THE MATRIX VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS

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1 Misklc Mathematical Ntes HU ISSN Vl. 13 (2012), N. 2, pp THE MATRI VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS RABİA AKTAŞ, BAYRAM ÇEKIM, AN RECEP ŞAHI Received 4 May, 2011 Abstract. I this paper, the matrix extesi f the multivariable Humbert plymials is itrduced. Varius families f liear, multiliear ad multilateral geeratig matrix fuctis f these matrix plymials are preseted. Miscellaeus applicatis are als discussed Mathematics Subect Classificati: 33C25; 15A60 Keywrds: Humbert plymials, Cha-Chya-Srivastava plymials, Lagrage-Hermite plymials, geeratig matrix fucti, matrix fuctial calculus 1. INTROUCTION It is well-kw that special matrix fuctis appear i the study f may areas. Geeralizati f the prperty f rthgality [11, 12], Rdrigues frmula [6, 10], a secd-rder Sturm-Liuville differetial equati [10], a three-term matrix recurrece [6, 7], relati betwee differet rthgal matrix plymials [21] are theretical examples. Statistics, grup represetati thery [17], scatterig thery [15], differetial equatis [18, 19], Furier series expasis [9], iterplati ad quadrature [22, 23], splies [8], ad medical imagig [5] are areas f applicati f rthgal matrix plymials. Thrughut this paper, fr a matrix A 2 C N N, its spectrum is deted by.a/. The tw-rm f A; which will be deted by kak; is defied by kaxk kak sup 2 ; x 0 kxk 2 where, fr a vectr y 2 C N, kyk 2 y T y 1=2 is the Euclidea rm f y: I ad will dete the idetity matrix ad the ull matrix i C N N, respectively. We say that a matrix A i C N N is a psitive stable if <./ > 0 fr all 2.A/, where.a/ is the set f the eigevalues f A. If A 0 ;A 1 ;:::;A are elemets f C N N ad A, the we call P.x/ A x C A 1 x 1 C ::: C A 1 x C A 0 c 2012 Misklc Uiversity Press

2 198 RABİA AKTAŞ, BAYRAM ÇEKIM, AN RECEP ŞAHI a matrix plymial f degree i x. Frm [20], e ca see.p / P.P C I /.P C 2I /:::.P C. 1/I /I 1I.P / 0 I: (1.1) Fr ay matrix A i C N N, the authrs explited the fllwig relati due t [20].1 x/ A 1 0.A/ x ;x < 1: (1.2) Š Guld [16] preseted a systematic study f a iterestig geeralizati f the Humbert ad the Gegebauer plymials ad several ther plymial systems, that is called geeralized Humbert plymials ad defied by C mxt C yt m p 1 P.m;x;y;p;C /t (1.3) 0 where m is a psitive iteger ad the ther parameters are urestricted (see als [26, p. 77, 86] ). Aktas et al. [1] preset a systematic ivestigati f a multivariable extesi f the Humbert plymials geerated by rq.ci m i x i t C y i t m i / i 1 P.m i x i t 0 P. 1;:::; r / t y i t m i < C i I i 1;2;:::;r/ (1.4) where x.x 1 ;:::;x r / ;y.y 1 ;:::;y r /; C.C 1 ;:::;C r / ; m.m 1 ;:::;m r /, m i 1;2;:::.i 1;2;:::;r/ ad the ther parameters are urestricted. The mai bective f this paper is t cstruct a matrix versi f the multivariable Humbert plymials give by (1.4) ad the derivati f varius families f multiliear ad mixed multilateral geeratig matrix fuctis fr these matrix plymials. We preset sme special cases f ur results ad als btai several recurrece relatis fr these matrix plymials. 2. MATRI ETENSION OF THE MULTIVARIABLE HUMBERT POLYNOMIALS The mai bect f this secti is t preset a systematic ivestigati f the matrix extesi f the multivariable Humbert plymials geerated by rq.c i m i x i t C y i t m i / A P i 1 t 0 (2.1).m i x i t y i t m i < C i I i 1;2;:::;r/ where A i 2 C N N ;x.x 1 ;:::;x r / ;y.y 1 ;:::;y r /; C.C 1 ;:::;C r /; m.m 1 ;:::;m r /; m i 1;2;:::.i 1;2;:::;r/ ad the ther parameters are urestricted:

3 THE MATRI VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS 199 (2.1) yields the fllwig explicit represetati: m 1 k 1 C:::Cm r k r C 1 C:::C r.a 1 / 1 Ck 1 C A Ck 1 /I :::.A r / r Ck r C A r r. r Ck r /I 1 Š::: r Šk 1 Š:::k r Š m 1 1 :::m r r. 1/k 1C:::Ck r x 1 1 :::x r r yk 1 1 :::yk r r ry m 1 k 1 C:::Cm r k r C 1 C:::C r p1 8 < : A p p Ck p C A p. p Ck p /I p p Šk p Š m p p. 1/ k p x p p y k p p 9 = ; (2.2) where, as usual,.a/ detes the Pchhammer symbl give by (1.1). We tice that the case r 1 i (2.1) reduces t the matrix versi f the geeralized Humbert plymials itrduced by Guld [16]. I this case, it is geerated by C mxt C yt m A 1 0 P.A/.m;x;y;C /t (2.3) where mxt yt m < C ; A 2 C N N ; m is a psitive iteger ad the ther parameters are urestricted. Fr the special cases f (2.3), icludig Gegebauer matrix plymials, we refer [19]. It is clear that the case C i 1; m i 1; y i 0 ; i 1;2;:::;r f the plymials f (2.1) reduces t matrix versi f the Cha-Chya-Srivastava multivariable plymials, which is geerated by [14] rq.1 x i t/ A P i 1 g.a 1;:::;A r / 0 A i 2 C N N.i 1;2;:::;r/ I t < mi.x 1 ;:::;x r /t x 1 1 ;:::;x r 1 : (2.4) Sice A i i 2 C fr N 1 i (2.4), we btaied the geeratig fucti f the Cha-Chya-Srivastava multivariable plymials [3]. O the ther had, if we chse C i 1; m i i; x i 0; y i x i ; i 1; 2;:::; r i (2.1), we get a matrix versi f the multivariable Lagrage-Hermite

4 200 RABİA AKTAŞ, BAYRAM ÇEKIM, AN RECEP ŞAHI plymials, which is geerated by [14] rq 1 x i t i A i 1 P 0 h.a 1;:::;A r /.x 1 ;:::;x r /t A i 2 C N N.i 1;2;:::;r/ I t < mi x 1 1 ;x 2 1=2 ;:::;x r 1=r : (2.5) Sice A i i 2 C fr N 1 i (2.5), we have the multivariable Lagrage-Hermite plymials preseted by Altı ad Erkuş [2]. Furthermre, we shuld remark that the case r 2 f the plymials crrespds t the familiar (tw-variable) Lagrage-Hermite plymials csidered by attli et al. [4]. Mrever, the special case C i 1; x i 0; y i x i ; i 1;2;:::;r gives the matrix versi f the Erkus-Srivastava multivariable plymials geerated by [14] rq.1 x i t m i / A P i 1 0 u.a 1;:::;A r /.x 1 ;:::;x r /t ; A i 2 C N N.i 1;2;:::;r/; 1=m t < mi x 1 1 1=m ;x 2 2 ;:::;x r 1=m r : (2.6) Sice A i i 2 C fr N 1 i (2.6), we have the Erkus-Srivastava multivariable plymials geerated by [13]. 3. AN APPLICATION OF SRIVASTAVA S THEOREM ON MIE GENERATING FUNCTIONS Srivastava [25] (see als the subsequet treatise the subect by Srivastava ad Macha [26, p. 378, Therem 12]) btaied a family f mixed geeratig fuctis fr certai geeral multivariable ad multiparameter sequeces f fuctis. Our geeratig fucti (2.1) fits easily it the geeral settig f Srivastava s therem. Thus, by applyig this geeral result t the geeratig fucti (2.1), we btai the

5 THE MATRI VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS 201 fllwig family f mixed geeratig fuctis fr matrix versi f the multivariable Humbert plymials give by (2.1) 1 0 P.A 1C 1 I;:::;A r C r I / t rq rp 1 i m i.c i C y i m i / 1.t/ W t ry C i C i 0 ad A i 2 C N N.i 1;:::;r/.C i m i x i C y i m i / A i y i m i 1.C i m i y i m i C y i m ; i / C x i.c i m i x i C y i m i / (3.1) m i x i C y i m i i I i 2 C; where all f the matrices cmmute with each ther. I a special case, it is easily see that (3.1) wuld at ce reduce t the geeratig fucti (2.1) whe i 0.i 1;:::;r/: Fr the special case f N 1; (3.1) gives mixed geeratig fucti fr the multivariable Humbert plymials give by [1]. Furthermre, the special case f N 1 ad r 1 f (3.1) reduces t the mixed geeratig fucti fr the geeralized Humbert plymials i [24]. 4. BILINEAR AN BILATERAL GENERATING MATRI FUNCTIONS I this secti, we derive several families f biliear ad bilateral geeratig matrix fuctis fr matrix versi f the multivariable Humbert plymials which are geerated by (2.1) ad give explicitly by (2.2). We begi by statig the fllwig therem. Therem 1. Crrespdig t a idetically -vaishig fucti.z/ f s cmplex variables 1;:::; s.s 2 N/ ad f cmplex rder, let ;.ziw/ W 1 a k Ck.z/w k (4.1) where.a k 0; ; 2 C/ ; z. 1;:::; s/ ad Œ=p ;p;;.x;yizi/ W a k pk Ck.z/ k (4.2)

6 202 RABİA AKTAŞ, BAYRAM ÇEKIM, AN RECEP ŞAHI where ;p 2 N; A i 2 C N N I x.x 1 ;:::;x r /I y.y 1 ;:::;y r /IC.C 1 ;:::;C r /I m.m 1 ;:::;m r /; m i 1;2;:::.i 1;2;:::;r/: The we have 1 ;p;; x;yizi ry t p t C i m i x i t C y i t m A i i ;.zi/ (4.3) 0 prvided that each member f (4.3) exists. Prf. Let T dete the left-had side f the equality (4.3) f Therem 1. The, up substitutig the plymials ;p;; x;yizi t p frm defiiti (4.2) it the left-had side f (4.3), we fid T 1 Œ=p 0 a k pk Ck.z/ k t pk : (4.4) Replacig by C pk; we ca write 1 1 T a k Ck.z/ k t t 1 a k Ck.z/ k ry C i m i x i t C y i t m A i i ;.zi/; which cmpletes the prf. I a similar maer, we ca give the ext result. Therem 2. Fr a -vaishig fucti.z/ f s cmplex variables 1;::: s.s 2 N/ ad fr p 2 N, ; 2 C, z. 1;:::; s/; A W.A 1 ;:::;A r /; B W.B 1 ;:::;B r /; A i ;B i 2 C N N fr i 1;2;:::;r; let Œ=p ;;C;m.x;yIzIw/ W a k P.A 1CB 1 ;:::;A r CB r / pk Ck.z/w k (4.5) ;p where a k 0I ;k 2 N 0 ; N 0 W N[f0g. The we have Œk=p l0 a l k P.B 1;:::;B r / k pl Cl.z/w l ;p ;;C;m.x;yIzIw/ (4.6)

7 THE MATRI VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS 203 prvided that each member f (4.6) exists where the matrices cmmute with each ther. 5. SPECIAL CASES AN SOME FURTHER PROPERTIES It is pssible t give may applicatis f the therems btaied i the previus sectis with the help f apprpriate chices f the multivariable fuctis Ck.z/; z. 1;:::; s/ ; k 2 N 0 ; s 2 N. Fr example, if we set s r ad Ck.z/ h.b 1;:::;B r /.z/ Ck i Therem 1, where the matrix versi f the multivariable Lagrage-Hermite plymials h.b 1;:::;B r /.x/ are geerated by (2.5), the we btai the fllwig result which prvides a class f bilateral geeratig matrix fuctis fr the matrix versi f the multivariable Lagrage-Hermite plymials ad fr the matrix versi f the multivariable Humbert plymials give explicitly by (2.2). Crllary 1. If ;.ziw/ W 1 P z. 1;:::; r/ ad a k h.b 1;:::;B r / Ck.z/w k ;a k 0; ; 2 N 0 ; Œ=p ;p;;.x;yizi/ W a k h.b 1;:::;B r /.z/ k pk Ck where 2 N 0 I p 2 N; A i ;B i 2 C N N ; x.x 1 ;:::;x r / I y.y 1 ;:::;y r /I C.C 1 ;:::;C r / Im.m 1 ;:::;m r /; m i 1;2;:::.i 1;2;:::;r/, the 1 ;p;; x;yizi ry t p t C i m i x i t C y i t m A i i ;.zi/ (5.1) 0 prvided that each member f (5.1) exists. Remark 1. Usig the geeratig relati (2.5) fr the matrix versi f the multivariable Lagrage-Hermite plymials ad settig a k 1; 0; 1; we btai where 1 Œ=p 0 ry pk h.b 1;:::;B r / k.z/ k t pk C i m i x i t C y i t m i A i! ry.1 i i / B i < mi 1 1 ; 2 1=2 1=r ;:::; r ;! ;

8 204 RABİA AKTAŞ, BAYRAM ÇEKIM, AN RECEP ŞAHI m i x i t y i t m i < C i I i 1;2;:::;r: Als, if we chse s 2r ad Ck.z/ P.E 1;:::;E r /.m;t;!;c/; ; 2 N Ck 0 ; t.t 1 ;:::;t r /;!.! 1 ;:::;! r / i Therem 2, we btai the fllwig class f biliear geeratig matrix fuctis fr the matrix versi f the multivariable Humbert plymials give explicitly by (2.2). Crllary 2. If ;p ;;C;m.x;yIt;!Iw/ Œ=p W a k P.A 1CB 1 ;:::;A r CB r / P.E 1;:::;E r /.m;t;!;c/w k pk Ck.a k 0I p 2 NI;k;; 2 N 0 / where A i ;B i ;E i 2 C N N fr i 1;2;:::;r; the Œk=p l0 a l k P.B 1;:::;B r / P.E 1;:::;E r /.m;t;!;c/w l k pl Cl ;p ;;C;m.x;yIt;!Iw/ (5.2) prvided that each member f (5.2) exists where A i B B A i fr i; 1;2;:::;r: Fr example, if we set s 1 ad Ck.y / L.E;/ Ck.y/ i Therem 1, where the th Laguerre matrix plymials L.E;/.x/ are defied by [18] L.E;/ 0.x/. 1/ k k kš. k/š.e C I / Œ.E C I / k 1 x k ; where E is a matrix i C N N, E C I is ivertible fr every iteger 0 ad is a cmplex umber with <./ > 0 ad they are geerated by 1 xt L.E;/.x/t.1 t/.eci / exp ; (5.3) 1 t t < 1; 0 < x < 1; the we btai the fllwig result which prvides a class f bilateral geeratig matrix fuctis fr the matrix versi f the multivariable Humbert ad Laguerre matrix plymials.

9 ad THE MATRI VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS 205 Crllary 3. If ;. Iw/ W 1 P 0 a k L.E;/ Ck. /wk where.a k 0; ; 2 N 0 /I Œ=p ;p;;.x;yi I/ W a k L.E;/ pk Ck. /k where ;p 2 N: The we have 1 ;p;; xiyi I ry t p t C i m i x i t C y i t m A i i ;. I/ (5.4) prvided that each member f (5.4) exists. Remark 2. Usig the geeratig relati (5.3) fr the Laguerre matrix plymials ad takig a k 1; 0; 1; we have 1 Œ=p 0 ry.1 C i where < 1; 0 < < 1: pk L.E;/ k. / k t pk m i x i t C y i t m i A i /.ECI / exp ; (5.5) 1 Remark 3. Fr r 1 i (5.5), we have a bilateral geeratig matrix fucti f the Humbert (2.3) ad Laguerre matrix plymials: 1 Œ=p 0 P.A/.m;x;y;C /L.E;/. / k t pk pk k C mxt C yt m A.1 /.ECI / exp : 1 Remark 4. Fr r 1 ad s 2 i Therem 1, settig Ck. / P.B/.m;x;y;C /.B 2 C N N / ad takig a k 1; 0; 1;we have biliear geeratig matrix fucti fr the Humbert matrix plymials: 1 Œ=p 0 P.A/.B/.m;x;y;C /P.m;x;y;C / k t pk pk k C mxt C yt m A C mx C y m B :

10 206 RABİA AKTAŞ, BAYRAM ÇEKIM, AN RECEP ŞAHI Furthermre, fr every suitable chice f the cefficiets a k.k 2 N 0 /; if the multivariable fucti Ck.y/; y.y 1 ;:::;y s /;.s 2 N/; is expressed as a apprpriate prduct f several simpler fuctis, the the assertis f Therems 1 ad 2 ca be applied i rder t derive varius families f multiliear ad multilateral geeratig matrix fuctis fr the matrix versi f the multivariable Humbert plymials give explicitly by (2.2). We w discuss sme further prperties f matrix versi f the multivariable Humbert plymials give by (2.2). First f all, the geeratig matrix relati (2.1) yields the fllwig additi frmula fr these multivariable plymials: P.A 1CB 1 ;:::;A r CB r / P.B 1;:::;B r / k k where A i ;B i 2 C N N ; A i B B A i fr i; 1;2;:::;r: O the ther had, the multivariable Humbert matrix plymials satisfy the fllwig x C 1 : (5.6) If we differetiate each member f the geeratig fucti (2.1) with respect t x ad y. 1;2;:::;r/; we btai the fllwig (differetial) recurrece relatis fr the matrix versi f the multivariable Humbert plymials: fr 1 Œk=m l0. 1/ l k C l lm ŠA m k lm C1 k lm ŠlŠC k l.m 1/C1 x k lm y l k 1.m; x; y; m Œk=m l0. 1/ l k C l lm ŠA m k lm k lm ŠlŠC k l.m 1/C1 x k lm y l k m (5.8)

11 THE MATRI VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS 207 where m ad m. 1;2;:::;r/ is a psitive iteger ad all matrices are cmmutative. By applyig (5.6), (5.7) ad (5.8), the fllwig recurrece relati fr the matrix plymials (give explicitly by (2.2)) ca be easily derived: r 1 Œk=m 1 x k lm C1 l0. 1/ l k C l lm ŠA m k lm C1 k lm lšc k l.m 1/C1 y l k 1 x k r m 1 lm Œk=m l0. 1/ l k C l lm ŠA m k lm C1 y lc1 k m, fr m k lm ŠlŠC k l.m 1/C1 where m. 1;2;:::;r/ is a psitive iteger ad all matrices cmmute. ACKNOWLEGEMENT The authrs are grateful t the referee(s) fr their valuable cmmets ad suggestis which imprved the quality ad the clarity f the paper. REFERENCES [1] R. Aktaş, R. Şahi, ad A. Altı, O a multivariable extesi f the Humbert plymials, Appl. Math. Cmput., vl. 218,. 3, pp , [2] A. Alti ad E. Erku, O a multivariable extesi f the Lagrage-Hermite plymials, Itegral Trasfrms Spec. Fuct., vl. 17,. 4, pp , [3] W. C. C. Cha, C. J. Chya, ad H. M. Srivastava, The Lagrage plymials i several variables, Itegral Trasfrms Spec. Fuct., vl. 12,. 2, pp , [4] G. attli, P. E. Ricci, ad C. Cesara, The Lagrage plymials, the assciated geeralizatis, ad the umbral calculus, Itegral Trasfrms Spec. Fuct., vl. 14,. 2, pp , [5] E. efez, A. Hervás, A. Law, J. Villaueva-Oller, ad R. J. Villaueva, Prgressive trasmissi f images: PC-based cmputatis, usig rthgal matrix plymials, Math. Cmput. Mdellig, vl. 32,. 10, pp , [6] E. efez, L. Jódar, ad A. Law, Jacbi matrix differetial equati, plymial slutis, ad their prperties, Cmput. Math. Appl., vl. 48,. 5-6, pp , [7] E. efez, L. Jódar, A. Law, ad E. Psda, Three-term recurreces ad matrix rthgal plymials, Util. Math., vl. 57, pp , [8] E. efez, J. Villaueva-Oller, R. Villaueva, ad A. Law, Matrix cubic splies fr prgressive 3 imagig, J. Math. Imagig Vis., vl. 17,. 1, pp , [9] E. efez ad L. Jódar, Sme applicatis f the Hermite matrix plymials series expasis, J. Cmput. Appl. Math., vl. 99,. 1-2, pp , [10] E. efez ad L. Jódar, Chebyshev matrix plymials ad secd rder matrix differetial equatis, Util. Math., vl. 61, pp , 2002.

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