A New Extension of Humbert Matrix Function and Their Properties

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doi:10.4236/apm.2011.16057 Published Online November 2011 (http://www.SciRP.org/journal/apm)

A New Extension of Humbert Matrix Function and

Their Properties

Ayman Shehata, Mohamed Abul-Dahab* 1

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt 2

Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt E-mail:*[email protected]

Received August 15, 2011; revised October 22, 2011; accepted October 30, 2011

Abstract

This paper deals with the study of the composite Humbert matrix function with matrix arguments J_{A B}_,

 

z .

The convergence and integral form this function is established. An operational relation between a Humbert matrix function and Kummer matrix function is studied. Also, integral expressions of this relation are de-duced. Finally, we define and study of the composite Humbert Kummer matrix functions.

Keywords:Hypergeometric Matrix function, Humbert Matrix Function, Kummer Matrix Function, Integral

Representations

1. Introduction

Special matrix functions appear in the literature related to statistics [1-4] and more recently in connection with matrix analogues of Laguerre, Hermite and Legendre differential equations and the corresponding poly- nomial families [5-7]. The connection between the Humbert matrix function and modified Bessel matrix function has been established in [8,9]. In recent papers [10,11], we defined and studied the Humbert matrix functions. The Kummer’s confluent hypergeometric function belongs to an important class of special functions of the mathematical physics with a large number of applications in different branches of the quantum mecha- nics atomic physics, quantum theory, nuclear physics, quantum electronics, elasticity theory, acoustics, theory of oscillating strings, hydrodynamics, random walk theory, optics, wave theory, fiber optics, electromagnetic field theory, plasma physics, the theory of probability and the mathematical statistics, the pure and applied mathematics in [3,4,12-14]. Recently, an extension to the Kummer matrix function of complex variable is appeared in [15]. The first author has earlier studied the certain Kummer matrix function of two complex variables under certain differential and integral operators [16]. The primary goal of this paper is to consider a new system of matrix functions, namely the composite Humbert matrix function, Humbert Kummer matrix function and composite

Humbert Kummer matrix function.

The paper is organized as follows: Section 2 is define and study of the composite Humbert matrix function. The convergence and integral form is established. In Section 3 an operational relation between a Humbert matrix function and Kummer matrix function is given. Integral expressions of Humbert Kummer matrix functions are deduced. In Section 4 we defined and studied of the composite Humbert Kummer matrix functions.

Throughout this paper 0 will denote the complex

plane. A matrix is a positive stable matrix in D

P _CN N

if Re

 

 > 0 for all  

 

P where 

 

P is the set of all eigenvalues of and its two-norm denoted by

P

2

0 ₂

=sup ,

x

Px P

x



where for a vector in y CN,

 

1 2 2 =

T

y y y is the

Euclidean norm of y.

Let 

 

P and 

 

P be the real numbers which were defined in [17] by

 



 



 



 



= max : ,

= min : .

P Re z z P

 

 



 (1.1)

If f z

 

and g z

 

z

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A. SHEHATA ET AL. 316

N N

C  such that , then from the properties of the matrix functional calculus [18], it follows that

 

P

  

   

=

   

.

f P g P g P f P (1.2)

Hence, if Q in _CN N

=

PQ QP

  

is a matrix for which and if , then

 

Q

  



=

   

.

f P g Q g Q f P (1.3)

The reciprocal Gamma function denoted by

 

_{ }

1 z





1 =

z



z

is an entire function of the complex va-

riable . Then the image of acting on denoted by is a well-defined matrix. Further- more, if

 

1 z



 P

 

P

invertible fo





  

1

P I P

P nI  P

 

 







=lim 1

1





is r all integer 0.

PnI n (1.4)

The Pochhammer symbol or shifted factorial defined [17] by

 







 

₀

= 1

; 1; =

n

P P n I

n P I



   (1.5)

Jódar and Cortés have proved in [17] that

 

1

! P.

n

P 

 

 

n

P n



 

Q

n (1.6) Let and be two positive stable matrices in P

N N

C  . The gamma matrix function and the beta matrix function have been defined in [19] as follows

 

P





,



B P Q

 

= ₀e = exp

t P

P t

t P

  









1

0

, = P I

Q



t 









d ;

ln

I

P I

t I





t

.

(1.7)

and



1



Q Id

B P t  t (1.8) The Schur decomposition of , was given by [18] in the form:

P 1 2

; 0,

!

s

tP

P r t t s

 

 

 

 ( ) 1 

=0

e et P r s 

_



and

1 2 1 ( )

=0

ln ; !

s

r

P P

s

P r n

n n n

s

 

 

 

 





2. Composite Humbert Matrix Function

Let us introduce the following notation (see [20])





 





 





 



 









 







1 2 1 2 1 2 1 2

1 2 1 2 1 2

1 2

1 ₁ 2 ₂

= , , , ,

= ,

= , , , ,

= ,

= , , , ,

= ,

=

s s k

k k

k _s

s s

s

k k k _ks

s

k k k _ks

k k k k

k k k k

z z z z

A A A A

A A A A

B B B B

B B B B

A I A I A I A I

B I B I B I B I

  

   

  



 

(2.1)

and





, = _{1 1}, , _{2 2}, , , ,

A B A B A B A B_s _s

J J J  J .

Suppose that

 

1



 

1



,

3

0 2 =

3

, ; , ; ; = 1, 2, , 27

A_i B_i i

A B_i _i i i i

i i

z

J z A I B I

z

F A I B I i s



 

  _ _{ } _

   

 

 _     _

  

(2.2) are s Humbert matrix function with square complex matrices A A1, 2, , As and B B1, 2, , Bs of the same

order N. Construct the function

 

1



 

1



,

3 0 2

3 3

=0

3

, ; , ; 27

,

A B A B

A B kI A B kI k

z

J z A I

z

F A I B I

U z



 



   

 

B I

_{ }    

 

 

 _     _

 





(2.3)

where

 

1





1





3 3

1 1

= .

3 !

k

A B kI A B kI

1

A k I B k I

U

k

 

   

      

1.

 (1.9)

This function, will be called the composite Humbert matrix function of several complex variables

1, , ,2 s

z z  z . Now we prove that the matrix power series (2.3) convergence for all z30. Using the ratio test, we obtain



















 











3( 1)

1 1

3( 1) 3( 1)

3 3( 1)

( ) ₃ ( )

1 ₃

1 3

1 1

3 3

( )

2 2

1 _{= limsup}

limsup

3 1 !

1 ( 1 )

limsup .

1

3 !

A B k I A B k I

A B k I

A B kI A B k I

k _{A B} _kI k

A B kI

k

A k I B k I z

U z

R U z k

A k I B k I z

A kI B kI z

z k

k

  

 

     

    

 _{ } 

 _

 

 

 



     





   

   

 

(3)

Note that if k1 is large enough so that k1 > A , then by perturbation lemma, [13], we can write









1 1 1

1 =

1 1

A

A k I I

k k



  

  _  _

   

1

1 1

=

1 1 1

A I

k k k A



 _  _

 

     

(2.4)

hence







 







( ) ₁ ₁

1 1

3

1 1

= limsup

1.. 1 1 ..

1

.. 1 1 ..

| |

.

.. 1

k _s

s s

k k k A

k A k B

z

k B



 

   

 

   



 

  _

(2.5)

For positive numbers 1

R

i

 and positive integer , we can write

k

= , = 1, 2, 2 .

i i

k k i s (2.6) Substitute from (2.6) into (2.5) one gets

3( 1) 3( 1)

1

= limsup

A B k

A B k I

U _{ } _ z   

3

( ) ₃

I

A B kI

k _{A B} _kI

R  U _{ } z  

Thus, the power series (2.3) is absolutely convergent for all

= 0.

3 _<

z .

l Form he Composite Humbert Matrix unction

ert ction, we use the following formulas (see [10,

Integra of t

F

  

1 1 1

1

1 2 2

2 2 2 2 2

2

1 1

1

1 = exp d ,

2π

1

1 = exp d ,

2π

C

A k I C

A_s k_s I 1

1 1 1

1 1

1

1 = exp( ) d ,

2π

A k I

s s s s s

Cs

A k I r r r

i

A k I r r r

i

A k I r r r

i

   

  





(2.7) and

   

  

_









 

  









 

  









 

   

1

2

1

1 1 1

1 1 1 1 1

1

1 2 2

2 2 2 2 2

1 1

1

1 = exp d ,

2π

1

1 = exp d ,

2π

1

1 = exp d .

2π s

B k I C

B_s k_s I

s s _C s s s

B k I t t t

i

B k I t t t

i

B k I t t t

i

   

   



   



 

  





(2.8)

From (2.7) and (2.8) into the series expression of the composite Humbert matrix function given in (2.3), it follows that





 

   

 

  

1

3

1 1 1

, 2 ₁ 1 1

=0

1 1 1

1 3

= exp d

! 2π

.. exp d

exp d ..

A B kI k

A k I

A B s _C

k

A_s k I

s s

Cs

B k I

C

 

  1  .. exp d ,

s

B_s k_s I

s s

z

J ₁..

s C

z r r

k i

r r r

t t t

 

 _ _ _

  

   

 

 _{ }

 





(2.9) interchanging the order of the integral and summation,

r

t t   t





 

1

, 2 1 1 1

1

1 1 1

3

=0 3

= exp d ..

2

.. exp d

exp d ..

1

!

s

A B

A I

A B s C

A_s I

s s

Cs

B I C

k k

C

k

z

J z r r r

i

r r r

t t t

z

k





 

  

     

 

  



(2.10)

thus,

27 .. exp B_s I d ,

s s s

r t

t t t

   

 





 

   

_

_

1

3

, 2

1

1 1

3

= ( exp

27r t 2π

d d d d ,

s

A B

A B s C C_s C C

A I B I

s s

z

J z r t

i

r t r r t t



 

   

 

  _ _

  _{ }

 

 

3

1 2

1 1 2 2

= ...

27 27 27 27

s _.

s s z

z z

z

r t r t r t r t

 

  

 

 

Therefore, the following result has been established. Theorem 2.1 Let be matrices in CN N .

on for several and B

A

(4)

complex variables z z1, , ,2 zs

3. Humbert Kummer Matrix Fun

we will dedu mbert ma

satisfies integral in (2.11).

ction

section, ce a new matrix function that is a mixed from Hu trix function is in the form In this

 













1 1

, = 1

A nI

3 ,

0 2

; , 1 ;

27 3

A nI z

J z A



  _ _

   



x function i

I n I

n

 _ _

z

  (3.1)

F A I I

 _    _

 

and Kummer matri s in the form





   

=0 , ; ; = ! 3 n n n

K B C z

n  _{ }  



( n 

B C  z

3.2)



this function given the following





   

 

   

 









  



 





 

    





, , , ; = n n n A nI

B C _z

M A B C z J z

  _{ }  



=0 =0 3 1 1 2 =0 3 1 2 =0 1 =0 ! 3 3 !

1 1 1

3 ! 1 3 3 ! 1 , ! n A n n n k k k n k k k

n n n

n n B C z n z

k I A k I

k k

z I k

B C k I

n   _          _{ }          _{ }                   



by Theorem 2.1 in [9], we know that

n

  





1 1

A A k

z     

  



    















 

















1 =0 2 1 1 1 1 !

= , ; 1 ;1

= 1 1

1 1

n n n

n

B C k I

n F B C k I

k k I B C

k I B k I C

                      



, (3.3)

0. Hence







k 1 I B C



>

   





   

 



 



,

=0 ! 3

n n

1 1 1

3 1 3 3 3 =0 , , ; = 3 ; , , ; , 27 . n n n A nI A A kI A kI k

B C z

M A B C z J z

z

I B C A I I B I C

z

F I B C A I I B I C

U z              _{ }               _       _   



(3.4)

Provided that 0.



 

(B C) >

  Where A, B and are matrices in

C

N N

C  such that A(k1)I, (k1)IC and (k1)IB

This function, nction of are invertib will be complex va le fo riable r every the Hum

simplicity, we can write the Humber Kummer matri function in the form HKMF.

We define the radius of regularity of the function ber For integer k 

Kummer ma 1 . trix fu called z. x ( , , ; )

M A B C z in the form













 

1 3 3 ( ( ) ( ) 1 = limsup limsu A kI A kI k

I B B I C I

k U R k k            

This means MF is an entire function.

Integral Expressions of Humbert Kummer Matrix Function

In this section, we provide integral expressions of HKMF.

Suppose that )

1

)) ( ) ( 3

2

1 3

3 p

1 ! !

limsup = 0.

!

C A I I A k

A A kI

k k k k k k k             _         (3.5)  

that the the HK

(I(BC) and (AI) are matrices in

N N

C  such that











 























= ,

, and are positive stable.

I B C A I A I I B C

I B C A I A B C

     

    

(3.6) By (1.5) and (3.6) one gets













.

(3.7) By lemma 2 of [19] and (3.7), we see that























 















1 1 1 1 1 = 1 1

B C k I B C

A k

A I A B C

k I B C A B C A k I

                        



1 = k k

I B C A I

I      _  _  

















1 1

A I I

I B C

       



 



















1 1 ₍ ₎

0

= _k _{B C} 1 A B C Id .

C A k I

t t t

           





1 (3.8) From relation (3.7) and (3.8), we get



1k I BC  AB





























1 1 1

1 ₍ ₎

0 1 d .

k k

A B C I k B C

I B C A I

I B C A I A B C

t t t

(5)

Hence, for z <, we can write







 





  









 







  







1 1 1

0 =0

3

1 1

1 ( ) 0

3

1 1

=0 , , ;

3

1

3 !

3

1 d

1

3 A

k

k k

A B C I B C

k

k k

k

M A B C z

z

1 ₍ ₎

k B C 1 A B C Id

1 1 1

A

I B I C A B C

z

I B I C

k

t t t

z

I B A B C

t t t

z t

I B I C

  

 

    

 



 

_{ }       

 

 

   

 _  _{ }  _{ }

   

  

  

 

   

 _  _{ }  _{ } 

 







I C

  

 

_{ }    

 _{  }

  _





 







3

1 1 1

1 ( ) 0

3 3

0 2

!

3

1

; , ; d .

3

k

A

A B C I B C

k

z

I B I C A B C

t t

z t

F I B I C t

  

    



 

_{ }       

  

 _ _ 

 

 _   _ _ _

 



(3.10)

Summarizing, the following result has been established:

Theorem 3.1 Let A, B and C be matrices in CN N such that ( ) > 1,A

(I(BC))(AI)= (AI I)( (BC)) and

)

(I(BC) , (AI), (A B C) are positive stable. Then for z < it follows that







 





1 1 1

1 ( ) 0

3 3 0 2

, , ;

= 3

1

; , ; d .

3

A

A B C I B C

M A B C z

z

I B I C A B C

t t

z t

F I B I C t

  

    

        

   



 

 



 _ 

 

 _

 

    _ _ _

 _ _ 



 

 

 



Another integral representation of HKMF can be established starting from the formula in (2.7), we find that









 

1 ₁ ₌ 1 _exp ( _d

2π

A k I

C

( 1) )

A k I r r

i

 

  

_

  r

and substituting the above expression into the series expression of the HKMF given in (3.4), it follows that







 



 













 

1 1

3

1 1

=0

( ( 1) )

, , ;

= 3

1

3

! 1

exp d ,

2π

A

k k

k

A k I

C

M A B C z z

I B C I B I C

z

I B C I B I C

k

r r r

i

 



  

        

   

 

   

   _  _{ }  _{ }

  







interchanging the order of the integral and summation,







 



 













1 1 (

3

1 1

3 ) 1

, , ; = exp d

2 3

1

3

,

!

A

A I C

k k

k z

M

=0 k

A B C z I B C I B I C r r r

i

z

I B C I B I C

r k

   

 



        

    

 

   

   _  _{ }  _ _

  





i.e.,



 



 

1 1

3 ( )

1 2 ₃

1 =

2πi 3

exp ; , ; d .

3

A

A I C

z

M I B C I B I C

z

r r F I B C I B I C r

r

 

 

      

   

  _ _  

      _ _  

 

 _ _ _ _ 







(3.11)

Therefore, we obtain the following theorem:

Theorem 3.2 Let B and C be matrices in

N N

C  such that (B C ) <1. Then for z < , expre- .11) hold

(6)

mposite HKMF

Let

4. Co







 







1 1

3 1

1 3 , , ;

3

; , , ;

27 i i i i i

Ai i

i i i i

i i

i i i i i i

M A B C z

z

I B C A I I B

z

I C F I B C A I I B I C

 



 

_{ }       

 

 

  _       _,

ar rt m

 

e composite Humbe Ku mer matrix functions with square complex matrices Ai, Bi and Ci e same

< 1

of th order N, provided that (B_iC_i) .

ert Kum

Construct the composite Humb mer matrix s of these functions for any mode of arrangement,

function we put















 











1 1 1

3 1 3

, , ;

3

; , , ;

27

A

M A B C z

z

3

1 3 ; , , ;

I B C A I I B I C

z

F I B C A I I B I C

I B

  

 

_{ }         

 

 

 _       _

 



z

F I B C A I I C 

 _       _



where











 









 









 







1 2 1 2

1 1 1

1 2

= , , , ,

= ,

s

s s

k k _ks

s

k k k _ks

s

k k k _ks

B B B B

C C C C

I B C I B C I B C

I B I B I B I B

I C I C I C I C

     

   

 



and



1 2



= , , , s

M M M  M .

Then M is the composite Humbert Kummer matrix Now we calculate the radius of convergence of this function as follows

functions.

   

 

1 3 3

1 3

1 2

3

3 3

1 2

1

= limsup

limsup

! ! !

A kI A kI

k _k

A kI A

A A _s

s k

s

U R

k k k



  

 

 

 

 

 

 

 

 _ _

 





1 2

... 2 ... 2 ... 2

1 1 1 _, ₀

1 2

1 ,

= .

k

k k _s

k k_s k k_s k k_s k

k k _ks

k k



 

         

  

   

     

     

    



=0

as above putting ki=ik and using relation (1.9) and

the following relation

 

 



where

 





1 2

1 1

1

=0

1 ln

ln

!

ln e , j

1

=0 =0

ln

! !

j

r r

r

j j

j r

j

r A

A r k

A

r k

j

r k

 

 



  

  

  



we get

A

r k

j j



  

 



1 3 3 1

=limsup 0

A kI A kI

k _k

U

R 

  

 



 

  .

Then the composite Humbert Kummer matrix functions is an entire function.

5. References

tine and

ypergeometric Function of Two Ar-gument Matrix,” Journal of Multivariate Analysis, Vol. 2, No. 3, 1972, pp. 332-338.

doi:10.1016/0047-259X(72)90020-6

[1] A. G. Constan R. J. Mairhead, “Partial Differen-tial Equations for H

[2] A. T. James, “Special Functions of Matrix and Single Argument in Statistics in Theory and Application of Spe-cial Functions,” Academic Press, New York, 1975. [3] A. M. Mathai, “A Handbook of Generalized Special

Functions for Statistical and Physical Sciences, University Press, Oxford, 1993.

[4] A. M. Mathai, “Jacobians of Matrix Transformations and Matrix Argument,” World Scientific Pub-ng, New York, 1997.

[5] L. Jodar and E. Defez, “A Connection between Lagurre’s and Hermite’s Matrix Polynomials,” Applied Mathemat-ics Letters,Vol. 11, 1998, pp. 13-17.

[6] E. Defez and L. Jódar, “Chebyshev Matrix Polynomails and Second Order Matrix Differential Equations,”

Utili-“Some Applications of the Her-ynomials Series Expansions,” Journal of and Applied Mathematics, Vol. 99, No.

05-117.

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815-01-2

plied Mathematics Letters, Vol. 820. doi:10.1016/S0893-9659(03)900

, pp.

ied Mathematics Letters

Matrix Function

Approxi-s and Their

unctions and

he- [9] L. Jódar, R. Company and E. Navarro, “Bessel Matrix

Functions: Explict Solution of Coupled Bessel Type Equations,” Utilitas Mathematics, Vol. 46, 1994 129-141.

[10] Z. M. G. Kishka, A. Shehata and M. Abul-Dahab, “On Humbert Matrix Function,” Appl , [1 Article in Press.

[11] S. Z. Rida, M. Abul-Dahab, M. A. Saleem and M. T. Mohammed, “On Humbert

Ψ1(A,B;C,C';z,w) of Two Complex Variables under Dif-ferential Operator,” International Journal of Industrial Mathematics, Vol. 32, 2010, pp. 167-179.

[12] N. N. Lebedev, “Special Functions and Their Applica-tions,” Dover Publications Inc., New York, 1972. [13] L. Y. Luke, “The Special Functions and Their

mations,” Vol. 2, Academic Press, New York, 1969. [14] H. M. Srivastava and P. W. Karlsson, “Multiple Gaussian

Hypergeometric Series,” Ellis Horwood, Chichester, 1985.

[15] M. S. Metwally, “On p-Kummers Matrix Function of Complex Variable under Differential Operator

Properties,” Southeast Asian Bulletin of Mathematics, Vol. 35, 2011, pp. 1-16.

[16] A. Shehata, “A Study of Some Special F

Polynomials of Complex Variables,” Ph.D. Thesis, Assiut University, Assiut, 2009.

7] L. Jódar and J. C. Cortés, “On the Hypergeometric Matrix Function,” Journal of Computational and Applied Mat matics, Vol. 99, No. 1-2, 1998, pp. 205-217.

doi:10.1016/S0377-0427(98)00158-7

[18] G. Golub and C. F. Van Loan, “Matrix Computations,”

.

The Johns Hopkins University Press, Baltimore, 1989. [19] L. Jódar and J. C. Cortés, “Some Properties of Gamma

and Beta Matrix Functions,” Applied Mathematics Letters, Vol. 11, No. 1, 1998, pp. 89-93

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