Advances in Pure Mathematics, 2011, 1, 315-321
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*
1Department of Mat hem at i cs, Faculty of Science, Assiut University, Assiut, Egypt
2Department of Mat hem at i cs, Faculty of Science, South Valley University, Qena, Egypt
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


,AB
J
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 func-
tions 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 func-
tion, 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 func-
tions 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
N
N
C

if


>0Re  for all where

P  


P  is the
set of all eigenvalues of and its two-norm denoted
by
P
2
02
=, sup
x
Px
Px

where for a vector in
y
N
C,

1
2
2=T
y
yy is the
Euclidean norm of .
y
Let


P  and


P  be the real numbers which
were defined in [17] by






  
=max :,
=min :.
PRezzP
PRezzP



 (1.1)
If


f
z and


g
zz are holomorphic functions of
the complex variable which are defined in an open
set

of the complex plane and is a matrix in P

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
PgPgPfP (1.2)
Hence, if in Q
N
N
C
=PQ QP

is a matrix for which
and if , then

Q  
 
=.
f
PgQgQf P (1.3)
The reciprocal Gamma function denoted by
 
1z 

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

1z 
P

P
invertible fo
 
 
1
PIP
PnI P


 

 
=1
lim
1

israll integer0.PnI n (1.4)
The Pochhammer symbol or shifted factorial defined
[17] by
  
 
0
=1
;1;=
n
PP nI
nP I

 (1.5)
Jódar and Cortés have proved in [17] that
 
1
! .
P
n
P



n
Pn


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

,BPQ

0
=e
=exp
tP
Pt
tP




1
0
,=
PI
Q t

 
d;
ln
I
PI
t
I


t
.
(1.7)
and

1 d
QI
BPt t

 (1.8)
The Schur decomposition of , was given by [18] in
the form:
P
1
2
; 0,
!
s
tP
Prt
t
s





1
()
=0
ee
r
t P
s



and
1
2
1
()
=0
ln
;
!
s
r
PP
s
Pr n
nn n
s








2. Composite Humbert M atrix Function
Let us introduce the following notation (see [20])







 
 
12
12
12
12
12
12
12
12
12
12
12
12
=,,,,
=,
=,
=,,,,
=,
=,,,,
=,
=,
=
s
s
k kk ks
s
s
s
s
s
s
kkk k
s
s
kkk k
s
kkk k
kkkk
zzz z
AAA A
AAA A
BBB B
BBB B
AIA IAIAI
BIBIBIBI



 
 









(2.1)
and


,,, ,
1122
=,,,
ABA BABAB
ss
JJJ J.
Suppose that
 
11
,
3
02
=3
,;,;;=1,2,,
27
AB
ii
i
AB iii
ii
ii
ii
z
JzAI BI
z
F
AIBI is


 







(2.2)
are
s
Humbert matrix function with square complex
matrices 12 ,,,
s
A
AA and 12 ,,,
s
BB B of the same
order N. Construct the function
 
11
,
3
02
3
3
=0
3
,;,;27
,
AB
AB
AB kI
AB kI
k
z
J
zAI
z
FAIBI
Uz




 BI









(2.3)
where
 


 
11
33
11
=.
3!
k
AB kIAB kI
1
A
kI BkI
Uk

 
 
1 . (1.9)
This function, will be called the composite Humbert
matrix function of several complex variables
12
,,,
s
zz z. Now we prove that the matrix power series
(2.3) convergence for all 30z. Using the ratio test, we
obtain


 



 


3( 1)
11 3( 1)
3( 1)
33(1)
() ()
3
13
1
3
11
3
3()
22
1=limsuplimsup 31!
1( 1)
. limsup 1
3!
AB k I
AB k I
AB kI
AB kIABkI
kk
AB kI
AB kI
AB kIk
Ak IBk Iz
Uz
RUz k
AkI BkIz
AkI BkIzz
k k
 
  
 
 
 




 


 




A. SHEHATA ET AL.317
Note that if is large enough so that 1k 1>kA,
then by perturma, [13], we can write bation lem
 
1
11
1=
11
A
Ak II
kk


 



1
11
=11 1
AI
kkk A




 

(2.4)
hence

 

() 11
11
3
1
1
=limsup 1.. 11..
1
.. 11..
||
.
.. 1
ks
ss
ss
kkk A
kAkB
z
kB





 


 

(2.5)
For positive numbers
1
R
i

and positive integer , we
can write
k
=,=1,2,2.
ii
kk si

 (2.6)
Substitute from (2.6) into (2.5) one gets
3( 1)
3( 1) 1=
limsup
AB k
AB kI
Uz
 
 
3
() 3
I
AB kI
kAB kI
RUz

 
Thus, the power series (2.3) is absolutely convergent
for all
=
0.
3<z.
l Formhe Composite Humbert Matrix
unction
ert
ction, we use the following formulas (see [10,
Integra of t
F
To get an integral form for the composite Humb
matrix fun
12, p. 115, No. (5.10.5)]).







 
  
 
11 1
1
1
122
22222
2
1
1
1
1= expd,
2π
1
1= expd,
2π
C
Ak I
C
AkI
ss
1
111
11
1
1= exp()d,
2π
AkI
s
ssss
Cs
Ak Irrr
i
A
kIrr r
i
A
kIrr r
i




 




(2.7)
and


 
  
 
  


  


1
2
1
111
11111
1
122
2222 2
1
1
1
1= expd,
2π
1
1= expd,
2π
1
1= expd.
2πs
Bk I
C
Bk I
C
Bk I
ss
s
ssss
C
Bk Ittt
i
Bk Ittt
i
Bk Ittt
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
11
,11
21
=0
1
1
1
1
11
11 1
13
=expd
!2π
..expd
expd..
AB kI
k
AkI
AB sC
k
Ak I
s
ss
Cs
Bk I
C

 
1
..expd,
s
Bk I
ss
ss
z
J
1
..
s
C
zrr
ki
rr r
tt t











(2.9)
interchanging the order of the integral and summation,
r
tt t



  








1
1
,111
21
1
1
11 1
3
=0
3
=expd..
2
..expd
expd..
1
!
s
AB
AI
AB sC
AI
s
ss
Cs
BI
C
k
k
Ck
z
Jzrr r
i
rr r
tt t
z
k













(2.10)
thus,
27
..expd,
BI
s
ss s
rt
tt t





 


1
3
,21
11
3
=( exp
27rt
2π
dddd,
s
AB
AB sCCCC
s
AI BIss
z
z
Jz rt
i
rt rrtt


 

 
 






(2.11)
where


12
=,,,,
s
rrr r


12
=,,,
s
ttt t ,


12
=,
s
rrr r




12
=,
s
ttt t




1
=,
s
A
IAIAI




1
=s BIBI BI


and
3 33 3
12
112 2
=...
2727 2727
s.
s
s
z zz
z
rtrtrtrt
 


Therefore, the following result has been established.
Theorem 2.1 Let be matrices in
N
N
C

.
on for several
and B
A
Then the composit Humbert matrix functi

A. SHEHATA ET AL. 318
complex variables 12
,,,
s
zz z
3. Humbert Kummer Matrix Fun
we will dedu
mbert ma
satisfies integral i n (2.11).
ction
section, ce a new matrix function that
is a mixed from Hutrix function is in the form
In this
  

11
,=1 An I
3
,
02
;,1;27
3
AnI
z
J
zA

 



x function i
InI
n

 
z

(3.1)
FA I I


and Kummer matris in the form


=0
,;;=!3
nn
n
KB C zn 

 (
n
BC z
 3.2)

this function given the following

 

     
 
   

  
,
,,; =
n
nn AnI
BC z
MABCzJ z




=0
=0
3
11
2
=0
3
1
2
=0
1
=0
!3
3!
11 1
3
!1
3
3!
1
,
!
n
A
nn
n
k
k
kn
k
k
k
nn n
n
n
BC
z
n
z
kI AkI
kk
z
I
k
BCk I
n











 














by Theorem 2.1 in [9], we know that
n


11
AAk
z 


 
 
 
   
1
=0
21
11
1
!
=,;1;1
=1 1
11
nn n
n
BCkI
n
FBCk I
kkIBC
kIB kIC







 
  

,
(3.3)
0. Hence


1>kIBC  



,
=0 !3 nn
111
3
13
3
3
=0
,,; =
3
;,,; ,
27
.
n
nn AnI
A
AkI
AkI
k
BC z
MABCzJ z
z
I
BCAIIB IC
z
FIBCAIIBI C
Uz

















(3.4)
Provided that 0.


()>BC


Where
A
, B and
are matrices in
C
N
N
C

such that (1)
A
kI,
(1)kIC

 and (1)kIB


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
ABCz in the form




1
3
3
(() ()
1=limsup
limsu
AkI
AkI
k
IBB ICI
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! !
= 0. limsu p
!
CAIIAk
AAkI
k
k k
kk
k
k
 









(3.5)

that the the HK
(( )
I
BC

 and ()
A
I are matrices in
N
N
C

such that







  
   
=,
,andare positive stable.
IBCAI AIIBC
IBC AIABC
 
 
(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
=
11
B C kIBC
A k
AI ABC
kIB CA B CAkI




 
  

1
=
k
k
IBC AI
I


 


 



1
1
AI I
IBC

 











1
1()
0
=1 d.
ABC I
kBC
CA kI
ttt







1
(3.8)
From relation (3.7) and (3.8), we get

1kIB CA B  
  




1
11
1()
01d.
k
k
ABC I
kBC
IBC AI
I
BC AIABC
ttt




  


 


(3.9)

A. SHEHATA ET AL. 319
Hence, for <z, we can write



  

 


111
0
=0
3
11
1()
0
3
11
=0
,,;
3
13
!
3
1d
13
A
k
k
k
kk
ABC I
BC
k
kk
k
MABCz
z
1()
1d
ABC I
kBC
111
A
I
BICABC
z
IB IC
k
ttt
z
I
B ABC
ttt
zt
IB IC
 










  



 












IC
 

 


 

 

3
111
1()
0
3
3
02
!
3
1
;,; d.
3
k
A
ABC I
BC
k
z
I
BICABC
tt
zt
FIBIC t
 


















(3.10)
Summarizing, the following result has been esta-
blished:
Theorem 3.1 Let A, B and C be matrices in
N
N
C

such that ()>1,A

((=()(())))( )
I
BAIIBC C AI and
)(( )
I
BC

, ()
A
I

, ()
A
BC are positive stable.
Then for <z

it follows that





111
1()
0
3
3
02
,,;
=3
1
;,; d.
3
A
ABC I
BC
MABCz
z
I
BICABC
tt
zt
FIBICt





 























Another integral representation of HKMF can be
established starting from the formula in (2.7), we find
that



1(
1
1= expd
2π
AkI
C
(1))
A
kI rr
i

 r

and substituting the above expression into the series
expression of the HKMF given in (3.4), it follows that



 

 

11
3
11
=0
((1))
,,;
=3
13
!
1expd ,
2π
A
k
k
kk
k
k
Ak I
C
MABCz
zIBC IB IC
z
IBC IBIC
k
rr r
i









   
 




interchanging the order of the integral and summation,
 
   
11 (
3
11
3
)
1
,,; =expd
23
13
,
!
AAI
C
k
k
kk
k
z
M
=0k
ABCzI BCI BICrrr
i
z
IBC IBICr
k
 



 




 




i.e.,


11
3
()
12 3
1
=2πi3
exp;,;d.
3
A
AI
C
z
MIBC
IBIC
z
rrFIBCIBICr
r



 






  







(3.11)
Therefore, we obtain the following theorem:
Theorem 3.2 Let and be matrices in B C
N
N
C

such that 1()<BC


. Then for <,z

expre-
.11) hold ssion (3true.
A
,

A. SHEHATA ET AL. 320
mposite HKMF
Let
4. Co



11
3
1
13
,,;
3
;,,;
27
iiiii
Ai
iii ii
ii
iiiiii
MABCz
zIBCAI IB
z
ICFIB CAIIBIC







 

,
arrt m

e composite HumbeKumer matrix functions with
square complex matrices i
A
, i B and Ce same
i
<1
of th
order N, provided that ()
ii
BC


.
ert KumConstruct the composite Humbmer matrix
s of these functions for any mode of arrangement, function
we put

  


111
3
13
,,;
3
;,,;
27
27
A
MABCz
z
3
13
;,,;
I
BCAIIB IC
z
FI BCAIIBIC
IB











z
FI BCAI IC

 



where


 
 
 
12
12
11
1
12
12
12
12
=,,,,
=,,,,
=,
=,
=,
s
s
ss
kk k
s
s
kkk ks
s
kkk ks
BBB B
CCCC
IBCIBCIB C
IBIB IBIB
ICICICIC
 
 
 





and

12
=,,,
s
MMM 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
12
12
333
12
1=limsup
limsup
!! !
AkI
AkI
kk
AkI
A AA s
s
ks
U
R
kkk
kk k





 












12
......... 2
22
11 1
,0
12
1,
=.
k kk s
kk kkkk
ss s
k
kk k
s
kk

 
 

 
 
 








=0
as above putting =
ii
kk

and using relation (1.9) and
the following relation






where



1
2
11
1
=0
1
ln
ln!
lne,
j
1
=0 =0
ln
!!
j
rr
r
jj
j
r
j
rA
Ar k
A
rk j
rk














we get
A
rk
jj






1
3
3
1=0
limsup
AkI
AkI
kk
U
R 








.
Then the composite Humbert Kummer matrix func-
tions is an entire function.
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