Matrix Analogues of Some Properties for Bessel Matrix Functions

Transcription

1 J. Math. Sci. Univ. Tokyo (15, Matrix Analogues of Some Properties for Bessel Matrix Functions By Bayram Çek ım and Abdullah Altın Abstract. In this study, we obtain some recurrence relations for Bessel matrix functions of the first kind. Then we derive a new integral representation of these matrix functions. We lastly give new relations between Bessel matrix functions of the first kind and Laguerre matrix polynomials. 1. Introduction In recent years, the theory of special matrix functions has become an active research area in applied mathematics. In this theory, Gamma and Beta matrix functions [1], Hermite matrix polynomials [], Laguerre matrix polynomials [9], Jacobi matrix polynomials [4], Chebyshev matrix polynomials [3] and Bessel matrix polynomials [1] were studied. Furthermore, Bessel matrix functions were introduced by Jódar et al. in [6, 8, 7, 14]. It was considered that these function satisfy Bessel type differential equation t X (t+tx (t+ ( t I A X(t =, <t< where A is a matrix in C r r and unknown X(t isac r 1 -valued function. Jódar et al. gave different solutions of Bessel type differential equation according to a matrix A. After that study, Jódar et al. defined Bessel matrix functions of first kind and second kind. By using the kinds of Bessel matrix functions, the general solutions of this equation were obtained. Then, Çekim and Erkuş-Duman derived some integral representations of Bessel matrix functions in [1]. In current study, after having obtained the various properties by using these functions, we give relations between Bessel matrix functions of the first kind and Laguerre matrix polynomials. 1 Mathematics Subject Classification. Primary 33C45, 33C1; Secondary 15A6. Key words: Bessel matrix functions, Jordan block, Gamma matrix function, Beta matrix function, Laguerre matrix polynomials. 519

2 5 Bayram Çek ım and Abdullah Altın. Preliminaries and Some Facts For the sake of clarity, in the presentation of the next results, we recall some concepts and properties of the matrix functional calculus. Throughout this paper, we denote the set of all eigenvalues for a matrix A C r r by σ(a. Definition 1. A matrix A C r r is called a positive stable matrix if Re(λ > for all λ σ(a. Theorem 1 ([5]. If f(z and g(z are holomorphic functions in an open set Ω of the complex plane, and if A is a matrix in C r r for which σ(a Ω, then f(ag(a =g(af(a. Definition ([9]. For a positive stable matrix P in C r r, the Gamma matrix function Γ (P is defined by (.1 Γ(P = where t P I = exp [(P Ilnt]. e t t P I dt, Furthermore, the reciprocal scalar Gamma function, Γ 1 (z =1/Γ(z, is an entire function of the complex variable z. Thus, for any A C r r, the Riesz-Dunford functional calculus [5] shows that Γ 1 (A is well defined and, indeed, it is the inverse of Γ(A. Hence, if σ(a (A C r r doesnot contain or a negative integer, (. Γ 1 (A =A(A + I(A +I...(A + kiγ 1 (A +(k +1I. Definition 3 ([9]. For positive stable matrices P and Q in C r r, the Beta matrix function B (P, Q is defined by (.3 1 B (P, Q = t P I (1 t Q I dt.

3 Bessel Matrix Functions of the First Kind 51 Theorem ([9]. and PQ = QP. Then Let P,Q, P + Q be positive stable matrices in C r r (.4 B (P, Q =Γ(P Γ(QΓ 1 (P + Q. Theorem 3 ([11]. Let P and Q be matrices in C r r where Then, we have PQ = QP and P + ni, Q + ni, P + Q + ni are invertible for all n N. (.5 B (P, Q =Γ(P Γ(QΓ 1 (P + Q. Moreover, let us consider the Bessel functions of the first kind of order υ, defined by J υ (t = ( t υ m ( 1 m Γ 1 (υ + m +1 ( t m, <t< which is an entire function of the parameter υ [13]. Thus, if H is a Jordan block of the following form υ υ 1... H = (.6 C r r, υ we can write the image by means of the matrix functional calculus acting on the matrix H and the function of υ, J υ (t, the Bessel matrix functions of the first kind of order H is defined as follows: (.7 J H (t = ( t H m ( 1 m Γ 1 (H +(m +1I ( t m, <t<

4 5 Bayram Çek ım and Abdullah Altın where υ/ Z [8]. Let us take A C r r satisfying (.8 σ(a Z = φ. In [8], Jódar et al. defined Bessel matrix functions of the first kind of order A as the following: (.9 J A (t = ( t A m ( 1 m Γ 1 (A +(m +1I ( t m, <t<. Now, let us consider the general case. Let A be a matrix satisfying the condition (.8 and H = diag(h 1,..., H k be the Jordan canonical form of A, where H i is a Jordan block such that υ i υ i 1... H i = (.1 C p i p i, υ i p i 1, p 1 + p p k = r. Here if p i =1, we can write H i =(υ i where υ i / Z for 1 i k. Thus, the Bessel matrix functions of the first kind can be written in the form J H (t = diag 1 i k (J Hi (t. Furthermore, if P is an invertible matrix in C r r and H = diag(h 1,..., H k is the Jordan canonical form of A such that H = diag(h 1,..., H k =PAP 1, we have the Bessel matrix functions of the first kind of order A in (.9 (see [8]. 3. Some Properties for Bessel Matrix Functions of the First Kind In this section, we derive some recurrence relations for Bessel matrix functions of the first kind. Let H be a Jordan block defined as in (.6.

5 Bessel Matrix Functions of the First Kind 53 Thus, if H satisfies the condition (.8, and < z <, wehavethe following properties. Using (.7 and differentiating with respect to z, we have d dz [zh J H (z] ( = d dz z H ( z H ( 1 m ( z m Γ 1 (H +(m +1I = H I ( 1 m ( z Γ 1 (H +(m +1I(mI +H (m 1I+H ( (z H I = z H ( 1 m ( z m Γ 1 (H + mi = z H J H I (z. Thus, for the Bessel matrix functions of the first kind, we have the first property d (3.1 dz [zh J H (z] = z H J H I (z where / σ(h. Similarly, we obtain ( d dz [z H J H (z] = d ( z H z H ( 1 m ( z m Γ 1 (H +(m +1I dz ( 1 m = mγ 1 (H +(m +1I mi H z m 1 m=1 ( 1 m = Γ 1 (H +(m +I (m+1i H z m+1 ( = z H ( 1 m = z H J H+I (z. That is, we have the second property ( z (m+1i+h Γ 1 (H +(m +I (3. d dz [z H J H (z] = z H J H+I (z.

6 54Bayram Çek ım and Abdullah Altın Then from (3.1, we obtain (3.3 zj H(z =zj H I (z HJ H (z where / σ(h. From (3., we derive (3.4 zj H(z = zj H+I (z+hj H (z where / σ(h. Also, by (3.3 and (3.4, we get (3.5 J H(z =J H I (z J H+I (z and (3.6 HJ H (z =z[j H I (z+j H+I (z] where / σ(h. Furthermore, using (.7, we find (3.7 J H (z = z (H IJ H I(z J H I (z where, 1 / σ(h. Now, let us give a generalization for these properties. Let H i be a matrix defined by (.1 and H = diag(h 1,..., H k be a matrix in C r r. Here v i is not a negative integer for 1 i k. For the matrix H = diag(h 1,..., H k, the properties (3.1 (3.7 can easily be obtained. Also, if P is an invertible matrix in C r r and H = diag(h 1,..., H k is the Jordan canonical form of A such that H = diag(h 1,..., H k =PAP 1, we can prove the following theorem for the matrix A. Theorem 4. If A is a matrix in C r r which satisfies the condition (.8, and <z<, we obtain d dz [za J A (z] = z A J A I (z where / σ(a, d dz [z A J A (z] = z A J A+I (z, zj A(z =zj A I (z AJ A (z where / σ(a,

7 Bessel Matrix Functions of the First Kind 55 zj A(z = zj A+I (z+aj A (z, J A(z =J A I (z J A+I (z where / σ(a, AJ A (z =z[j A I (z+j A+I (z] where / σ(a, J A (z = z (A IJ A I(z J A I (z where, 1 / σ(a. 4. An Integral Representation for Bessel Matrix Functions of the First Kind In this section, we derive a new integral representation for Bessel matrix functions of the first kind. The integral is given as (4.1 t ( x(t x H J H ( x(t x dx where H is a Jordan block matrix satisfying the condition (.8, <t<. Substituting (.7 in (4.1 and choosing x = ty, it follows that t ( x(t x H J H ( x(t x dx = t H+I 1 ( y(1 y H ( t H y(1 y ( ( 1 m Γ 1 t m y(1 y (H +(m +1I dy ( t H = t H+I ( 1 m ( t m Γ 1 (H +(m +1I 1 y H+mI (1 y H+mI dy.

8 56 Bayram Çek ım and Abdullah Altın From (.3 and (.5, we have t ( x(t x H J H ( x(t x dx = t H+I ( 1 m ( t mi+h Γ 1 (H +(m +1I B(H +(m +1I,H +(m +1I { = t H+I ( 1 m ( t mi+h Γ 1 (H +(m +1I Γ (H +(m +1IΓ 1 (H +(m +I } where H +ni is invertible for all n N. Also, by (. and (.7, we obtain t ( x(t x H J H ( x(t x dx = { πt H+I ( 1 m ( t mi+h Γ 1 (H +(m +1I Γ (H +(m +1I H (m+1i Γ 1 (H +(m +1IΓ 1 ( H + 1 I +(m +1I} = ( πt H+ 1 t H+ 1 I H I ( 1 m 4 = πt H+ 1 I H J H+ 1 I ( t. This result is summarized below: Γ 1 (H + 1 I +(m +1I ( t 4 m Theorem 5. If H is a Jordan block matrix in C r r which satisfies condition (.8 and H + ni is invertible for all n N, <t<, we obtain the following integral representation: t ( x(t x H J H ( x(t x dx = πt H+ 1 I H J H+ 1 I ( t.

9 Bessel Matrix Functions of the First Kind 57 Now, we can give the generalization of Theorem 5. Corollary 1. If A is a matrix in C r r which satisfies condition (.8 and A + ni is invertible for all n N, <t<, we obtain the following integral representation: t ( x(t x A J A ( x(t x dx = πt A+ 1 I A J A+ 1 I ( t. 5. New Relations of Bessel Matrix Functions of the First Kind and Laguerre Matrix Polynomials In this section, we obtain the new relations between Bessel matrix functions of the first kind and Laguerre matrix polynomials. Let A be a matrix in C r r satisfying the condition (.8 and λ be a complex number whose real part is positive. Then, the Laguerre matrix polynomials L (A,λ n (x are defined by [9]: (5.1 L (A,λ n (x = n k= ( 1 k k!(n k! (A + I n [(A + I k ] 1 (λx k, n N. These matrix polynomials have the following generating matrix function: (5. n= λxt L (A,λ n (xt n =(1 t A I e 1 t ; x C,t C, t < 1. Recall that, if M C r r is a positive stable matrix and n Z +, from [1], we have (5.3 Γ(M = lim (n 1! (M 1 n n nm. We are in a position to give the first relation of Bessel matrix functions of the first kind and Laguerre matrix polynomials. Theorem 6. Bessel matrix functions of the first kind and Laguerre matrix polynomials satisfy the relation { ( x (5.4 n H L n n} (H,λ =(λx 1 H J H ( λx lim n

10 58 Bayram Çek ım and Abdullah Altın where x and λ are positive real numbers, n Z +, and H is both Jordan block and positive stable matrix. Proof. Using (5.1,(5.3 and (.7, respectively, we get { ( x lim n H L (H,λ n n { n} n H (H + I n n ( n k λ k n! k! = lim n =Γ 1 (H + I k= =(λx 1 H (λx 1 H k= ( 1 k (H + I 1 k! k (λxk k= =(λx 1 H J H ( λx. Hence, the proof is completed. } ( x k (H + I 1 k n ( 1 k Γ 1 (H +(k +1I(λx k k! Corollary. Bessel matrix functions of the first kind and Laguerre matrix polynomials satisfy the following relation { ( x n A L (A,λ n =(λx n} 1 A J A ( λx lim n where x and λ are positive real numbers, n Z +, and A is a positive stable matrix. Lastly, we present the second relation of Bessel matrix functions of the first kind and Laguerre matrix polynomials. Theorem 7. Bessel matrix functions of the first kind and Laguerre matrix polynomials satisfy the relation (5.5 n= Γ 1 (H +(n +1IL (H,λ n (xt n = e t (λxt 1 H J H ( λxt where H is a Jordan block satisfying the condition (.8, and x, λ and t are positive real numbers.

11 Bessel Matrix Functions of the First Kind 59 Proof. Using (5.1 and (.7, we obtain = = n= n= n,k= Γ 1 (H +(n +1IL (H,λ n (xt n Γ 1 (H + I n! n k= ( n k k! (H + I 1 k (λxk t n ( 1 k n! k! Γ 1 (H +(k +1I(λx k t n+k = e t (λxt 1 H J H ( λxt, which completes the proof. Corollary 3. Bessel matrix functions of the first kind and Laguerre matrix polynomials satisfy the relation n= Γ 1 (A +(n +1IL (A,λ n (xt n = e t (λxt 1 A J A ( λxt where x, λ and t are positive real numbers, and A is a matrix satisfying the condition (.8. Acknowledgements. The authors would like to thank to editors and referees for their valuable comments and suggestions, which have improved the quality of the paper. References [1] Çekim, B. and E. Erkuş-Duman, Integral representations for Bessel matrix functions, Gazi Universtiy Journal of Science 7(1 (14, [] Defez, E. and L. Jódar, Some applications of the Hermite matrix polynomials series expansions, J. Comput. Appl. Math. 99 (1998, [3] Defez, E. and L. Jódar, Chebyshev matrix polynomials and second order matrix differential equations, Util. Math. 61 (, [4] Defez, E., Jódar, L. and A. Law, Jacobi matrix differential equation, polynomial solutions and their properties, Comput. Math. Appl. 48 (4,

12 53 Bayram Çek ım and Abdullah Altın [5] Dunford, N. and J. Schwartz, Linear operators, part I, Interscience, New York, [6] Jódar, L., Legua, M. and A. G. Law, A matrix method of Frobenius, and application to a generalized Bessel equation, Congressus Numerantium 86 (199, [7] Jódar, L., Company, R. and E. Navarro, Solving explicitly the Bessel matrix differential equation, without increasing problem dimension, Congressus Numerantium 9 (1993, [8] Jódar, L., Company, R. and E. Navarro, Bessel matrix functions: explicit solution of coupled Bessel type equations, Utilitas Mathematica 46 (1994, [9] Jódar, L., Company, R. and E. Navarro, Laguerre matrix polynomials and system of second-order differential equations, Appl. Num. Math. 15 (1994, [1] Jódar, L. and J. C. Cortés, Some properties of Gamma and Beta matrix functions, Appl. Math. Lett. 11(1 (1998, [11] Jódar, L. and J. C. Cortés, On the hypergeometric matrix functions, J. Comput. Appl. Math. 99 (1998, [1] Kishka, Z. M. G., Shehata, A. and M. Abul-Dahab, The generalized Bessel matrix polynomials, J. Math. Comput. Sci. ( (1, [13] Lebedev, N. N., Special functions and their applications, Dover, New York, 197. [14] Sastre, J. and L. Jódar, Asymptotics of the modified Bessel and incomplete Gamma matrix functions, Appl. Math. Lett. 16(6 (3, (Received February 8, 13 (Revised December 6, 13 Bayram Çek ım Gazi University Faculty of Science Department of Mathematics Teknikokullar, TR-65 Ankara, Turkey bayramcekim@gazi.edu.tr Abdullah Altın Ankara University Faculty of Science Department of Mathematics Tandoğan, TR-61 Ankara, Turkey altin@science.ankara.edu.tr