioal etesios of the Beroulli ad Appell Polyomials 417 The multivariate Hermite-KampédeFériet polyomials appear as a iterestig tool for itroducig ad studyig multidimesioal geeralizatios of the Appell polyomials too, icludig the Beroulli ad Euler oes, startig from the correspodig geeratig fuctios. A first approach i this directio was give i [7. I this article, we will study i some detail properties of the geeralized D Appell polyomials, cosiderig first the case of the D Beroulli polyomials, i order to itroduce the subject i a more friedly way. The relevat etesios to the multidimesioal Beroulli ad Appell case ca be derived almost straightforwardly, but the relevat equatios are rather ivolved. We will show that for every iteger j it is possible to defie a class of D Beroulli polyomials deoted by B j), y) geeralizig the classical Beroulli polyomials. Furthermore, i sect. 5, the bivariate Appell polyomials R j), y) are itroduced, by meas of the geeratig fuctio At) e t+ytj = R j), y) t!. Eploitig the factorizatio method, itroduced i [15 ad recalled i [13, we derive differetial equatios satisfied by these D polyomials. The differetial equatio for the classical Appell polyomials was first obtaied i [17, ad recetly recovered i [14, by usig the factorizatio method. It is worth otig that, i geeral, the differetial operators satisfied by the multidimesioal Appell polyomials are of ifiite order. This is a quite geeral situatio, sice the Appell type polyomials satisfyig a differetial operator of fiite order ca be cosidered as a eceptioal case [7. I a forthcomig article we will cosider further geeralizatios as the multiide polyomials defied by meas of the geeratig fuctios or, more geerally: At, τ)e tl +yτ j = At 1,...,t r )e 1t j rt jr r = 0,+,m 0,+ R l,j),m, y) t τ m! m! 1,..., r R j 1,...,jr) 1,..., r 1,..., r ) t 1 1 tr r 1! r!, which belog to the set of multidimesioal special fuctios recetly itroduced by G. Dattoli ad his group.

4 418 Gabriella Bretti ad Paolo E. Ricci. RECALLING BERNOULLI AND APPELL POLYNOMIALS The Beroulli polyomials B ) are defied see [1, p. i) by the geeratig fuctio:.1) G, t):= tet e t 1 = B ) t, t < π,! ad cosequetly, the Beroulli umbers B := B 0) satisfy.) t e t 1 = t B!. It is well kow that as B 0) = B 1) = B, 1, ) B ) = B k k k, B ) =B ). The Beroulli umbers see [4-16) eter i may mathematical formulas, such the Taylor epasio i a eighborhood of the origi of the circular ad hyperbolic taget ad cotaget fuctios, the sums of powers of atural umbers, the residual term of the Euler-MacLauri quadrature rule. The Beroulli polyomials, first studied by Euler [10, are employed i the itegral represetatio of differetiable periodic fuctios, sice they are employed for approimatig such fuctios i terms of polyomials. They are also used for represetig the remaider term of the composite Euler-MacLauri quadrature rule see [19). The Appell polyomials [1 ca be defied by the geeratig fuctio.3) where.4) G A, t) =At) e t = At) = R ) t! R k k! tk, A0) 0)

5 Multidimesioal etesios of the Beroulli ad Appell Polyomials 419 is aalytic fuctio at t =0, ad R k := R k 0). It is easy to see that for ay At) the derivatives of R ) satisfy.5) R ) =R ), ad furthermore if At) = t e t 1, the R ) =B ), if At) = e t +1, the R ) =E ), i.e. the Euler polyomials, if At) =α 1 α m t m [e α1t 1) e αmt 1) 1, the the R ) are the Beroulli polyomials of order m [3, if At) = m [e α 1t +1) e αmt +1) 1, the the R ) are the Euler polyomials of order m [3, If At) =e ξ 0+ξ 1 t+ ξ d+1 t d+1,ξ d+1 0, the the R ) are the geeralized Gould-Hopper polyomials [9 icludig the Hermite polyomials whe d =1 ad classical -orthogoal polyomials whe d =. 3. THE D BERNOULLI POLYNOMIALS B ), y) Startig from the Hermite KampédeFériet or Gould-Hopper) polyomials H ), y), we defie the D Beroulli polyomials B ), y) by meas of the geeratig fuctio: 3.1) G ), y; t):= t e t 1 et+yt = B ), y) t! It is possible to fid the eplicit form of the polyomials B ), y) i terms of the Hermite Kampé defériet polyomials H ), y) ad vice-versa. Theorem 3.1. The followig represetatio formulas hold true: ) B ), y) = B h h H ) h, y)= 3.) [ B h h h s y s =! h)! h s)!s!, s=0 where B k deote the Beroulli umbers;

6 40 Gabriella Bretti ad Paolo E. Ricci 3.3) H ), y)= ) h 1 h +1 B) h, y). Proof. Eq. 3.) is obtaied startig from the geeratig fuctio 3.1) by usig the Cauchy product of the series epasios.) ad 1.1), for j =,ad the usig the idetity priciple of power series. Eq. 3.3) is obtaied i the same way, startig from the equatio e t+yt = et 1 B ), y) t t!. A recurrece relatio for the polyomials B ) is give by the followig theorem Theorem 3.4. For ay itegral 1 the followig liear homogeeous recurrece relatio for the geeralized Beroulli polyomials B ), y) holds true: 3.4) B ) 0, y) = 1, B ), y) = 1 + 1)yB ), y), where B h deote the Beroulli umbers. ) B k k B ) k, y)+ 1 ) B ), y) Proof. Differetiatig both sides of eq. 3.1) with respect to t, recallig the geeratig fuctios.) of the Beroulli umbers, ad usig some elemetary algebra ad the idetity priciple of power series, recursio 3.4) easily follows. We are ow i coditio to prove the followig theorem, which gives differetial equatios satisfied by the B ), y): Theorem 3.4. The D Beroulli polyomials B ), y) satisfy the differetial or itegro-differetial equatios: 3.5) 3.6) [ B! D + + B 4 ) B +! y D + [ 1 ) k=1 D y +D 1 B k+1 4! D4 k +1)! D k) 1 ) D y +yd 1 Dy D + B ), y)=0, D k+1 y +1)D B ), y)=0,

7 Multidimesioal etesios of the Beroulli ad Appell Polyomials ) [ 1 ) k=1 D ) B k+1 k +1)! Dk 1 D y + 1)D Dy k+1 +1)D D y +D ) Dy + ydy) B ), y)= 0, ). Proof. We start provig eq. 3.5), which gives a differetial equatio satisfied by the B ), y) with respect to the variable, assumig y as a parameter. It is easily see that a lowerig operator L for the polyomials B), y) is give by L = 1. I fact, for ay fied y, the B ), y) belog to the Appell class, assumig i eq..3) At) = t e t 1 eyt, ad therefore eq..5) holds true. Furthermore, by usig the recurrece relatio of Theorem 3., we fid the correspodig icreasig operator L + see [13, which is give by L + = 1 ) B k+1 +yd k +1)! D k. The, by eploitig the factorizatio method see [15, [13), equatio 3.5) immediately follows usig L +1 L+ B ), y)=b ), y). I order to fid the itegro-differetial equatio 3.6), icludig derivatives with respect to y, ote that differetiatig the geeratig fuctio 1.1) assumig j =), with respect to y, yields: B ), y) = 1)B ), y) 3.8), y)= B), y so that 3.9) D 1 D y B ), y)=b ), y) ad therefore, we ca assume: 3.10) L := 1 D 1 D y. Usig agai the recurrece relatio of Theorem 3., we obtai the correspodig icreasig operator L + : 3.11) L + := 1 ) +yd 1 D y B k+1 k +1)! D k) D k y

8 4 Gabriella Bretti ad Paolo E. Ricci ad cosequetly the itegro-differetial equatio 3.6) follows. Therefore, the differetial equatio 3.7) immediately follows differetiatig both sides of eq. 3.6) 1)-times with respect to. 4. THE D BERNOULLI POLYNOMIALS B j), y) I a similar way, startig from the Hermite Kampé defériet or Gould-Hopper) polyomials H j), y), we defie the D Beroulli polyomials B j), y) by meas of the geeratig fuctio: 4.1) G j), y; t):= t e t 1 et+ytj = B j), y)t! It is worth otig that the polyomial H j), y), beigisobaric of weight, caot cotai the variable y, for every =0, 1,..., j 1. Usig the same procedure as before, the followig results for the B j), y) polyomials ca be easily derived. Eplicit forms of the polyomials B j) i terms of the Hermite Kampé de Fériet polyomials H j) ad vice-versa. Theorem 4.1. The followig represetatio formulas hold true: ) B j), y) = B h h H j) h, y) =! B h h)! [ h j r=0 h jr y r h jr)!r!, where B k deote the Beroulli umbers: ) H j) 1, y)= h h +1 Bj) h, y). Recurrece relatio. Theorem 4.. For ay itegral 1 the followig liear homogeeous recurrece relatio for the geeralized Beroulli polyomials B j), y) holds true: B j) 0, y) = 1, B j) ), y) = 1 B k k B j) k, y) + 1 ) B j) 1)!, y)+jy j)! Bj) j, y).

9 Multidimesioal etesios of the Beroulli ad Appell Polyomials 43 Shift operators. L := 1 D L + := 1 ) B k+1 k +1)! D k + jyd j 1 L := 1 D j 1) D y L + := 1 ) + jyd j 1) Dy j 1 Differetial or itegro-differetial equatios. B k+1 k +1)! D j 1) k) D k y Theorem 4.3. The D Beroulli polyomials B j), y) satisfy the differetial or itegro-differetial equatios: [ B! D + + B ) j+1 Bj j +1)! Dj+1 + j! jy D j + B ) j 1 1 j 1)! Dj D + B j), y)=0, [ 1 ) D y + jd j 1) Dy j 1 +jyd j 1) Dy j k=1 B k+1 +1)D j 1) B j), y)=0, [ 1 ) D j 1)) +jd j 1) j) D j 1 y + ydy j ) k +1)! D j 1) k) D k+1 y D y +j 1) 1)D j 1)) 1 D y k=1 B k+1 +1)D j 1) B j), y)=0, j). k +1)! Dj 1)k 1) D k+1 y Note that the last equatio is derived by differetiatig j 1))-times with respect to both sides of the precedig oe, ad does ot cotai ati-derivatives for j.

10 44 Gabriella Bretti ad Paolo E. Ricci 5. D APPELL POLYNOMIALS For ay j, the D Appell polyomials R j), y) are defied by meas of the geeratig fuctio: 5.1) G j) A, y; t):=at) et+ytj = R j), y) t! Eve i this more geeral case, the polyomial R j), y), isisobaric of weight, so that it does ot cotai the variable y, for every =0, 1,..., j 1. Eplicit forms of the polyomials R j) i terms of the Hermite Kampé de Fériet polyomials H j) ad vice-versa. Theorem 5.1. The followig represetatio formulas hold true: ) R j), y) = R h h H j) h, y) =! R h h)! [ h j r=0 h jr y r h jr)!r!, where the R k are the Appell umbers appearig i eq..4), ) H j), y) = Q k k R j) k, y), where the Q k are the coefficiets of the Taylor epasio i a eighborhood of the origi of the reciprocal fuctio 1/At). Recurrece relatio. It is suitable to itroduce the coefficiets of the Taylor epasio: 5.) A t) At) = t α!. Theorem 5.. For ay itegral 1 the followig liear homogeeous recurrece relatio for the geeralized Appell polyomials R j), y) holds true: R j) 0, y) = 1, R j), y) = + α 0 )R j) 1, y)+ j 1 ) + 1 k α k 1 R j), y). k ) jy R j) j, y)

11 Multidimesioal etesios of the Beroulli ad Appell Polyomials 45 Shift operators. L : = 1 D L + : = + α 0 )+ L : = 1 D j 1) D y j j 1)! ydj 1 + L + j : = + α 0 )+ j 1)! yd j 1) α k + k)! D j 1) k) Differetial or itegro-differetial equatios. Theorem 5.3. or itegro-differetial equatios: [ α 1)! D + + α j j! Dj [ k=1 D j 1 y D k y. α k k)! D k The D Appell polyomials R j), y) satisfy the differetial αj 1 + jy j 1)! + α j j )! Dj α 0 ) D [ + α 0 ) D y + α k k)! D j 1) k) + α 0 ) D j 1)) + j j 1)! Dj 1) j) +1)D j 1) j j 1)! D j 1) D k+1 y ) D j yd j y + Dy j 1 ) +1)D j 1 D y +j 1) 1)D j 1)) 1 yd j y + Dy j 1 ) + k=1 R j), y)=0, j). α k R j), y)=0, R j), y)=0, D y k)! Dj 1)k 1) Dy k+1 6. EXAMPLE We give i this sectio a particular eample of D Appell polyomials, correspodig to a suitable choice of the fuctio At), already cosidered i [5.

12 46 Gabriella Bretti ad Paolo E. Ricci Fi the itegral N, ad defie: At) := 1 e ep ep t N ) ), so that the ormalizig coditio A0) = R 0 =1 is satisfied. The, recallig eq. 5.), the umerical values α k = 1)k+1 N k+1 are easily foud. Furthermore the Appell umbers ca be computed by meas of the recurrece relatio: R h+1 = 1 N h ) h 1) h k k N h k R k The first values of the R k are cosequetly: R 0 = A0) = 1, R 1 = 1 N, R = N, R 3 = 5 N 3,... adsoo. Assumig j =, the first D relevat Appell polyomials are R ) 0, y) = 1, R ) 1, y) = 1 N, R ), y) = N +y + N, R ) 3, y) = 3 3 N +6y + 6 N 6 N y 5 N 3, adsoo. Followig methods of the above sectios we have foud the recurrece relatio: R ) 0, y) = 1, R ), y) = 1 + ) ) R, y)+)yr), y) 1) k N k R) k, y),

13 Multidimesioal etesios of the Beroulli ad Appell Polyomials 47 ad the differetial equatios: [ 1) 1)! N D + 1! N 3 D3 + + ad, for : [ 1 ) N 1 N D ) D D y + 1)D + 1) 1)! N D y R ), y)=0, N D ) 1 N +y D ) D y +D yd y + D y D y +1)D R ), y) =0. ACKNOWLEDGEMENTS This article was cocluded i the framework of the Italia Natioal Group for Scietific Computatio G.N.C.S.). The authors epress their sicere gratitude to Dr. G. Dattoli for iterestig discussios ad for his friedly itroductio to the properties of the Hermite-Kampé de Fériet polyomials. REFERENCES 1. P. Appell, Sur ue classe de polyômes, A. Sci. Ecole Norm. Sup. ) ), P. Appell ad J. Kampé defériet, Foctios hypergéométriques et hypersphériques. Polyômes d Hermite, Gauthier-Villars, Paris, H. Batema ad A. Erdélyi, Higher Fuctios. The Gamma fuctio. The hypergeometric fuctio. Legedre fuctios, Vol. 1, McGraw-Hill, New York, J. Beroulli, Ars cojectadi, Thurisiorum, Basel, G. Bretti, M. X. He ad P.E. Ricci, O quadrature rules associated with Appell polyomials, Iterat. J. Appl. Math ), G. Dattoli, H. M. Srivastava ad P.E. Ricci, Two-ide multidimesioal Gegebauer polyomials ad their itegral represetatios,math. Comput. Modellig, ), G. Dattoli, P. E. Ricci ad C. Cesarao, Differetial equatios for Appell type polyomials, Fract. Calc. Appl. Aal., 5 00),

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