A New Extension of Bessel Matrix Functions

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1 Southeast Asian Bulletin of Mathematics (16) 4: Southeast Asian Bulletin of Mathematics c SEAMS. 16 A New Extension of Bessel Matrix Functions Ayman Shehata Deartment of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egyt Deartment of Mathematics, College of Science an Arts in Unaizah, Qassim University, Qassim, Kingom of Saui Arabia rshehata6@yahoo.com Receive 16 February 14 Accete 9 March 15 Communicate by H.M. Srivastava AMS Mathematics Subject Classification(): 33C1, 33C45, 33C6, 15A6 Abstract. The main aim of this aer is to efine a new family of secial matrix functions, say, -Hyergeometric, -Bessel, -moifie Bessel an -Tricomi matrix functions, where is a ositive integer. The convergence roerties of the -Hyergeometric matrix function are establishe an the matrix ifferential equation is constructe. The ifferential roerties, integral reresentations, integrals an various articular cases on the -Bessel matrix function have been obtaine. The -moifie Bessel matrix functions, an some roerties of this function are investigate. Finally, the -Tricomi matrix function an several results of the function are efine an stuie. Keywors: -Hyergeometric matrix function; -Bessel; -moifie Bessel an -Tricomi matrix functions; A Joran block; Differential roerties; Integral reresentations. 1. Introuction an Preliminaries Secial matrix functions aear in connection with statistics [4], mathematical hysics, theoretical hysics, grou reresentation theory, Lie grous theory [11], orthogonal matrix olynomials are closely relate [14]. The Bessel functions are among the most imortant secial functions with very iverse alications to hysics, engineering an mathematical analysis [1, 17, 4]. In [15, 16, 19, 4, 33], the hyergeometric matrix function has been introuce as a matrix ower series, hyergeometric matrix ifferential equation an an integral reresentation. In [1,, 3, 1, 13, 1], the authors introuce the

2 66 A. Shehata Bessel matrix functions an gave some results with Bessel matrix functions. In [8, 9,, 6, 7, 8, 3, 31, 3, 34, 35, 36, 37, 38, 39, 41], extension to the matrix function framework of the classical families of -Kummers, an q-aell, Humbert, Tricomi an Hermite-Tricomi matrix functions, an Gegenbauer, Rices, Humbert,Rainville s, Bessel an Konhauser matrix olynomials have been roose. The author has earlier stuie the an q-horn s H, l(m, n)-kummer matrix functions of two comlex variables uner ifferential oerators [5, 9]. The reason of interest for this family of Bessel function is ue to their intrinsic mathematical imortance an the fact that these olynomials have alications in hysics. The resent investigation is motivate essentially by several recent works [, 3, 1, 13,, 1, 5, 6, 9, 3]. From this motivation, we rove some new roerties for the new class of -hyergeometric, -Bessel, -moifie Bessel an -Tricomi matrix functions with comlex analysis. The outline of this aer is organize as follows: Section some basic relations involving the -hyergeometric matrix function, convergence roerties an matrix ifferential equation have been obtaine. Some new roerties, integral reresentations an integral exressions of -Bessel matrix function are given in Section 3. Results of Sections an 3 are use in Section 4 an 5 to stuy the roerties of the new class of -moifie Bessel an -Tricomi matrix functions. Here, we iscuss the roerties of the new class of -hyergeometric, -Bessel, -moifie Bessel an -Tricomi matrix functions an fix the notation in orer to make the section self-consistent. The following secialize versions of the efinitions an results of the revious works, which are useful in our stuy, are rehrase. Throughout this aer, for a matrix A in C N N, its sectrum σ(a) enotes the set of all eigenvalues of A. Its two-norm is enote by A, an is efine by Ax A su x x where x (x T x) 1 enotes the usual Eucliean norm of a vector x in C N. We say that a matrix A in C N N is a ositive stable matrix if R(µ) > for all µ σ(a) (see [14]). In this exression, R(z) is the real art of the comlex number z. The matrices I an O will enote the ientity matrix an the null matrix in C N N, resectively. For the urose of this work, we enote by µ(a) the logarithmic norm of A, which is efine by [6, 7] } µ(a) max {z;z eigenvalue of A+AT (1.1) where A T enotes the conjugate transose. We enote by the number µ(a) } µ(a) min {z;z eigenvalue of A+AT. (1.)

3 A New Extension of Bessel Matrix Functions 67 From [7], it follows that e At e tµ(a) for t >. Hence, t A ex(alnt) satisfies { t t A µ(a) if t 1, t µ(a) (1.3) if t 1. In [5], if f(z) an g(z) are holomorhic functions of the comlex variable z, which are efine in an oen set Ω of the comlex lane, an A, B are matrices in C N N with σ(a) Ω an σ(b) Ω, such that AB BA, then f(a)g(b) g(b)f(a). (1.4) Let P be a ositive stable matrix in C N N, then Gamma matrix function Γ(P) is efine by [14] ( ) Γ(P) e t t P I t; t P I ex (P I)lnt. (1.5) If A is a matrix in C N N such that A+nl is an invertible matrix for all integers n, then Γ(A) is an invertible matrix, an Pochhammer symbol (shifte factorial) is efine by (see [18]) (A) n A(A+I)(A+I)(A+3I)...(A+(n 1)I) Γ(A+nI)Γ 1 (A); n 1, (A) I. (1.6) For the urose of this our resent work, we recall here the following some concets an roerties of the matrix functional calculus. H is the N-imensional Joran block efine by n 1... n 1... n... H... C N N. (1.7) n 1... n Thus, if H is a Joran block of the form (1.7), z >, we can write the image by means of the matrix functional calculus acting on the matrix H an the function of n, J n (z), one can obtain Bessel matrix function of the first kin of orer H as follows: ( ) H ) z J H (z) Γ 1 (A+I) F 1 ( ;H +I; z ; z < ; arg(z) < π 4 ( 1) k ( ) H+kI z Γ 1 (H +(k +1)I). k! k (1.8)

4 68 A. Shehata for z <, arg(z) < π, where λ σ(h) is not a negative integer (see [1]). Let us take A C N N satisfying the conition λ is not a negative integer for every λ σ(a). (1.9) In [13], the Bessel matrix function J A (z) of the first kin of orer A was efine as follows: ( 1) k ( ) A+kI z J A (z) Γ 1 (A+(k +1)I) k! k ( ) A ) (1.1) z Γ 1 (A+I) F 1 ( ;A+I; z, 4 Now, we consier the general case. Let A be a matrix satisfying the conition (1.9) an H iag(h 1,...,H k ) be the Joran canonical form of A, where H i is a Joran block efine in the following form, n i 1... n i 1... n i... H i... C Ni Ni, (1.11) n i 1... n i for N i 1,N 1 +N N k N an H i (n i ) if H i is a Joran block of size 1 1 for n i is not negative integer for 1 i k. Bessel matrix function of the first kin can be written as the following J(t,H) [iag 1 i k (J Hi (t))]. (1.1) Furthermore, if P be an invertible matrix in C N N such that H iag(h 1,H,...,H k ) PAP 1, (1.13) then we have the same Bessel matrix function of the first kin of orer A in (1.1) (see [13]). Let C be a matrix in C N N such that C+nI is an invertible matrix for every integer n [, 3]. Then the hyergeometric matrix function can be efine by the matrix ower series in the form F 1 ( ;C;z) [(C) n ] 1zn n!. (1.14) n Let P an Q be ositive stable matrices in C N N, then Beta matrix function B(P,Q) is efine by [14] B(P,Q) 1 t P I (1 t) Q I t. (1.15)

5 A New Extension of Bessel Matrix Functions 69 If P, Q an P +Q are ositive stable matrices in C N N for which PQ QP, an P +ni, Q+nI an P +Q+nI are invertible matrices for n [15], then B(P,Q) Γ(P)Γ(Q)Γ l (P +Q). (1.16). Proerties of -Hyergeometric Matrix Function Let H be a Joran block as in (1.7). The -hyergeometric matrix function of the comlex variable z is efine by the ower series in the form ( ) F 1 1 ;H;z (k)! zk [(H) k ] 1 U k z k ; 1 (.1) k where U k 1 (k)! [(H) k] 1, (H) k is efine by (1.6) an 1. If k > H, then by erturbation lemma [5], we can write ( H k +I) 1 k k k H, (.) (H +ki) 1 k( H k +I) 1, (.3) for k > H. The convergence roerties of -hyergeometric matrix function, we use the ratio test with (.1), (.) an (.3) as in the form lim U k+1 z k+1 k U k z k lim (k)![( H k +I)] 1 z k+1 k k((k +1))! z k z (H k +I) 1 lim k k +1 (+ 1 k )(+ k )(+ k )...(+ 1 k ) 1 lim k k +1 (+ 1 k )(+ k )...(+ 1 H k )(1 k ) z for all z. Thus, the ower series (.1) is convergent for all comlex number z. Hence, the -hyergeometric matrix function is an entire function. Now, we aen this section by introucing the ifferential oerator θ z to the entire function in successive manner as the following θ(θ 1 )(θ )(θ 3 )(θ 4 ( ) 1 )...(θ ) F 1 ;H;z 1 k(k 1)...(k +1) z k [(H) k ] 1 (k)! k1 1 1 (k)! zk+1 [H +ki] 1 [(H) k ] 1 z ( ) H 1 F 1 ;H +I;z k

6 7 A. Shehata i.e, equivalently θ(θ 1 )(θ 1 )...(θ ) F 1( ;H;z) z ( ) H 1 F 1 ;H +I;z. Similarly to (.4), we can write F 1 ( ) ;H;z, then θ(θ 1 )(θ )(θ 3 ( ) 1 )...(θ )(θ I +H I) F 1 ;H;z 1 k1 (k I +H I) z k [(H) k ] 1 1 (k )! k z F 1( ;H;z). These results are summarize in the following. 1 (k)! zk+1 [(H) k ] 1 (.4) Theorem.1. Let H be a Joran block as in (1.7), then the -hyergeometric matrix function F 1( ;H;z) is a solution of the matrix ifferential equation of orer +1 [ θ(θ 1 )(θ 1 )...(θ )(θ I +H I) z ] ( ) F 1 ;H;z. (.5) Remark.. Putting 1, we get the hyergeometric matrix function F 1 ) 1 F 1 ( ;H;z k! zk [(H) k ] 1. k The revious roerties can be generalize as the following: Let H i is efine as in (1.7) an H iag(h 1,H,...,H k ) be a matrix in C N N. Here N i is not negative integer for 1 i k. The matrix H iag(h 1,H,...,H k ), the roerties ( ) can be easily rovie. Also, if P be an invertible matrix in C N N such that H iag(h 1,H,..., H k ) PCP 1, we can give the following theorem for the matrix C. Theorem.3. Let C be a matrix in C N N such that the matrix C + ni is an invertible matrix for every integer n. Then we have [ θ(θ 1 )(θ 1 )...(θ )(θ I +C I) z ] ( ) F 1 ;C;z. where F 1 ( ) ;C;z k 1 (k)! zk [(C) k ] 1 ; 1.

7 A New Extension of Bessel Matrix Functions 71 Remark.4. For 1, the -hyergeometric matrix function F 1 an entire function. ( ) ;C;z is 3. On -Bessel Matrix Function The basic Bessel function is efine when < 1. If 1 we try to efine a new class of matrix function, say, -Bessel matrix function; 1; of the first kin as in the following form ( ) H ) z J H (z) Γ 1 (H +I) F 1 ( ; H +I; z 4 an F 1 k ) ( ; H +I; z 4 ( 1) k (k)! Γ 1 (H +(k +1)I) Γ(H +I) k ( ) H+kI (3.1) z ( 1) k ( z (k)! Γ 1 (H +(k +1)I) where H is a Joran block as in (1.7), an the -Bessel matrix function is an entire function. For the -Bessel matrix function which is efine in (3.1), we have: Theorem 3.1. Let H be a Joran block as in (1.7) an J H (z) is efine in (3.1). Then the following ifferential roerties are satisfie [ ] z H J H (z) z H J H I (z), ( ) m [ ] 1 z H J H (z) z H mi J H mi (z), z z J H (z) z J H I (z) H J H (z), ( )( ) ( ) z A J A+I (z). ( 1 )[ ] z H J H (z) ) k (3.) Proof. From (3.1), we have [ ] z H J H (z) [ ( 1) k ] n+k (k)! Γ 1 (H +(k +1)I)z H+kI. k Differentiating term by term of the above equality, we get [ ] z H ( 1) k (H +ki) J H (z) n+k Γ 1 (H +(k +1)I)z H+(k 1)I. (k)! k1

8 7 A. Shehata On setting k r+1 an simlification, gives [ ] z H J H (z) z H ( 1) r H+(r 1)I (k)! Γ 1 (H I +(r+1)i)z H+(r 1)I. r This can be written as [ ] z H J H (z) z H J H I (z). (3.3) From (3.3) an mathematical inuction, we obtain ( 1 z ) m [ ] z H J H (z) z H mi J H mi (z). (3.4) Develoment of the left han sie of (3.3), we have the relation z J H (z) z J H I (z) H J H (z). (3.5) This relation is calle a matrix ifferential recurrence relation. Insert the factor z H on each sie of equation (3.1), an ifferentiating term by term, we get ( )( ) ( )[ ] z H J H (z) ( ) k1 ( 1) k (k)(k 1)...(k + 1) H+kI Γ 1 (H +(k +1)I)z k 1 (k)! 1 ( ) z H J H+I (z). Thus the roof is comlete. (3.6) Remark 3.. Taking 1, we rouce the Bessel matrix function J H (z) (see [, 3]). The revious roerties can be generalize as the following: Let H i is efine as in (1.7) an H iag(h 1,H,...,H k ) be a matrix in C N N. Here N i is not negative integer for 1 i k. The matrix H iag(h 1,H,...,H k ), the roerties ( ) can be easily rovie. Furthermore, if P be an invertible matrix in C N N such that H iag(h 1, H,...,H k ) PAP 1, then we can give the following theorem for the matrix A. Theorem 3.3. Let A be a matrix in C N N satisfy the conition (1.9) an J A (z)

9 A New Extension of Bessel Matrix Functions 73 is efine in (3.1). Then the following ifferential roerties are satisfie ( ) A ) z J A (z) Γ 1 (A+I) F 1 ( ; A+I; z 4 ( 1) k (k)! Γ 1 (A+(k +1)I)( z )A+kI, k [ ] z A J A (z) z A J A I (z), ( ) m [ ] 1 z A J A (z) z A mi J A mi (z), z z J A(z) z J A I (z) A J A (z), ( )( ) ( ) z A J A+I (z). ( 1 )[ ] z A J A (z) Remark 3.4. Putting 1, we rouce the Bessel matrix function J A (z) (see [1]). Now, the integral reresentations of the -Bessel matrix function efine in (3.1) is given thrash the following theorem. Theorem 3.5. Let H be a Joran block matrix as in (1.7) such that µ(h) > 1 an argz < π. Then the integral reresentation for -Bessel matrix function is given by J H (z) where F 1 ( ) H z Γ 1 (H + 1 I) 1 π 1 (1 t ) H 1 I F 1 ( ) ; 1 t ; z 4 ( ) k ( 1) k 1 zt k (k)! ( 1 ) k. Proof. Using the Gauss s multilication theorem i1 ( ; 1 ; z t 4 ) t (3.7) m Γ(mH) (π) 1 m m mh 1 I Γ(H + i 1 I) m 1,,3,... (3.8) m By the Lemma of [15], if k > 1, one gets the ientity Γ 1 (H +(k +1)I) Γ 1 (H + 1 I) Γ(k + 1 ) 1 1 t k (1 t ) H 1 I t. (3.9)

10 74 A. Shehata From (3.1), (3.8) an (3.9), we obtain ( 1) k ( ) H+kI z Γ 1 (H + 1 J H (z) I) 1 (k)! Γ(k + 1 k ) t k (1 t ) H 1 I t 1 ( ) H z Γ 1 (H + 1 I) 1 ( 1) k ( ) k 1 zt π (k)! ( 1 ) (1 t ) H 1 I t. k 1 k Theorem 3.6. Let H be a Joran block as in (1.7) such that µ is not a negative integer for every µ σ(h). Then the another integral reresentation for -Bessel matrix function is where F ( z J H (z) ( ) ; ; z 4s k ) H 1 e s s (H+I) π i F c ( 1) k (k)! ( z 4s) k. ) ( ; ; z s (3.1) 4s Proof. Another integral reresentation of J H (z) can be establishe starting from the formula Γ 1 (H +(k +1)I) 1 e s s (H+(k+1)I) s. (3.11) π i Using (3.1) an (3.11), we have ( 1) k ( z J H (z) (k)! k ( ) H z 1 π i c ) H+kI 1 e s s (H+(k+1)I) s π i c ( ) z e s s (H+I) k s. c k ( 1) k (k)! 4s These integral reresentations can be generalize through the following results. Theorem 3.7. (i) Let A be a matrix in C N N such that µ(a) > 1 an argz < π. Then we obtain ( ) A z Γ 1 (A+ 1 J A (z) I) 1 (1 t ) A 1 I π F 1 ( ; 1 t ) ; z t. 4 (ii) If A is a matrix in C N N satisfy the conition (1.9). Then we obtain ( ) A ) z 1 J A (z) e s s (A+I) π i F ( ; ; z s. 4s c 1

11 A New Extension of Bessel Matrix Functions 75 Here, we erive some integrals exressions involving -Bessel matrix function efine in (3.1). These are containe in the next theorem. Theorem 3.8. If H, M an H M are Joran blocks as in (1.7) satisfying the conitions µ(h) > ν(m) > 1, η(h M) > 1 where η µ ν, we obtain ( ) H M z 1 J H (z) Γ 1 (H M) (1 t ) H M I t M+I J M (zt)t. (3.1) Proof. Consier the integral matrix functional 1 (1 t ) H M I t M+I J M (zt)t an substitute the series (3.1) for J M (z) to obtain 1 k (1 t ) H M I t M+I J M (zt)t ( 1) k ( ) M+kI z 1 (k)! Γ 1 (M +(k +1)I) (1 t ) H M I t M+(k+1)I t. Let u t in the last integral, we get k k ( 1) k (k)! Γ 1 (M +(k +1)I)( z )M+kI1 1 ( 1) k (k)! (z )M+kI1 Γ(H M)Γ 1 (H +(k +1)I) 1 Γ(H M)(z )M H J H (z) this is the require result. (1 u) H M I u M+kI u One can generalize the above result as follows. Theorem3.9. If A, B an A B are matrices in C N N satisfying the conitions, ñ(a) > m(b) > 1, r(a B) > 1 for all r n m, then we obtain ( ) A B z 1 J A (z) Γ 1 (A B) (1 t ) A B I t B+I J B (zt)t. This class of integrals forms is consiere in the next theorem. Theorem 3.1. If H, M an H + M are Joran blocks as in (1.7) satisfying the conitions µ(h) > 1, η(h +M) > for all η µ+ν, a, b are arbitrary

12 76 A. Shehata numbers with R(a±ib) > an HM MH, then e az z M I J H (bz) ) H ( b Γ(H +M)Γ 1 (H +I)a M a ( 1 F 1 (H +M), 1 ) (H +M +I);H +I; b a (3.13) where ( 1 F 1 k (H +M), 1 (H +M +I);H +I; b a ( 1) k ( 1 (H +M)) k( 1 (H +M +I)) k (k)! ( b a ) ) k [(H +I) k ] 1 ; b a < 1. Proof. First assume that b a < 1, substitute the series for J A (bz) in the left han sie of the integral (3.13) to get k k e az z M I J H (bz) ( 1) k Γ 1 (H +(k +1)I) (k)! ( b )H+kI e az z H+M+(k 1)I ( 1) k Γ 1 (H +(k +1)I) ( b +M +ki) )H+kIΓ(H (k)! a H+M+kI. Alying the ulication formula for the Gamma matrix function, we have Γ(H +M) H+M I Γ( 1 π (H +M))Γ(1 (H +M +I)), Γ(H +M +ki) H+M+(k 1)I Γ( 1 (H +M)+kI) π (3.14) Γ( 1 (H +M +I)+kI) an Γ( 1 (H +M)+kI) (1 (H +M)) kγ( 1 (H +M)), Γ( 1 (H +M +I)+kI) (1 (H +M +I)) kγ( 1 (H +M +I)). (3.15)

13 A New Extension of Bessel Matrix Functions 77 Using the equations (3.14) an (3.15), we get e az z M I J H (bz) Γ(H +M)Γ 1 (H +I) ( 1) k b H+kI ( 1 (H +M)) k( 1 (H +M +I)) k[(h +I) k ] 1 H a H+M+kI. (k)! k Hence, the roof is comlete. Furthermore, if P is an invertible matrix in C N N such that H iag(h 1, H,...,H k ) PAP 1, then one can get the following theorem for the matrix A. Theorem3.11. If A, B an A+B are matrices in C N N satisfying the conitions ñ(a) > 1, r(a + B) > for all r n + m, a, b are arbitrary numbers with R(a±ib) > an AB BA, then e az z B I J A (bz) ) A ( b Γ(A+B)Γ 1 (A+I)a B a ( 1 F 1 (A+B), 1 ) (A+B +I);A+I; b a ; b a < 1. When M H +I or H +I, the F 1 in (3.13) reuces to a 1 F, which can be summe by the binomial theorem for b a < 1. Corollary 3.1. If H is a Joran block as in (1.7) such that µ(h) > 1 b are arbitrary numbers with R(a±ib) >, then e az z H J H (bz) (b)h Γ(H + 1 I) πa H+I 1 F (H + 1 I, ; ; b a ) an a, (3.16) where 1 F (H + 1 I; ; b a ) k ( 1) k (H + 1 I) k (k)! ( ) k b ; b a a < 1. Proof. From (3.1) an b a < 1, in the left han sie of the integral (3.16) can be

14 78 A. Shehata written k e az z H J H (bz) e az z H ( 1) k Γ 1 ( ) H+kI (H +(k +1)I) bz (k)! k ( 1) k Γ 1 ( ) H+kI (H +(k +1)I) b e az z H+kI. (k)! On the other han, with the hel of Gamma matrix function in (1.5), we get e az z H+kI 1 a H+(k+1)I e t t H+kI t 1 ah+(k+1)iγ(h +(k +1)I). (3.17) Using (3.18) an the ulication formula for the Gamma matrix function, we obtain k e az z H J H (bz) ( 1) k Γ 1 (H +(k +1)I) (k)! Thus, the roof is establishe. ( ) H+kI b Γ(H +(k +1)I). a H+(k+1)I Corollary3.13. If H is a Joran block as in (1.7) satisfying the conition µ(h) > 3 an a, b are arbitrary numbers with R(a±ib) >, then e az z H+I J H (bz) (b)h Γ(H + 3 I) πa H+I 1 F (H + 3 I, ; ; b a ) (3.18) where 1 F (H + 3 I; ; b a ) k ( 1) k (H + 3 I) k (k)! ( b a ) k ; b a < 1. Proof. The result (3.18) follows immeiately form (3.16) if we ifferentiate both sies with resect to a. Thus, the roof is comlete. The next corollary gives another integral for -Bessel matrix function.

15 A New Extension of Bessel Matrix Functions 79 Corollary3.14. If H is a Joran block as in (1.7) satisfying the conition µ(h) > 1, an a an b are arbitrary numbers with R(a±ib) > an b a < 1, then 1 a e az J H (bz) ( ) H ( b 1 a F 1 (H +I), 1 ) (H +I);H +I; b a. (3.19) Proof. From the efinition of J H (z), we have k e az J H (bz) ( 1) k Γ 1 (H +(k +1)I) (k)! e az ( 1) k Γ 1 (H +(k +1)I) (k)! k ) H+kI e az z H+kI. ( b ( ) H+kI bz Using the Gamma matrix function, we get the integral on the right han sie equals an e az z H+kI Γ(H +(k +1)I) a H+(k+1)I Γ(H +(k +1)I) H+kI π Γ((k + 1 )I + 1 H)Γ((k +1)I + 1 H). Evaluating the inner integral, we get k bn πa H+I 1 a ( b a )H e az J H (bz) ( 1) k Γ 1 (H +(k +1)I) (k)! ( ) H+kI b Γ(H +(k +1)I) a H+(k+1)I Γ((k+ 1 )I + 1 H)Γ((k+1)I+ 1 H)Γ 1 (H+(k+1)I) (k)! k ( 1 (H +I)) k( 1 (H +I)) k[(h +I) k ] 1 ) k ( b (k)! a. k Thus, the result is establishe. ) k ( b a Corollary3.15. If H is a Joran block as in (1.7) satisfying the conition µ(h) >

16 8 A. Shehata, an a, b are arbitrary numbers with R(a±ib) > an b a < 1, then e az z 1 J H (bz) ( ) H ( b H 1 1 a F 1 H, 1 ) (H +I);H +I; b a. (3.) Proof. The roof of the corollary is very similar to corollary Now, let P be an invertible matrix in C N N such that H iag(h 1,H,..., H k ) PAP 1, we give a generalization for the above results in the following corollary for the matrix A. Corollary (i) If A is a matrix in C N N such that ñ(a) > 1 an a, b are arbitrary numbers with R(a±ib) >, then we have for b a e az z A J A (bz) (b)a Γ(A+ 1 I) πa A+I < 1. 1 F (A+ 1 I, ; ; b a (ii) If A is a matrix in C N N such that ñ(a) > 3, a, b are arbitrary numbers with R(a±ib) > an b a < 1, then we have e az z A+I J A (bz) (b)a Γ(A+ 3 I) πa A+I 1 F (A+ ), 3 I, ; ; b a (iii) If A is a matrix in C N N satisfying the conition ñ(a) > 1, an a, b are arbitrary numbers with R(a±ib) > an b a < 1, then we obtain e az J A (bz) 1 a ). ( ) A ( b 1 a F 1 (A+I), 1 ) (A+I);A+I; b a. (iv) If A is a matrix in C N N satisfying the conition ñ(a) >, a an b are arbitrary numbers with R(a±ib) > an < 1, then we obtain ( ) A ( b e az z 1 J A (bz) A 1 1 a F 1 A, 1 ) (A+I);A+I; b a. b a

17 A New Extension of Bessel Matrix Functions 81 The next theorem gives another infinite integral of a -Bessel matrix function ue to Hankel integral. Theorem For H, M an H+M are Joran block as in (1.7) satisfying the conitions µ(h) > 1, r(h +M) > for all r µ+ν, a an b are arbitrary numbers with R(a ) >, then we have e a z z M I J H (bz) 1 a H Γ( 1 (H +M))Γ 1 (H +I) ( ) H b 1 a F 1 ( 1 b (H +M);H +I; 4a ), (3.1) where ( ) 1 1 F b 1 (H +M);H +I; 4a ( 1) k ( 1 (H +M)) k[(h +I) k ] 1 ( ) k b. (k)! a k Proof. The integral can be evaluate using term by term integration as follows k e a z z M I J H (bz) ( 1) k Γ 1 (H +(k +1)I) (k)! an using the Gamma matrix function Thus, the result is establishe. ( ) H+kI b e a z z H+M+(k 1)I e a z z H+M+(k 1)I Γ(1 (H +M)+kI) a H+M+kI. Corollary3.18. If H is a Joran block as in (1.7) satisfying the conition µ(h) > 1 an a is an arbitrary number with R(a ) >, then we obtain e a z z H+I b H ) J H (bz) (a ) H+I F ( ; ; b (3.) 4a where ) F ( ; ; b 4a k ( 1) k ( ) k b. (k)! a

18 8 A. Shehata Proof. From (3.1), we have k e a z z H+I J H (bz) ( 1) k Γ 1 (H +k +1) (k)! Since the integral on the right han sie equals to e a z z H+(k+1)I ( ) H+kI b e a z z H+(k+1)I. Γ(H +(k +1)I) a H+(k+)I. Interchange the orer of the integral an summation, we get e a z z H+I J H (bz) Thus, we have the result. k ( 1) k (k)!a H+(k+)I ( ) H+kI b. Generalization for the above roerties are given in the following theorem an corollary. Theorem For A, B an A+B are matrices in C N N satisfying the conition ñ(a) > 1, r(a + B) > for all r n + m an a is an arbitrary number with R(a ) >, then the -Bessel matrix function satisfies the following reresentation e a z z B I J A (bz) 1 a A Γ( 1 ( ) A ( ) b (A+B))Γ 1 1 (A+I) 1 a F b 1 (A+B);A+I; 4a. Corollary 3.. If A is a matrix in C N N satisfying the conition ñ(a) > 1 an a is an arbitrary number with R(a ) >, then we obtain e a z z A+I J A (bz) b A ) (a ) A+I F ( ; ; b 4a. 4. On -moifie Bessel Matrix Function The urose of this section is to introuce an stuy a new class of matrix function, namely the -moifie Bessel matrix function an erive some of their

19 A New Extension of Bessel Matrix Functions 83 roerties. IfH isajoranblockasin(1.7)satisfyingtheconitions µ(h) > 1 an arg(z) < π. Then the -moifie Bessel matrix function I H (z) can be efine as in the following: I H (z) i H J H (iz) ( ) H z Γ 1 (H + 1 I) 1 (1 t ) H 1 I π F 1 ( ; 1 ; z t ) t 4 1 (4.1) an ( z I H (z) k ) H Γ 1 (H +I) F 1 1 (k)! Γ 1 (H +(k +1)I) ) ( ; H +I; z 4 ( ) H+kI (4.) z As in the same receing manner we can fin the recurrence relations of I H (z) as follows: ( 1 z [ ] z H I H (z) z H I H I (z), (4.3) z I H (z) z I H I (z) H I H (z), ] (4.4) z H mi I H mi (z) (4.5) ) m [ z H I H (z) an ( )( ) ( )[ ] z H I H (z) ( (k)(k 1)...(k + 1) ) H+kI Γ 1 (H +(k +1)I)z k 1 (k)! k1 1 ( ) z H I H+I (z). (4.6) These relations can be generalize as in the following: Let H i is efine by (1.7) an H iag(h 1,H,...,H k ) be a matrix in C N N. Here N i is not negative integer an zero for 1 i k. For matrix H iag(h 1,H,...,H k ), the roerties ( ) can be easily rovie. Also, if P be an invertible matrix in C N N such that H iag(h 1,H,..., H k ) PAP 1, we can the following theorem for the matrix A.

20 84 A. Shehata Theorem 4.1. If A is a matrix in C N N such that (1.9), we obtain [ ] z A I A (z) z A I A I (z), an ( 1 z z I A (z) z I A I (z) A I A (z), ] z A mi I A mi (z) ) m [ z A I A (z) ( )( ) ( )[ ] z A I A (z) 1 ( ) z A I A+I (z). Remark 4.. For 1, reuces to efinition for the moifie Bessel matrix function I A (z), see [1]. 5. On -Tricomi Matrix Function The urose of this section is to introuce an stuy a new class of -Tricomi matrix function of comlex variable z which can be state in the following form: C H (z) k ( 1) k Γ 1 (H +(k +1)I)z k (k)! (5.1) where H is a matrix in C N N such that z is not a negative integer for every z σ(h), this is known as -Tricomi matrix function an is characterize by the following link with -Bessel matrix function of comlex variables: C H (z) z 1 H J H ( z). (5.) With the sam way, we can euce recurrence relations for C H (z) as in the following: [ ] z H C H (z) z H C H I (z), (5.3) an ( 1 z z C H (z) z C H I (z) H C H (z), ] (5.4) z H mi C H mi (z) (5.5) ) m [ z H C H (z) ( 1 )( ) (... 1 )[ ] z H C H (z) ( 1 )z H C H+I (z). (5.6)

21 A New Extension of Bessel Matrix Functions 85 Theorem5.1. Let H be a Joran block as in (1.7), then the C H (z) is a solution of the matrix ifferential equation of orer +1 [ θ(θ 1 )(θ 1 )...(θ )(θ I +H)+ z ] C H (z). (5.7) Proof. From (5.1) an the ifferential oerator θ, we get θ(θ 1 )(θ )(θ 3 1 )...(θ )(θ I +H) C H (z) 1 1 (k)! zk+1 [(H) k ] 1 z C H (z). k Remark 5.. Putting 1, we get the Tricomi matrix ifferential equation of secon orer z C H(z)+(H +I) C H(z)+C H (z). (5.8) Theorem 5.3. The -Tricomi matrix function satisfies the matrix recurrence relations [ ] z A C A (z) z A C A I (z), z C A (z) z C A I (z) A C A (z), ( 1 )( ) (... 1 )[ ] z A C A (z) ( 1 )z A C A+I (z) an ( 1 z ) m [ ] z A C A (z) z A mi C A mi (z) where A is a matrix in C N N, P be an invertible matrix in C N N such that H iag(h 1,H,...,H k ) PAP 1 satisfy z is not a negative integer for every z σ(h). Theorem 5.4. Let A be a matrix in C N N satisfying the conition (1.9), then the -Tricomi matrix function is a solution of the matrix ifferential equation [ θ(θ 1 )(θ 1 )...(θ )(θ I +A)+ z ] C A (z). (5.9)

22 86 A. Shehata Remark 5.5. Putting 1, we get the Tricomi matrix ifferential equation of secon orer z C A(z)+(A+I) C A(z)+C A (z). (5.1) In a similar manner as in the roof of Theorem 3.5, we erive in the following theorem. Theorem 5.6. Let H be a Joran block as in (1.7) such that µ(h) > 1, the integral reresentation of -Tricomi matrix function is given as C H (z) Γ 1 (H + 1 I) 1 (1 t ) H 1 I π F 1 ( ; 1 ) ; zt t. (5.11) 1 The following theorem can be rove using the same technique as in Theorem 3.7. Theorem 5.7. Let H be a Joran block as in (1.7) satisfy the conition µ is not a negative integer for every µ σ(h). The -Tricomi matrix function hols the following integral reresentation: C H (z) 1 ( e s s (H+I) π i F c ; ; z s ) s. (5.1) The revious results can be generalize as in the following. Let H i is efine by (1.7) an H iag(h 1,H,...,H k ) be matrix in C N N. Here N i is not negative integer for 1 i k. For matrix H iag(h 1,H,...,H k ), the roerties ( ) can be easily rovie. Also, if P be an invertible matrix in C N N such that H iag(h 1,H,..., H k ) PAP 1, we give the following theorem. Theorem 5.8. (i) Let A be a matrix in C N N such that µ(a) > 1, the integral reresentation of -Tricomi matrix function is given as C A (z) Γ 1 (A+ 1 I) 1 (1 t ) A 1 I π F 1 ( ; 1 ) ; zt t. 1 (ii) Let A be a matrix in C N N satisfy the conition (1.9). The -Tricomi matrix function hols the following integral reresentation: C A (z) 1 ( e s s (A+I) π i F ; ; z ) s. c s

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