On generalized Laguerre matrix polynomials

Transcription

1 Acta Univ. Sapientiae, Matheatica, 6, On genealized Laguee atix polynoials Raed S. Batahan Depatent of Matheatics, Faculty of Science, Hadhaout Univesity, 50511, Mukalla, Yeen eail: batahan@hotail.co A. A. Bathanya Depatent of Matheatics, Faculty of Education, Shabwa, Aden Univesity, Yeen eail: abathanya@yahoo.co Abstact. The ain object of the pesent pape is to intoduce and study the genealized Laguee atix polynoials fo a atix that satisfies an appopiate spectal popety. We pove that these atix polynoials ae chaacteized by the genealized hypegeoetic atix function. An explicit epesentation, integal expession and soe ecuence elations in paticula the thee tes ecuence elation ae obtained hee. Moeove, these atix polynoials appea as solution of a diffeential equation. 1 Intoduction Laguee, Heite, Gegenbaue and Chebyshev atix polynoials sequences have appeaed in connection with the study of atix diffeential equations [8, 7, 20, 4]. In [13], the Laguee and Heite atix polynoials wee intoduced as exaples of ight othogonal atix polynoial sequences fo appopiate ight atix oent functionals of integal type. The Laguee atix polynoials wee intoduced and studied in [11, 14, 16, 17]. In [22], it is shown that these atix polynoials ae othogonal with espect to a non-diagonal Sobolev-Laguee atix polynoials atix oent functional. Recently, the 2010 Matheatics Subject Classification: 15A15, 33C45, 42C05 Key wods and phases: Laguee atix polynoials, thee tes ecuence elation, genealized hypegeoetic atix function and Gaa atix function 121

2 122 R. S. Batahan, A. A. Bathanya nueical invesion of Laplace tansfos using Laguee atix polynoials has been given in [18]. A genealized fo of the Gegenbaue atix polynoials is pesented in [2]. Moeove, two genealizations of the Heite atix polynoials have been given in [1, 19]. The ain ai of this pape is to conside a new genealization of the Laguee atix polynoials. The stuctue of this pape is the following. Afte a section intoducing the notation and peliinay esults, we chaacteize, in Section 3, the definition of the genealized Laguee atix polynoials and an explicit epesentation and integal expession ae given. Finally, Section 4 deals with soe ecuence elations in paticula the thee tes ecuence elation fo these atix polynoials. Futheoe, we pove that the genealized Laguee atix polynoials satisfy a atix diffeential equation. 2 Peliinaies Thoughout this pape, fo a atix A in C N N, its spectu σa denotes the set of all eigenvalues of A. We say that a atix A is a positive stable if Reµ> 0 fo evey eigenvalue µ σa. If fz and gz ae holoophic functions of the coplex vaiable z, which ae defined in an open set Ω of the coplex plane and A is a atix in C N N with σa Ω, then fo the popeties of the atix functional calculus [5, p. 558], it follows that faga = gafa. The ecipocal gaa function denoted by Γ 1 z = 1/Γz is an entie function of the coplex vaiable z. Then, fo any atix A in C N N, the iage of Γ 1 z acting on A, denoted by Γ 1 A is a well-defined atix. Futheoe, if A + ni is invetible fo evey intege n 0, 1 whee I is the identity atix in C N N, then ΓA is invetible, its invese coincides with Γ 1 A and it follows that [6, p. 253] A n = AA + I...A + n 1I; n 1, 2 with A 0 = I. Fo any non-negative integes and n, fo 2, one easily obtains A n+ = A n A + ni, 3 and A n = n 1 A + s 1I. 4 n

3 On genealized Laguee atix polynoials 123 Let P and Q be couting atices in C N N such that fo all intege n 0 one satisfies the condition P + ni, Q + ni, and P + Q + ni ae invetible. 5 Then by [10, Theoe 2] one gets BP, Q = ΓPΓQΓ 1 P + Q, 6 whee the gaa atix function, ΓA, and the beta atix function, BP, Q, ae defined espectively [9] by and ΓA = BP, Q = In view of 7, we have [10, p. 206] exp tt A I dt, 7 t P I 1 t Q I dt. 8 A n = ΓA + niγ 1 A; n 0. 9 If λ is a coplex nube with Reλ > 0 and A is a atix in C N N with A + ni invetible fo evey intege n 1, then the n-th Laguee atix polynoials L A,λ n x is defined by [8, p. 58] L A,λ n x = n 1 k λ k k!n k! A + I n[a + I k ] 1 x k, 10 and the geneating function of these atix polynoials is given [8] by λxt Gx, t, λ, A = 1 t A+I exp = L A,λ n xt n t Accoding to [8], Laguee atix polynoials satisfy the thee-te ecuence elation n + 1L A,λ n+1 x + [ λxi A + 2n + 1I ] L A,λ n x + 12 A + nil A,λ n 1 x = θ; n 0, with L A,λ 1 x = θ and L A,λ 0 x = I whee θ is the zeo atix in C N N.

4 124 R. S. Batahan, A. A. Bathanya Definition 1 [2] Let p and q be two non-negative integes. The genealized hypegeoetic atix function is defined in the fo: pf q A 1,..., A p ; B 1..., B q ; z 13 = A 1 n... A p n [B 1 n ] 1 1 zn... [B q n ], whee A i and B j ae atices in C N N such that the atices B j ; 1 j q satisfy the condition 1. With p = 1 and q = 0 in 13, one gets the following elation due to [11, p. 213] 1 z A = 1 A nz n, z < The following lea povides esults about double atix seies. The poof ae analogous to the coesponding fo the scala case c.f [15, p. 56] and [21, p. 101]. Lea 1 [2, 3, 19] If Ak, n and Bk, n ae atices in C N N fo n 0 and k 0, then it follows that: and k 0 Ak, n = n/ Ak, n = k 0 Bk, n = n/ n Ak, n k, 15 Ak, n + k, 16 k 0 Bk, n k ; n >, 17 whee a is the standad floo function which aps a eal nube a to its next sallest intege. It is obviously desiable, by 2, to have the following: 1 n k! I = 1k ni k 18 = 1k p n 1 k I ; 0 k n. k

5 On genealized Laguee atix polynoials Definition of genealized Laguee atix polynoials Let A be a atix in C N N satisfying the spectal condition 1 and let λ be a coplex nube with Reλ > 0. Fo a positive intege, we can define the genealized Laguee atix polynoials [GLMPs] by By 14 one gets Fx, t, λ, A = 1 t A+I exp n=0 λx t 1 t 1 k λ k x k A + I + ki n t n+k = k! = n=0 n=0 L A,λ n, xt n. 19 L A,λ n, xt n, which by using 17 and 3 and equating the coefficients of t n, yields an explicit epesentation fo the GLMPs in the fo: n/ L A,λ 1 k λ k n, x = k!n k! A + I n[a + I k ] 1 x k. 20 It should be obseved that when = n, the explicit epesentation 20 becoes L A,λ n,n x = A + I n λx n I. If > n, then fo 20 one gets Moeove, it is evident that n, x = A + I n. L A,λ L A,λ n, 0 = A + I n and L A,λ n, x = L A,1 n, λ 1 x. Note that the expession 20 coincides with 10 fo the case = 1. In view of 4 and 18, we can ewite the foula 20 in the fo L A,λ n, x = A + I n [ n/ 1 A + si k 1 +1k λ k k! ] 1. x k p n 1 I k 21

6 126 R. S. Batahan, A. A. Bathanya Theefoe, in view of 13, the hypegeoetic atix epesentation of GLMPs is given in the fo: L A,λ n, x = A + I n F n I,, n + 1 I; A + I,, A + I ; 1+1 λx. 22 We give a geneating atix function of GLMPs. This esult is contained in the following. Theoe 1 Let A be a atix in C N N satisfying 1 and let λ be a coplex nube with Reλ > 0. Then [A + I n ] 1 L A,λ n, xt n = e t 0F ; A + I,, A + I xt ; λ. 23 Poof. By vitue of 20 and applying 16, we have [A + I n ] 1 L A,λ n, xt n t n 1 k λ k = [A + I k ] 1 x k t, k k! k 0 which, by using 4 and 13, educes to 23. It is clea that e t 0F ; A + I,, A + I e 1 yt e ty 0F xyt ; λ = ; A + I,, A + I xyt ; λ. Thus, by using 23 and applying 15, it follows that [A + I n ] 1 L A,λ n, xyt n = n 1 y n k y k n k! By equating the coefficients of t n, in the last seies, one gets L A,λ n, xy = A + I n n 1 y n k y k n k! [A + I k ] 1 L A,λ k, xtn. [A + I k ] 1 L A,λ k, x.

7 On genealized Laguee atix polynoials 127 Let B be a atix in C N N satisfying 1. Fo 3, 4, 14 and 16 and taking into account 20 we have B n [A + I n ] 1 L A,λ n, xt n = 1 t B λ n B n [A + I n ] 1 xt n t By using 4 and 13, the equation 24 gives the following geneating function of GLMPs: B n [A + I n ] 1 L A,λ n, xt n = 1 t B B F,, B + 1I ; A + I,, A + I xt ; λ. 1 t 25 Clealy, 25 educes to 19 when B = A + I. We now poceed to give an integal expession of GLMPs. Fo this pupose, we state the following esult. Theoe 2 Let A and B be positive stable atices in C N N such that AB = BA. Then L A+B,λ n, x = ΓA + B + n + 1IΓ 1 BΓ 1 A + n + 1I 1 0 t A 1 t B I L A,λ n, xtdt. 26 Poof. Accoding to 8 and 20, we can wite Ψ = = 1 0 n/ t A 1 t B I L A,λ n, xtdt 1 k λ k k!n k! A + I n[a + I k ] 1 x k BA + k + 1I, B, 27 and since the suation in the ight-hand side of the above equality is finite, then the seies and the integal can be peuted. Hence by 6 and 9 it

8 128 R. S. Batahan, A. A. Bathanya follows that Ψ = Γ A + n + 1I n/ 1 k λ k x k ΓB k!n k! Γ 1 A + B + k + 1I = Γ A + n + 1I ΓBΓ 1 A + B + n + 1I n/ 1 k λ k x k k!n k! A + B + I n[a + B + I k ] 1. Fo 20, 27 and 28, the expession 26 holds. We conclude this section giving an integal fo of GLMPs. 28 Theoe 3 Fo GLMPs the following holds 0 x A L A,λ n, xe x ΓA + n + 1I dx = n F 0,, n + 1 ; ; 1 +1 λ. 29 Poof. Fo 7, 9 and 20, it follows that 0 n/ x A L A,λ n, xe x 1 k λ k dx = k!n k! A + I n[a + I k ] 1 ΓA + k + 1I n/ 1 k λ k = ΓA + n + 1I k!n k!. Using 18 and taking into account 13 we aive at Recuence elations In addition to the thee tes ecuence elation, soe diffeential ecuence elations of GLMPs ae obtained hee.

9 On genealized Laguee atix polynoials 129 Theoe 4 The genealized Laguee atix polynoials satisfy the following elations: and =0 2 =0 2 1 n + 1 L A,λ n+1, x = A + I λx 2 1 =0 =0 1 λx = L A,λ n,x 1 L A,λ n +1, x 1 1 L A,λ n,x, 30 1 DL A,λ n,x = λx 1 L A,λ n,x. 31 Poof. Diffeentiating 19 with espect to t yields 1 t 2 n 1 nl A,λ n, xt n 1 = A + I1 t 2 1 L A,λ n, xt n λx t 1 1 t L A,λ n, xt n λx t 1 t 1 L A,λ n, xt n. With the help of the binoial theoe, it follows that 2 =0 2 1 n + 1L A,λ n+1, xtn+ 2 1 = A + I =0 [ λx + n 1 =0 n 1 = L A,λ n, xt n+ 1 L A,λ n +1, xtn+ 1 L A,λ n,xt n+ ]. Hence, by equating the coefficients of t n, equation 30 holds.

10 130 R. S. Batahan, A. A. Bathanya Now, by diffeentiating 19 with espect to x one gets DL A,λ n, xt n = λx 1 t 1 t 1 λx t t A+I exp 1 t. Thus, it follows that n =0 1 DL A,λ n,xt n = λx 1 L A,λ n,xt n, n which, by equating the coefficients of t n, gives 31. It is wothy to ention that 30 educes to 12 fo = 1. Also, fo the case = 1, the expession 31 gives the esult fo the Laguee atix polynoials in the fo DL A,λ n x = DL A,λ n 1 x λla,λ n 1 x. Diffeentiating 19 with espect to x again we obtains DL A,λ n, xt n = λx 1 t 1 t A++1I exp = λx 1 n L A+I,λ n, xt n. λx t 1 t Hence, by equating the coefficients of t n, we eadily obtain DL A,λ n, x = λx 1 L A+I,λ n, x. 32 It ay be noted that the foula 32 educes to the esult of [12, p. 16] fo Laguee atix polynoials, when = 1, in the fo DL A,λ n Using the fact that λx 1 t A+I t exp 1 t and 19, one gets L A,λ n, xt n = n x = λl A+I,λ x. n 1 λx = 1 t 1 t A++1I t exp 1 t, =0 1 L A+I,λ n, xt n.

11 Hence, we obtain that On genealized Laguee atix polynoials 131 L A,λ n, x = =0 1 L A+I,λ n, x. Let B be a atix in C N N satisfying 1 with AB = BA. Note that 1 t A+I λx t exp 1 t = 1 t A B 1 t B+I λx t exp 1 t. Using 14, 15 and 19, it follows that L A,λ n, xt n = n A B k L B,λ k! n k, xtn. Identifying the coefficients of t n, in the last seies, gives n L A,λ A B k n, x = L B,λ k! n k, x. 33 By evesing the ode of suation in 33, we obtain that n L A,λ n, x = A B n k n k! And finally, we pove the following esult. L B,λ k, x. 34 Theoe 5 The GLMPs is a solution of the following diffeential equation [ 1 Θ Θ 1I + 1 A + si + 1 λx ] 1 Θ + p n 1 35 I L A,λ n, x = θ, whee Θ = x d dx. Poof. It is clea that 1 Θxk = kx k. Accoding to 22 we can wite W = F n + 1 I,..., n I; A + I,..., A + I ; 1+1 λ x n/ [ p n 1 ] 1 1 = g A + si 1 +1k λ k xk k!. k k

12 132 R. S. Batahan, A. A. Bathanya It follows afte eplacing k by k + 1 and using 3 that 1 Θ 1 Θ 1I + 1 A + si W = n/ p n 1 k+1 [ 1 +1k+1 k+1 xk+1 λ k! n/ p n 1 = 1 +1 λ x 1 +1k λ k xk k! = 1 +1 λ x 1 Θ + p n 1 ] 1 1 A + si k k+1 [ W. g ] 1 1 A + si k Theefoe, W is a solution of the following diffeential equation [ 1 Θ 1 Θ 1I+ 1 A+sI λ x Θ+p n 1 ] W = θ. Since W = [A + I n ] 1 L A,λ n, x, then 35 follows iediately. It is woth noticing that taking = 1 in 35 gives the following [8] [ ] xi d2 dx 2 + A + 1 λxi d dx + λni L A,λ n x = θ. Acknowledgents The authos wish to expess thei gatitude to the unknown efeee fo seveal helpful suggestions.

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