Mathematical ad Computatioal Applicatios, Vol. 18, No. 3, pp. 33-39, 013 SOME RELATIONS ON HERMITE MATRIX POLYNOMIALS Levet Kargi ad Veli Kurt Departmet of Mathematics, Faculty Sciece, Uiversity of Adeiz TR-07058, Atalya, Turey. largi@adeiz.edu.tr, vurt@adeiz.edu.tr Abstract- I this study we give additio theorem, multiplicatio theorem ad summatio formula for Hermite matrix polyomials. We write Hermite matrix polyomials as hypergeometric matrix fuctios. We also obtai a ew geeratig fuctio for Hermite matrix polyomials ad usig this fuctio, we prove some ew results ad relatios. Key Words- Hypergeometric matrix fuctios, Hermite matrix polyomials, geeratig matrix fuctios. 1. INTRODUCTION I the recet two decades, orthogoal matrix polyomials comprise a emergig field of study, with importat results i both theory ad applicatios cotiuig to appear i the literature. Hermite matrix polyomials are itroduced by Jodar ad Compay i [9]. Moreover, some properties of the Hermite matrix polyomials are give i [3, 14, 15] ad a geeralized form of the Hermite matrix polyomials have bee itroduced ad studied i [16, 17, 19, 0,, 4]. Other classical orthogoal polyomials as Laguerre, Gegebauer, Chebyshev ad Jacobi polyomials have bee exteded to orthogoal matrix polyomials, ad some results have bee ivestigated, see for example [4, 5, 8, 1, 3]. From the coectio with orthogoal matrix polyomials, special matrix fuctios have bee itroduced ad studied by some mathematicias. Gamma matrix fuctio is itroduced ad studied i [7, 10] for matrices i C r r whose eigevalues are all i the right ope half-plae. Apart from the close relatioships with the well-ow beta ad gamma matrix fuctios, the emergig theory of orthogoal matrix polyomials ad its operatioal calculus suggest the study of hypergeometric matrix fuctio. Hypergeometric matrix fuctio F ; A; z has bee recetly itroduced i [13]. Explicit closed form for geeral solutios of the hypergeometric matrix differetial equatios is give i [1]. The paper is orgaized as follows. I the ext sectio we deal with importat properties of the Hermite matrix polyomials such as additio, multiplicatio theorems ad summatio formula. We obtai a geeratig fuctio for Hermite matrix polyomials ad write these polyomials as hypergeometric matrix fuctios. We obtai some results which follow from this geeratig fuctio. Throughout this paper, if A is a matrix i C r r, its spectrum A will deotes the set of all the eigevalues of A. Its -orm will be deoted by A ad defied by A sup Ax, x 0 x where for a y i C r, y y T, y 1 is the Euclidea orm of y. If f z ad g z are holomorphic fuctios of the complex variable z, which are defied i a ope set Ω of
34 L. Kargi ad V. Kurt the complex plae, ad A is a matrix i C r r such that σ A Ω, the from the properties of matrix fuctioal calculus [6, page 558], it follows that f A g A g A f A. If D 0 is the complex plae cut alog the egative real axis ad log z deotes the priciple logarithm of z, the z 1 represets exp 1 log z. If A is a matrix Cr r i which σ A D 0 the A 1 A deotes the image by z 1 of the matrix fuctioal calculus actig o the matrix A. Let A be a matrix i C r r such that Re z > 0 for every eigevalues z σ A, (1) The the Hermite matrix polyomials H x, A are defied by [9] H x, A 1! x A!!, 0 () ad satisfyig the three-terms recurrece relatio H x, A x AH 1 x, A 1 H x, A, 1. H 1 x, A 0, H 1 x, A I, where I is the uit matrix i C r r. Accordig to [9], we have 0 H x, A! t exp xt A t. (3) The Pochhammer symbol or shifted factorial is defied by [11] A A A + I A + ( 1 I), 1, (4) with A 0 I. By usig (4) it is easy to show that A A A + I. (5) The hypergeometric matrix fuctio F A, B; C; z has bee give i [11] A B C 1 F A, B; C; z z,! z < 1, 0 where A, B, C are matrices i C r r such that C + I is ivertible for all itegers 0. Note that by (4) if A ii where i is a atural umber the A i+j 0 for j 1 ad F A, B; C; z becomes a matrix polyomial of degree i. Lemma 1: ([18]) Let. deotes ay matrix orm for which I 1. If M < 1 for a matrix M i C r r, the I + M c exists ad give by I M c c M,! where c is a positive iteger. We coclude this sectio recallig a result related to the rearragemet of the terms i iterated series. If A, ad B, are matrices i C r r for 0, 0, the i a aalogous way to the proof of Lemma 11 of [], it follows that 0
Some Relatios o Hermite Matrix Polyomials 35 ad 0 0 A, B, 0 0 A,, B,. (6) (7). SOME RELATIONS ON HERMITE MATRIX POLYNOMIALS Propositio : Hermite matrix polyomials satisfy the multiplicatio ad additio formula as follows: H x, A!!! + H z 1 + z + 1, A where ad are costats. 1 1 H x, A, (8) H z 1, A H z, A (9) Proof: Taig x for x ad t for t i (3), we have 0 H x, A t! exp xt A t exp xt A t + t t H x, A 1 1 t +!!. 0 By usig (6) ad comparig the coefficiets of t o both sides of the above equatio,! we get (8). We ca similarly prove the equatio (9). Corollary 3: The Hermite matrix polyomials have the followig relatios: H z 1 + z, A H x, A H x + y, A H z 1, A H z, A, (10) H x, A H y, A, (11) H x, A H y, A. (1) Propositio 4: The summatio formulas for H x, A are give as follows:
36 L. Kargi ad V. Kurt ad for x y, m 0 H x, A H m x, A m!! AH m x, A H m (y, A) m +1 m! mi m, H m+ x, A m!!! H +1 y, A H x, A H +1 x, A H y, A +1! (y x) (13). (14) Proof: Usig (3), we have 0 m 0 s H x, A H m x, A u! v m! exp ( Ax u + v (u + v) + uv m u m v m ı m! 0 m0 H ı x, A (u + v).! Maig ecessary arragemet ad comparig the coefficiets of u completes the! m! proof of (13). Equatio (14) ca be easily proved by usig the three-terms recurrece relatio for Hermite matrix polyomials. Propositio 5: Let A be a matrix i C r r satisfyig coditio (1) ad v m A < 1. The Hermite matrix polyomials have the followig geeratig fuctio: (c) H (x, A) t (I xt A) c F ci!, c + 1 I ; ; 4t I xt A, 0 where c is a positive iteger ad t < 1, x < 1. Proof: Usig () ad (6), we have 0 (c) H (x, A) t! By usig Lemma 1, we have (c) H (x, A) t! 0 0 0 0 ( 1) (c)! ( 1) (c) (x A)!! t ( 1) (c + I) (xt A) (c) t.!! (I xt A) c+ t. Usig the relatio (5) ad maig ecessary arragemet, completes the proof. Theorem 6: Let A be a ivertible matrix i C r r satisfyig coditio (1). The Hermite matrix polyomials H x, A ca be write as hypergeometric matrix fuctios: H x, A x A F I, 1 I ; ; A 1 x. (15) Proof: From (), we have
Some Relatios o Hermite Matrix Polyomials 37 H x, A x A ( 1) ( I) (A)! x.! Sice ( I)!, usig the relatio (5) we get (15). Theorem 7: Let A be a ivertible matrix i C r r satisfyig coditio (1). For Z +, Hermite matrix polyomials have the followig geeratig fuctio: 0 H + x, A t! exp(xt A t )H x A Proof : By usig (7), we have 0 H + x, A t! u! 0 0 H x, A exp(xt A t ) H x, A t u!! t + u! 1 H x A u By comparig the coefficiets of, we obtai (16).! As a example of equatio (16), let us derive the followig theorem: t, A. (16) 1 t, A u!. Theorem 8: Let A be a ivertible matrix i C r r satisfyig coditio (1). Hermite matrix polyomials satisfy the followig relatio: t 1 H x, A H y, A! 1 4t exp A (xyt (x + y )t ) ı ı 1 4t, (17) where t < 1. Proof: By usig (6) ad (16), we have 0 Sice H x, A H y, A t! 0 0 0 ( 1) x A H (y, A)t! ( )! H + y, A x A! ı ( 1) t! exp Axyt Ax t H y xt, A! ı ı ( 1) t.
38 L. Kargi ad V. Kurt H y xt, A ad! 1! 1 0 H x, A H y, A t! s0 ( 1) s! y xt s A, it follows that s! s! (1 4t ) 1 exp Axyt Ax t exp Combiig the expoetial factors, we arrive at (17). s At (y xt) ı 1 4t. Theorem 9: Let A be a matrix i C r r satisfyig coditio (1), A < 1 ad c be a positive iteger. Hermite matrix polyomials satisfy the followig relatio: 0 F I, c; ; y H x, A t! exp (xt A t )(I + xyt A yt ) c F ci, c + 1 I ; ; 4y t (I + xyt A yt ). Proof: Applyig equatio (16) to Propositio 5, we complete the proof. 3. CONCLUSIONS I this paper, we carry the properties of classical scalar Hermite polyomials to the Hermite matrix polyomials. Equatio (1) is the matrix aalog of the Ruge additio formula of the scalar Hermite polyomials. For the case A 1 1, the expressio (13) cocides with the formula which was proved by Feldheim for classical scalar Hermite polyomials. Also Propositio 5 is the matrix aalogous of the Batema s geeratig relatio for classical scalar Hermite polyomials give i [1]. Replacig t with t i (17), we give aother proof for the equatio (41) i [14]. Theorem 9 is the matrix aalog of the Brafma s relatio for classical scalar Hermite polyomials i []. Acowledgemets- The preset ivestigatio was supported by the Scietific Research Project Admiistratio of Adeiz Uiversity. 4. REFERENCES 1. F. Brafma, Geeratig fuctios of Jacobi ad related polyomials, Proceedigs of the America Mathematical Society (6), 94-949, 1951.. F. Brafma, Some geeratig fuctios for Laguerre ad Hermite polyomials, Caadia Joural of Mathematics 9, 180-187, 1957. 3. E. Defez ad L. Jodar, Some applicatios of the Hermite matrix polyomials series expasios, Joural of Computatioal Applied Mathematics 99, 105-117, 1998. 4. E. Defez ad L. Jodar, Chebyshev matrix polyomials ad secod order matrix differetial equatios, Utilitas Mathematica 6, 107-13, 00.,
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