A New Extension of Bessel Matrix Functions · SoutheastAsian Bulletinof Mathematics c SEAMS.2016 Southeast Asian Bulletin of Mathematics (2016) 40: 265–288 A New Extension of Bessel

Southeast Asian

Bulletin of

Mathematicsc©SEAMS. 2016

Southeast Asian Bulletin of Mathematics (2016) 40: 265–288

A New Extension of Bessel Matrix Functions

Ayman Shehata

Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt

Department of Mathematics, College of Science and Arts in Unaizah, Qassim Univer-

sity, Qassim, Kingdom of Saudi Arabia

Email: [email protected]

Received 16 February 2014Accepted 9 March 2015

Communicated by H.M. Srivastava

AMS Mathematics Subject Classification(2000): 33C10, 33C45, 33C60, 15A60

Abstract. The main aim of this paper is to define a new family of special matrix func-

tions, say, p-Hypergeometric, p-Bessel, p-modified Bessel and p-Tricomi matrix func-

tions, where p is a positive integer. The convergence properties of the p-Hypergeometric

matrix function are established and the matrix differential equation is constructed. The

differential properties, integral representations, integrals and various particular cases

on the p-Bessel matrix function have been obtained. The p-modified Bessel matrix

functions, and some properties of this function are investigated. Finally, the p-Tricomi

matrix function and several results of the function are defined and studied.

Keywords: p-Hypergeometric matrix function; p-Bessel; p-modified Bessel and p-Tri-

comi matrix functions; A Jordan block; Differential properties; Integral representations.

1. Introduction and Preliminaries

Special matrix functions appear in connection with statistics [4], mathemati-cal physics, theoretical physics, group representation theory, Lie groups theory[11], orthogonal matrix polynomials are closely related [14]. The Bessel func-tions are among the most important special functions with very diverse ap-plications to physics, engineering and mathematical analysis [10, 17, 40]. In[15, 16, 19, 24, 33], the hypergeometric matrix function has been introducedas a matrix power series, hypergeometric matrix differential equation and anintegral representation. In [1, 2, 3, 12, 13, 21], the authors introduced the

266 A. Shehata

Bessel matrix functions and gave some results with Bessel matrix functions.In [8, 9, 20, 26, 27, 28, 30, 31, 32, 34, 35, 36, 37, 38, 39, 41], extension tothe matrix function framework of the classical families of p-Kummers, p andq-Appell, Humbert, Tricomi and Hermite-Tricomi matrix functions, and Gegen-bauer, Rices, Humbert,Rainville’s, Bessel and Konhauser matrix polynomialshave been proposed. The author has earlier studied the p and q-Horn’s H2,pl(m,n)-Kummer matrix functions of two complex variables under differentialoperators [25, 29]. The reason of interest for this family of Bessel function is dueto their intrinsic mathematical importance and the fact that these polynomialshave applications in physics. The present investigation is motivated essentiallyby several recent works [2, 3, 12, 13, 20, 21, 25, 26, 29, 32]. From this mo-tivation, we prove some new properties for the new class of p-hypergeometric,p-Bessel, p-modified Bessel and p-Tricomi matrix functions with complex anal-ysis. The outline of this paper is organized as follows: Section 2 some basicrelations involving the p-hypergeometric matrix function, convergence proper-ties and matrix differential equation have been obtained. Some new properties,integral representations and integral expressions of p-Bessel matrix function aregiven in Section 3. Results of Sections 2 and 3 are used in Section 4 and 5 tostudy the properties of the new class of p-modified Bessel and p-Tricomi matrixfunctions.

Here, we discuss the properties of the new class of p-hypergeometric, p-Bessel,p-modified Bessel and p-Tricomi matrix functions and fix the notation in orderto make the section self-consistent. The following specialized versions of thedefinitions and results of the previous works, which are useful in our study, arerephrased.

Throughout this paper, for a matrix A in CN×N , its spectrum σ(A) denotesthe set of all eigenvalues of A. Its two-norm is denoted by ‖A‖, and is definedby

‖A‖ = supx 6=0

‖Ax‖2‖x‖2

where ||x||2 = (xTx)1

2 denotes the usual Euclidean norm of a vector x in CN .We say that a matrix A in CN×N is a positive stable matrix if ℜ(µ) > 0 forall µ ∈ σ(A) (see [14]). In this expression, ℜ(z) is the real part of the complexnumber z. The matrices I and O will denote the identity matrix and the nullmatrix in CN×N , respectively.

For the purpose of this work, we denote by µ(A) the logarithmic norm of A,which is defined by [6, 7]

µ(A) = max

z; z eigenvalue ofA+AT

2

(1.1)

where AT denotes the conjugate transpose. We denote by the number µ(A)

µ(A) = min

z; z eigenvalue ofA+AT

2

. (1.2)

A New Extension of Bessel Matrix Functions 267

From [7], it follows that ‖eAt‖ ≤ etµ(A) for t > 0. Hence, tA = exp(A ln t)satisfies

‖tA‖ =

tµ(A) if t ≥ 1,

tµ(A) if 0 ≤ t ≤ 1.(1.3)

In [5], if f(z) and g(z) are holomorphic functions of the complex variable z,which are defined in an open set Ω of the complex plane, and A, B are matricesin CN×N with σ(A) ⊂ Ω and σ(B) ⊂ Ω, such that AB = BA, then

f(A)g(B) = g(B)f(A). (1.4)

Let P be a positive stable matrix in CN×N , then Gamma matrix function Γ(P )is defined by [14]

Γ(P ) =

∫ ∞

0

e−ttP−Idt; tP−I = exp

(

(P − I) ln t

)

. (1.5)

If A is a matrix in CN×N such that A+nl is an invertible matrix for all integersn ≥ 0, then Γ(A) is an invertible matrix, and Pochhammer symbol (shiftedfactorial) is defined by (see [18])

(A)n = A(A + I)(A+ 2I)(A+ 3I) . . . (A+ (n− 1)I)

= Γ(A+ nI)Γ−1(A); n ≥ 1, (A)0 = I.(1.6)

For the purpose of this our present work, we recall here the following some con-cepts and properties of the matrix functional calculus. H is the N -dimensionalJordan block defined by

H =

n 1 0 . . . 0 0 0 00 n 1 . . . 0 0 0 00 0 n . . . 0 0 0 00 0 0 . . . 0 0 0 00 0 0 . . . 0 0 0 00 0 0 . . . 0 0 n 10 0 0 . . . 0 0 0 n

∈ CN×N . (1.7)

Thus, if H is a Jordan block of the form (1.7), z > 0, we can write the image bymeans of the matrix functional calculus acting on the matrix H and the functionof n, Jn(z), one can obtain Bessel matrix function of the first kind of order Has follows:

JH(z) =

(

z

2

)H

Γ−1(A+ I) 0F1

(

−;H + I;−z2

4

)

; |z| < ∞; |arg(z)| < π

=

∞∑

k=0

(−1)k

k!Γ−1(H + (k + 1)I)

(

z

2

)H+2kI

.

(1.8)

268 A. Shehata

for |z| < ∞, |arg(z)| < π, where λ ∈ σ(H) is not a negative integer (see [12]).Let us take A ∈ CN×N satisfying the condition

λ is not a negative integer for every λ ∈ σ(A). (1.9)

In [13], the Bessel matrix function JA(z) of the first kind of order A was definedas follows:

JA(z) =

∞∑

k=0

(−1)k

k!Γ−1(A+ (k + 1)I)

(

z

2

)A+2kI

=

(

z

2

)A

Γ−1(A+ I) 0F1

(

−;A+ I;−z2

4

)

,

(1.10)

Now, we consider the general case. Let A be a matrix satisfying the condition(1.9) and H = diag(H1, . . . , Hk) be the Jordan canonical form of A, where Hi

is a Jordan block defined in the following form,

Hi =

ni 1 0 . . . 0 0 0 00 ni 1 . . . 0 0 0 00 0 ni . . . 0 0 0 00 0 0 . . . 0 0 0 00 0 0 . . . 0 0 0 00 0 0 . . . 0 0 ni 10 0 0 . . . 0 0 0 ni

∈ CNi×Ni , (1.11)

for Ni ≥ 1, N1 + N2 + . . . + Nk = N and Hi = (ni) if Hi is a Jordan block ofsize 1× 1 for ni is not negative integer for 1 ≤ i ≤ k. Bessel matrix function ofthe first kind can be written as the following

J(t,H) = [diag1≤i≤k(JHi(t))]. (1.12)

Furthermore, if P be an invertible matrix in CN×N such that

H = diag(H1, H2, . . . , Hk) = PAP−1, (1.13)

then we have the same Bessel matrix function of the first kind of order A in(1.10) (see [13]).

Let C be a matrix in CN×N such that C+nI is an invertible matrix for everyinteger n ≥ 0 [22, 23]. Then the hypergeometric matrix function can be definedby the matrix power series in the form

0F1(−;C; z) =

∞∑

n=0

[(C)n]−1 z

n

n!. (1.14)

Let P and Q be positive stable matrices in CN×N , then Beta matrix functionB(P,Q) is defined by [14]

B(P,Q) =

∫ 1

0

tP−I(1− t)Q−Idt. (1.15)


If P , Q and P +Q are positive stable matrices in CN×N for which PQ = QP ,and P + nI, Q+ nI and P +Q+ nI are invertible matrices for n ≥ 0 [15], then

B(P,Q) = Γ(P )Γ(Q)Γ−l(P +Q). (1.16)

2. Properties of p-Hypergeometric Matrix Function

Let H be a Jordan block as in (1.7). The p-hypergeometric matrix function ofthe complex variable z is defined by the power series in the form

p0F1

(

−;H ; z

)

=

∞∑

k=0

1

(pk)!zk[(H)k]

−1 =

∞∑

k=0

Ukzk; p ≥ 1 (2.1)

where Uk = 1(pk)! [(H)k]

−1, (H)k is defined by (1.6) and p ≥ 1. If k > ‖H‖, thenby perturbation lemma [5], we can write

‖(Hk

+ I)−1‖ ≤ k

k − ‖H‖ , (2.2)

(H + kI)−1 = k(H

k+ I)−1, (2.3)

for k > ‖H‖. The convergence properties of p-hypergeometric matrix function,we use the ratio test with (2.1), (2.2) and (2.3) as in the form

limk→∞

∣

∣

∣

∣

∥

∥

∥

∥

Uk+1

Uk

∥

∥

∥

∥

zk+1

zk

∣

∣

∣

∣

= limk→∞

∣

∣

∣

∣

∥

∥

∥

∥

(pk)![(Hk+ I)]−1

k(p(k + 1))!

∥

∥

∥

∥

zk+1

zk

∣

∣

∣

∣

≤ limk→∞

∣

∣

∣

∣

z

∥

∥

∥

∥

(Hk+ I)−1

∥

∥

∥

∥

kp+1(p+ pk)(p+ p−1

k)(p+ p−2

k) . . . (p+ 1

k)

∣

∣

∣

∣

≤ limk→∞

1

kp+1(p+ pk)(p+ p−1

k) . . . (p+ 1

k)(1 − ‖H‖

k)|z| = 0

for all z. Thus, the power series (2.1) is convergent for all complex number z.Hence, the p-hypergeometric matrix function is an entire function.

Now, we append this section by introducing the differential operator θ = z ddz

to the entire function in successive manner as the following

θ(θ − 1

p)(θ − 2

p)(θ − 3

p)(θ − 4

p) . . . (θ − p− 1

p) p0F1

(

−;H ; z

)

=1

pp

∞∑

k=1

pk(pk − 1) . . . (pk − p+ 1)

(pk)!zk[(H)k]

−1

=1

pp

∞∑

k=0

1

(pk)!zk+1[H + kI]−1[(H)k]

−1 =z

ppH−1 p

0F1

(

−;H + I; z

)

270 A. Shehata

i.e, equivalently

θ(θ − 1

p)(θ − 2

p) . . . (θ − p− 1

p) p0F1(−;H ; z)

− z

ppH−1 p

0F1

(

−;H + I; z

)

= 0.

(2.4)

Similarly to (2.4), we can write p0F1

(

−;H ; z

)

, then

θ(θ − 1

p)(θ − 2

p)(θ − 3

p) . . . (θ − p− 1

p)(θ I +H − I) p

0F1

(

−;H ; z

)

=1

pp

∞∑

k=1

(k I +H − I)

(pk − p)!zk[(H)k]

−1 =1

pp

∞∑

k=0

1

(pk)!zk+1[(H)k]

−1

=z

ppp0F1(−;H ; z).

These results are summarized in the following.

Theorem 2.1. Let H be a Jordan block as in (1.7), then the p-hypergeometricmatrix function p

0F1(−;H ; z) is a solution of the matrix differential equation oforder p+ 1[

θ(θ − 1

p)(θ − 2

p) . . . (θ − p− 1

p)(θ I +H − I)− z

pp

]

p0F1

(

−;H ; z

)

= 0. (2.5)

Remark 2.2. Putting p = 1, we get the hypergeometric matrix function 0F1

0F1

(

−;H ; z

)

=

∞∑

k=0

1

k!zk[(H)k]

−1.

The previous properties can be generalized as the following: Let Hi is definedas in (1.7) and H = diag(H1, H2, . . . , Hk) be a matrix in CN×N . Here Ni isnot negative integer for 1 ≤ i ≤ k. The matrix H = diag(H1, H2, . . . , Hk), theproperties (2.1-. . .-2.7) can be easily provided.

Also, if P be an invertible matrix in CN×N such that H = diag(H1, H2, . . . ,

Hk) = PCP−1, we can give the following theorem for the matrix C.

Theorem 2.3. Let C be a matrix in CN×N such that the matrix C + nI is an

invertible matrix for every integer n ≥ 0. Then we have[

θ(θ − 1

p)(θ − 2

p) . . . (θ − p− 1

p)(θ I + C − I)− z

pp

]

p0F1

(

−;C; z

)

= 0.

where p0F1

(

−;C; z

)

=∑∞

k=01

(pk)!zk[(C)k]

−1; p ≥ 1.


Remark 2.4. For p ≥ 1, the p-hypergeometric matrix function p0F1

(

−;C; z

)

is

an entire function.

3. On p-Bessel Matrix Function

The basic Bessel function is defined when |p| < 1. If |p| ≥ 1 we try to define anew class of matrix function, say, p-Bessel matrix function; p ≥ 1; of the firstkind as in the following form

pJH(z) =

(

z

2

)H

Γ−1(H + I) p0F1

(

−; H + I;−z2

4

)

=

∞∑

k=0

(−1)k

(pk)!Γ−1(H + (k + 1)I)

(

z

2

)H+2kI (3.1)

and

p0F1

(

−; H + I;−z2

4

)

= Γ(H + I)

∞∑

k=0

(−1)k

(pk)!Γ−1(H + (k + 1)I)

(

z

2

)2k

where H is a Jordan block as in (1.7), and the p-Bessel matrix function is anentire function.

For the p-Bessel matrix function which is defined in (3.1), we have:

Theorem 3.1. Let H be a Jordan block as in (1.7) and pJH(z) is defined in (3.1).Then the following differential properties are satisfied

d

dz

[

zH pJH(z)

]

= zH pJH−I(z),

(

1

z

d

dz

)m[

zH pJH(z)

]

= zH−mIpJH−mI(z),

z pJ′H(z) = z pJH−I(z)−H pJH(z),

d

dz

(

d

dz− 2

1

p

)(

d

dz− 2

2

p

)

. . .

(

d

dz− 2

p− 1

p

)[

z−HpJH(z)

]

=− 1

2(2

p)pz−A

pJA+I(z).

(3.2)

Proof. From (3.1), we have

d

dz

[

zH pJH(z)

]

=d

dz

[ ∞∑

k=0

(−1)k

2n+2k(pk)!Γ−1(H + (k + 1)I)z2H+2kI

]

.

Differentiating term by term of the above equality, we get

d

dz

[

zH pJH(z)

]

=

∞∑

k=1

(−1)k(2H + 2kI)

2n+2k(pk)!Γ−1(H + (k + 1)I)z2H+(2k−1)I .

272 A. Shehata

On setting k = r + 1 and simplification, gives

d

dz

[

zH pJH(z)

]

= zH∞∑

r=0

(−1)r

2H+(2r−1)I(pk)!Γ−1(H − I + (r + 1)I)zH+(2r−1)I .

This can be written as

d

dz

[

zH pJH(z)

]

= zH pJH−I(z). (3.3)

From (3.3) and mathematical induction, we obtain

(

1

z

d

dz

)m[

zH pJH(z)

]

= zH−mIpJH−mI(z). (3.4)

Development of the left hand side of (3.3), we have the relation

z pJ′H(z) = z pJH−I(z)−H pJH(z). (3.5)

This relation is called a matrix differential recurrence relation.

Insert the factor z−H on each side of equation (3.1), and differentiating termby term, we get

d

dz

(

d

dz− 2

1

p

)(

d

dz− 2

2

p

)

. . .

(

d

dz− 2

p− 1

p

)[

z−HpJH(z)

]

=(2

p)p

∞∑

k=1

(−1)k(pk)(pk − 1) . . . (pk + p− 1)

2H+2kI(pk)!Γ−1(H + (k + 1)I)z2k−1

=− 1

2(2

p)pz−H

pJH+I(z).

(3.6)

Thus the proof is completed.

Remark 3.2. Taking p = 1, we produce the Bessel matrix function JH(z) (see[2, 3]).

The previous properties can be generalized as the following: Let Hi is definedas in (1.7) and H = diag(H1, H2, . . . , Hk) be a matrix in CN×N . Here Ni is notnegative integer for 1 ≤ i ≤ k. The matrix H = diag(H1, H2, . . . , Hk), theproperties (3.1-. . .-3.6) can be easily provided.

Furthermore, if P be an invertible matrix in CN×N such that H = diag(H1,

H2, . . . , Hk) = PAP−1, then we can give the following theorem for the matrixA.

Theorem 3.3. Let A be a matrix in CN×N satisfy the condition (1.9) and pJA(z)


is defined in (3.1). Then the following differential properties are satisfied

pJA(z) =

(

z

2

)A

Γ−1(A+ I) p0F1

(

−; A+ I;−z2

4

)

=

∞∑

k=0

(−1)k

(pk)!Γ−1(A+ (k + 1)I)(

z

2)A+2kI ,

d

dz

[

zA pJA(z)

]

= zA pJA−I(z),

(

1

z

d

dz

)m[

zA pJA(z)

]

= zA−mIpJA−mI(z),

z pJ′A(z) = z pJA−I(z)−A pJA(z),

d

dz

(

d

dz− 2

1

p

)(

d

dz− 2

2

p

)

. . .

(

d

dz− 2

p− 1

p

)[

z−ApJA(z)

]

=− 1

2(2

p)pz−A

pJA+I(z).

Remark 3.4. Putting p = 1, we produce the Bessel matrix function JA(z) (see[21]).

Now, the integral representations of the p-Bessel matrix function defined in(3.1) is given thrash the following theorem.

Theorem 3.5. Let H be a Jordan block matrix as in (1.7) such that µ(H) > − 12

and |argz| < π. Then the integral representation for p-Bessel matrix function isgiven by

pJH(z) =

(

z

2

)H Γ−1(H + 12I)√

π

∫ 1

−1

(1− t2)H− 1

2I p

0F1

(

−;1

2;−z2t2

4

)

dt (3.7)

where p0F1

(

−; 12 ;− z2t2

4

)

=∑∞

k=0(−1)k

(pk)!1

( 1

2)k

(

zt2

)2k

.

Proof. Using the Gauss’s multiplication theorem

Γ(mH) =(2π)1−m

2 mmH− 1

2I

m∏

i=1

Γ(H +i− 1

mI)∀ m = 1, 2, 3, . . . . (3.8)

By the Lemma 2 of [15], if k > − 12 , one gets the identity

Γ−1(H + (k + 1)I) =Γ−1(H + 1

2I)

Γ(k + 12 )

∫ 1

−1

t2k(1− t2)H− 1

2Idt. (3.9)

274 A. Shehata

From (3.1), (3.8) and (3.9), we obtain

pJH(z) =

∞∑

k=0

(−1)k

(pk)!

(

z

2

)H+2kI Γ−1(H + 12I)

Γ(k + 12 )

∫ 1

−1

t2k(1 − t2)H− 1

2Idt

=

(

z

2

)H Γ−1(H + 12I)√

π

∫ 1

−1

∞∑

k=0

(−1)k

(pk)!

1

(12 )k

(

zt

2

)2k

(1− t2)H− 1

2Idt.

Theorem 3.6. Let H be a Jordan block as in (1.7) such that µ is not a negativeinteger for every µ ∈ σ(H). Then the another integral representation for p-Besselmatrix function is

pJH(z) =

(

z

2

)H1

2π i

∫

c

ess−(H+I) p0F0

(

−;−;−z2

4s

)

ds (3.10)

where p0F0

(

−;−;− z2

4s

)

=∑∞

k=0(−1)k

(pk)!

(

z2

4s

)k

.

Proof. Another integral representation of pJH(z) can be established startingfrom the formula

Γ−1(H + (k + 1)I) =1

2π i

∫

c

ess−(H+(k+1)I)ds. (3.11)

Using (3.1) and (3.11), we have

pJH(z) =∞∑

k=0

(−1)k

(pk)!

(

z

2

)H+2kI1

2π i

∫

c

ess−(H+(k+1)I)ds

=

(

z

2

)H1

2π i

∫

c

ess−(H+I)∞∑

k=0

(−1)k

(pk)!

(

z2

4s

)k

ds.

These integral representations can be generalized through the following re-sults.

Theorem 3.7.

(i) Let A be a matrix in CN×N such that µ(A) > − 12 and |argz| < π. Then

we obtain

pJA(z) =

(

z

2

)AΓ−1(A+ 12I)√

π

∫ 1

−1

(1− t2)A− 1

2I p

0F1

(

−;1

2;−z2t2

4

)

dt.

(ii) If A is a matrix in CN×N satisfy the condition (1.9). Then we obtain

pJA(z) =

(

z

2

)A1

2π i

∫

c

ess−(A+I) p0F0

(

−;−;−z2

4s

)

ds.


Here, we derive some integrals expressions involving p-Bessel matrix functiondefined in (3.1). These are contained in the next theorem.

Theorem 3.8. If H, M and H −M are Jordan blocks as in (1.7) satisfying theconditions µ(H) > ν(M) > −1, η(H −M) > −1 where η = µ− ν, we obtain

pJH(z) = 2Γ−1(H −M)

(

z

2

)H−M ∫ 1

0

(1− t2)H−M−I tM+IpJM (zt)dt. (3.12)

Proof. Consider the integral matrix functional∫ 1

0(1−t2)H−M−I tM+I

pJM (zt)dtand substitute the series (3.1) for pJM (z) to obtain

∫ 1

0

(1 − t2)H−M−I tM+IpJM (zt)dt

=

∞∑

k=0

(−1)k

(pk)!Γ−1(M + (k + 1)I)

(

z

2

)M+2kI ∫ 1

0

(1− t2)H−M−I t2M+(2k+1)Idt.

Let u = t2 in the last integral, we get

∞∑

k=0

(−1)k

(pk)!Γ−1(M + (k + 1)I)(

z

2)M+2kI 1

2

∫ 1

0

(1− u)H−M−IuM+kIdu

=∞∑

k=0

(−1)k

(pk)!(z

2)M+2kI 1

2Γ(H −M)Γ−1(H + (k + 1)I)

=1

2Γ(H −M)(

z

2)M−H

pJH(z)

this is the required result.

One can generalize the above result as follows.

Theorem 3.9. If A, B and A−B are matrices in CN×N satisfying the conditions,n(A) > m(B) > −1, r(A−B) > −1 for all r = n−m, then we obtain

pJA(z) = 2Γ−1(A−B)

(

z

2

)A−B ∫ 1

0

(1 − t2)A−B−ItB+IpJB(zt)dt.

This class of integrals forms is considered in the next theorem.

Theorem 3.10. If H, M and H + M are Jordan blocks as in (1.7) satisfyingthe conditions µ(H) > −1, η(H +M) > 0 for all η = µ + ν, a, b are arbitrary

276 A. Shehata

numbers with ℜ(a± ib) > 0 and HM = MH, then

∫ ∞

0

e−azzM−IpJH(bz)dz

=Γ(H +M)Γ−1(H + I)a−M

(

b

2a

)H

p2F1

(

1

2(H +M),

1

2(H +M + I);H + I;− b2

a2

)

(3.13)

where

p2F1

(

1

2(H +M),

1

2(H +M + I);H + I;− b2

a2

)

=

∞∑

k=0

(−1)k(12 (H +M))k(12 (H +M + I))k

(pk)!

(

b

a

)2k

[(H + I)k]−1; | b

a| < 1.

Proof. First assume that | ba| < 1, substitute the series for pJA(bz) in the left

hand side of the integral (3.13) to get

∫ ∞

0


=

∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!(b

2)H+2kI

∫ ∞

0

e−azzH+M+(2k−1)Idz

=∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!(b

2)H+2kI Γ(H +M + 2kI)

aH+M+2kI.

Applying the duplication formula for the Gamma matrix function, we have

Γ(H +M) =2H+M−I

√π

Γ(1

2(H +M))Γ(

1

2(H +M + I)),

Γ(H +M + 2kI) =2H+M+(2k−1)I

√π

Γ(1

2(H +M) + kI)

Γ(1

2(H +M + I) + kI)

(3.14)

and

Γ(1

2(H +M) + kI) =(

1

2(H +M))kΓ(

1

2(H +M)),

Γ(1

2(H +M + I) + kI) =(

1

2(H +M + I))kΓ(

1

2(H +M + I)).

(3.15)


Using the equations (3.14) and (3.15), we get

∫ ∞

0


=Γ(H +M)Γ−1(H + I)

×∞∑

k=0

(−1)kbH+2kI(12 (H +M))k(12 (H +M + I))k[(H + I)k]

−1

2HaH+M+2kI (pk)!.

Hence, the proof is completed.

Furthermore, if P is an invertible matrix in CN×N such that H = diag(H1,

H2, . . . , Hk) = PAP−1, then one can get the following theorem for the matrixA.

Theorem 3.11. If A, B and A+B are matrices in CN×N satisfying the conditions

n(A) > −1, r(A + B) > 0 for all r = n + m, a, b are arbitrary numbers withℜ(a± ib) > 0 and AB = BA, then

∫ ∞

0

e−azzB−IpJA(bz)dz

=Γ(A+B)Γ−1(A+ I)a−B

(

b

2a

)A

p2F1

(

1

2(A+B),

1

2(A+B + I);A+ I;− b2

a2

)

; | ba| < 1.

When M = H + I or H + 2I, the p2F1 in (3.13) reduces to a p

1F0, which canbe summed by the binomial theorem for | b

a| < 1.

Corollary 3.12. If H is a Jordan block as in (1.7) such that µ(H) > − 12 and a,

b are arbitrary numbers with ℜ(a± ib) > 0, then

∫ ∞

0

e−azzH pJH(bz)dz =(2b)HΓ(H + 1

2I)√πa2H+I

p1F0

(

H +1

2I,−;−;− b2

a2

)

(3.16)

where

p1F0

(

H +1

2I;−;− b2

a2

)

=

∞∑

k=0

(−1)k(H + 12I)k

(pk)!

(

b

a

)2k

; | ba| < 1.

Proof. From (3.1) and | ba| < 1, in the left hand side of the integral (3.16) can be

278 A. Shehata

written∫ ∞

0

e−azzHp JH(bz)dz

=

∫ ∞

0

e−azzH∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!

(

bz

2

)H+2kI

dz

=∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!

(

b

2

)H+2kI ∫ ∞

0

e−azz2H+2kIdz.

On the other hand, with the help of Gamma matrix function in (1.5), we get

∫ ∞

0

e−azz2H+2kIdz =1

a2H+(2k+1)I

∫ ∞

0

e−tt2H+2kIdt

=1

a2H+(2k+1)IΓ(2H + (2k + 1)I).

(3.17)

Using (3.18) and the duplication formula for the Gamma matrix function, weobtain

∫ ∞

0

e−azzH pJH(bz)dz

=

∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!

(

b

2

)H+2kIΓ(2H + (2k + 1)I)

a2H+(2k+1)I.

Thus, the proof is established.

Corollary 3.13. If H is a Jordan block as in (1.7) satisfying the condition µ(H) >− 3

2 and a, b are arbitrary numbers with ℜ(a± ib) > 0, then

∫ ∞

0

e−azzH+IpJH(bz)dz

=2(2b)HΓ(H + 3

2I)√πa2H+2I

p1F0

(

H +3

2I,−;−;− b2

a2

) (3.18)

where

p1F0

(

H +3

2I;−;− b2

a2

)

=

∞∑

k=0

(−1)k(H + 32I)k

(pk)!

(

b

a

)2k

;

∣

∣

∣

∣

b

a

∣

∣

∣

∣

< 1.

Proof. The result (3.18) follows immediately form (3.16) if we differentiate bothsides with respect to a. Thus, the proof is completed.

The next corollary gives another integral for p-Bessel matrix function.


Corollary 3.14. If H is a Jordan block as in (1.7) satisfying the condition µ(H) >

−1, and a and b are arbitrary numbers with ℜ(a± ib) > 0 and

∣

∣

∣

∣

ba

∣

∣

∣

∣

< 1, then

∫ ∞

0

e−azpJH(bz)dz

=1

a

(

b

2a

)Hp2F1

(

1

2(H + I),

1

2(H + 2I);H + I;− b2

a2

)

.

(3.19)

Proof. From the definition of pJH(z), we have

∫ ∞

0

e−azpJH(bz)dz =

∫ ∞

0

e−az

∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!

(

bz

2

)H+2kI

dz

=

∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!

(

b

2

)H+2kI ∫ ∞

0

e−azzH+2kIdz.

Using the Gamma matrix function, we get the integral on the right hand sideequals

∫ ∞

0

e−azzH+2kIdz =Γ(H + (2k + 1)I)

aH+(2k+1)I

and

Γ(H + (2k + 1)I) =2H+2kI

√π

Γ((k +1

2)I +

1

2H)Γ((k + 1)I +

1

2H).

Evaluating the inner integral, we get

∫ ∞

0

e−azpJH(bz)dz

=

∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!

(

b

2

)H+2kIΓ(H + (2k + 1)I)

aH+(2k+1)I

=bn√

πaH+I

∞∑

k=0

Γ((k+ 12 )I +

12H)Γ((k+1)I+ 1

2H)Γ−1(H+(k+1)I)

(pk)!

(

− b2

a2

)k

=1

a(b

2a)H

∞∑

k=0

(12 (H + I))k(12 (H + 2I))k[(H + I)k]

−1

(pk)!

(

− b2

a2

)k

.

Thus, the result is established.

Corollary 3.15. If H is a Jordan block as in (1.7) satisfying the condition µ(H) >

280 A. Shehata

0, and a, b are arbitrary numbers with ℜ(a± ib) > 0 and

∣

∣

∣

∣

ba

∣

∣

∣

∣

< 1, then

∫ ∞

0

e−azz−1pJH(bz)dz

=H−1

(

b

2a

)Hp2F1

(

1

2H,

1

2(H + I);H + I;− b2

a2

)

.

(3.20)

Proof. The proof of the corollary is very similar to corollary 3.14.

Now, let P be an invertible matrix in CN×N such that H = diag(H1, H2, . . . ,

Hk) = PAP−1, we give a generalization for the above results in the followingcorollary for the matrix A.

Corollary 3.16.

(i) If A is a matrix in CN×N such that n(A) > − 12 and a, b are arbitrary

numbers with ℜ(a± ib) > 0, then we have

∫ ∞

0

e−azzA pJA(bz)dz =(2b)AΓ(A+ 1

2I)√πa2A+I

p1F0

(

A+1

2I,−;−;− b2

a2

)

,

for

∣

∣

∣

∣

ba

∣

∣

∣

∣

< 1.

(ii) If A is a matrix in CN×N such that n(A) > − 32 , a, b are arbitrary numbers

with ℜ(a± ib) > 0 and

∣

∣

∣

∣

ba

∣

∣

∣

∣

< 1, then we have

∫ ∞

0

e−azzA+IpJA(bz)dz =

2(2b)AΓ(A+ 32I)√

πa2A+2I

p1F0

(

A+3

2I,−;−;− b2

a2

)

.

(iii) If A is a matrix in CN×N satisfying the condition n(A) > −1, and a, b

are arbitrary numbers with ℜ(a± ib) > 0 and

∣

∣

∣

∣

ba

∣

∣

∣

∣

< 1, then we obtain

∫ ∞

0

e−azpJA(bz)dz =

1

a

(

b

2a

)Ap2F1

(

1

2(A+ I),

1

2(A+ 2I);A+ I;− b2

a2

)

.

(iv) If A is a matrix in CN×N satisfying the condition n(A) > 0, a and b are

arbitrary numbers with ℜ(a± ib) > 0 and

∣

∣

∣

∣

ba

∣

∣

∣

∣

< 1, then we obtain

∫ ∞

0

e−azz−1pJA(bz)dz = A−1

(

b

2a

)Ap2F1

(

1

2A,

1

2(A+ I);A+ I;− b2

a2

)

.


The next theorem gives another infinite integral of a p-Bessel matrix functiondue to Hankel integral.

Theorem 3.17. For H, M and H+M are Jordan block as in (1.7) satisfying theconditions µ(H) > −1, r(H +M) > 0 for all r = µ + ν, a and b are arbitrarynumbers with ℜ(a2) > 0, then we have

∫ ∞

0

e−a2z2

zM−IpJH(bz)dz

=1

2a−HΓ(

1

2(H +M))Γ−1(H + I)

×(

b

2a

)Hp1F1

(

1

2(H +M);H + I;− b2

4a2

)

,

(3.21)

where

p1F1

(

1

2(H +M);H + I;− b2

4a2

)

=∞∑

k=0

(−1)k(12 (H +M))k[(H + I)k]−1

(pk)!

(

b

2a

)2k

.

Proof. The integral can be evaluated using term by term integration as follows

∫ ∞

0

e−a2z2

zM−IpJH(bz)dz

=

∞∑

k=0

(−1)kΓ−1(H + (k + 1)I)

(pk)!

(

b

2

)H+2kI ∫ ∞

0

e−a2z2

zH+M+(2k−1)Idz

and using the Gamma matrix function

∫ ∞

0

e−a2z2

zH+M+(2k−1)Idz =Γ(12 (H +M) + kI)

2aH+M+2kI.

Thus, the result is established.

Corollary 3.18. If H is a Jordan block as in (1.7) satisfying the condition µ(H) >−1 and a is an arbitrary number with ℜ(a2) > 0, then we obtain

∫ ∞

0

e−a2z2

zH+IpJH(bz)dz =

bH

(2a2)H+I

p0F0

(

−;−;− b2

4a2

)

(3.22)

where

p0F0

(

−;−;− b2

4a2

)

=

∞∑

k=0

(−1)k

(pk)!

(

b

2a

)2k

.

282 A. Shehata

Proof. From (3.1), we have

∫ ∞

0

e−a2z2

zH+IpJH(bz)dz

=∞∑

k=0

(−1)kΓ−1(H + k + 1)

(pk)!

(

b

2

)H+2kI ∫ ∞

0

e−a2z2

z2H+(2k+1)Idz.

Since the integral on the right hand side equals to

∫ ∞

0

e−a2z2

z2H+(2k+1)Idz =Γ(H + (k + 1)I)

2a2H+(2k+2)I.

Interchanged the order of the integral and summation, we get

∫ ∞

0

e−a2z2

zH+IpJH(bz)dz =

∞∑

k=0

(−1)k

2(pk)!a2H+(2k+2)I

(

b

2

)H+2kI

.

Thus, we have the result.

Generalization for the above properties are given in the following theoremand corollary.

Theorem 3.19. For A, B and A + B are matrices in CN×N satisfying the con-dition n(A) > −1, r(A + B) > 0 for all r = n + m and a is an arbitrarynumber with ℜ(a2) > 0, then the p-Bessel matrix function satisfies the followingrepresentation

∫ ∞

0

e−a2z2

zB−IpJA(bz)dz

=1

2a−AΓ(

1

2(A+B))Γ−1(A+ I)

(

b

2a

)Ap1F1

(

1

2(A+B);A+ I;− b2

4a2

)

.

Corollary 3.20. If A is a matrix in CN×N satisfying the condition n(A) > −1and a is an arbitrary number with ℜ(a2) > 0, then we obtain

∫ ∞

0

e−a2z2

zA+IpJA(bz)dz =

bA

(2a2)A+I

p0F0

(

−;−;− b2

4a2

)

.

4. On p-modified Bessel Matrix Function

The purpose of this section is to introduce and study a new class of matrixfunction, namely the p-modified Bessel matrix function and derive some of their


properties. IfH is a Jordan block as in (1.7) satisfying the conditions µ(H) > − 12

and |arg(z)| < π. Then the p-modified Bessel matrix function pIH(z) can bedefined as in the following:

pIH(z) = i−HpJH(iz)

=

(

z

2

)H Γ−1(H + 12I)√

π

∫ 1

−1

(1− t2)H− 1

2I p

0F1

(

−;1

2;z2t2

4

)

dt(4.1)

and

pIH(z) =

(

z

2

)H

Γ−1(H + I) p0F1

(

−; H + I;z2

4

)

=

∞∑

k=0

1

(pk)!Γ−1(H + (k + 1)I)

(

z

2

)H+2kI (4.2)

As in the same preceding manner we can find the recurrence relations of pIH(z)as follows:

d

dz

[

zH pIH(z)

]

= zH pIH−I(z), (4.3)

zd

dzpIH(z) = z pIH−I(z)−H pIH(z), (4.4)

(

1

z

d

dz

)m[

zH pIH(z)

]

= zH−mIpIH−mI(z) (4.5)

and

d

dz

(

d

dz− 2

1

p

)(

d

dz− 2

2

p

)

. . .

(

d

dz− 2

p− 1

p

)[

z−HpIH(z)

]

=(2

p)p

∞∑

k=1

(pk)(pk − 1) . . . (pk + p− 1)

2H+2kI(pk)!Γ−1(H + (k + 1)I)z2k−1

=− 1

2(2

p)pz−H

pIH+I(z).

(4.6)

These relations can be generalized as in the following: Let Hi is defined by (1.7)and H = diag(H1, H2, . . . , Hk) be a matrix in CN×N . Here Ni is not negativeinteger and zero for 1 ≤ i ≤ k. For matrix H = diag(H1, H2, . . . , Hk), theproperties (4.1-. . .-4.6) can be easily provided.


Hk) = PAP−1, we can the following theorem for the matrix A.

284 A. Shehata

Theorem 4.1. If A is a matrix in CN×N such that (1.9), we obtain

d

dz

[

zA pIA(z)

]

= zA pIA−I(z),

zd

dzpIA(z) = z pIA−I(z)−A pIA(z),

(

1

z

d

dz

)m[

zA pIA(z)

]

= zA−mIpIA−mI(z)

and

d

dz

(

d

dz− 2

1

p

)(

d

dz− 2

2

p

)

. . .

(

d

dz− 2

p− 1

p

)[

z−ApIA(z)

]

=− 1

2(2

p)pz−A

pIA+I(z).

Remark 4.2. For p = 1, reduces to definition for the modified Bessel matrixfunction IA(z), see [21].

5. On p-Tricomi Matrix Function

The purpose of this section is to introduce and study a new class of p-Tricomimatrix function of complex variable z which can be stated in the following form:

pCH(z) =∑

k≥0

(−1)kΓ−1(H + (k + 1)I)zk

(pk)! (5.1)

where H is a matrix in CN×N such that z is not a negative integer for everyz ∈ σ(H), this is known as p-Tricomi matrix function and is characterized bythe following link with p-Bessel matrix function of complex variables:

pCH(z) = z−1

2H

pJH(2√z). (5.2)

With the sam way, we can deduce recurrence relations for pCH(z) as in thefollowing:

d

dz

[

zH pCH(z)

]

= zH pCH−I(z), (5.3)

zd

dzpCH(z) = z pCH−I(z)−H pCH(z), (5.4)

(

1

z

d

dz

)m[

zH pCH(z)

]

= zH−mIpCH−mI(z) (5.5)

and

d

dz

(

d

dz− 1

p

)(

d

dz− 2

p

)

. . .

(

d

dz− p− 1

p

)[

z−HpCH(z)

]

=− (1

pp)z−H

pCH+I(z).

(5.6)


Theorem 5.1. Let H be a Jordan block as in (1.7), then the pCH(z) is a solutionof the matrix differential equation of order p+ 1

[

θ(θ − 1

p)(θ − 2

p) . . . (θ − p− 1

p)(θ I +H) +

z

pp

]

pCH(z) = 0. (5.7)

Proof. From (5.1) and the differential operator θ, we get

θ(θ − 1

p)(θ − 2

p)(θ − 3

p) . . . (θ − p− 1

p)(θ I +H) pCH(z)

=1

pp

∞∑

k=0

1

(pk)!zk+1[(H)k]

−1 = − z

pppCH(z).

Remark 5.2. Putting p = 1, we get the Tricomi matrix differential equation ofsecond order

z2d2

dz2CH(z) + (H + I)

d

dzCH(z) +CH(z) = 0. (5.8)

Theorem 5.3. The p-Tricomi matrix function satisfies the matrix recurrencerelations

d

dz

[

zA pCA(z)

]

= zA pCA−I(z),

zd

dzpCA(z) = z pCA−I(z)−A pCA(z),

d

dz

(

d

dz− 1

p

)(

d

dz− 2

p

)

. . .

(

d

dz− p− 1

p

)[

z−ApCA(z)

]

=− (1

pp)z−A

pCA+I(z)

and(

1

z

d

dz

)m[

zA pCA(z)

]

= zA−mIpCA−mI(z)

where A is a matrix in CN×N , P be an invertible matrix in CN×N such thatH = diag(H1, H2, . . . , Hk) = PAP−1 satisfy z is not a negative integer for everyz ∈ σ(H).

Theorem 5.4. Let A be a matrix in CN×N satisfying the condition (1.9), thenthe p-Tricomi matrix function is a solution of the matrix differential equation

[

θ(θ − 1

p)(θ − 2

p) . . . (θ − p− 1

p)(θ I +A) +

z

pp

]

pCA(z) = 0. (5.9)

286 A. Shehata

Remark 5.5. Putting p = 1, we get the Tricomi matrix differential equation ofsecond order

z2d2

dz2CA(z) + (A+ I)

d

dzCA(z) +CA(z) = 0. (5.10)

In a similar manner as in the proof of Theorem 3.5, we derive in the followingtheorem.

Theorem 5.6. Let H be a Jordan block as in (1.7) such that µ(H) > − 12 , the

integral representation of p-Tricomi matrix function is given as

pCH(z) =Γ−1(H + 1

2I)√π

∫ 1

−1

(1− t2)H− 1

2I p

0F1

(

−;1

2;−zt2

)

dt. (5.11)

The following theorem can be proved using the same technique as in Theo-rem 3.7.

Theorem 5.7. Let H be a Jordan block as in (1.7) satisfy the condition µ is nota negative integer for every µ ∈ σ(H). The p-Tricomi matrix function holds thefollowing integral representation:

pCH(z) =1

2π i

∫

c

ess−(H+I) p0F0

(

−;−;−z

s

)

ds. (5.12)

The previous results can be generalized as in the following. Let Hi is definedby (1.7) and H = diag(H1, H2, . . . , Hk) be matrix in CN×N . Here Ni is notnegative integer for 1 ≤ i ≤ k. For matrix H = diag(H1, H2, . . . , Hk), theproperties (5.1-. . .-5.7) can be easily provided.


Hk) = PAP−1, we give the following theorem.

Theorem 5.8.

(i) Let A be a matrix in CN×N such that µ(A) > − 12 , the integral representa-

tion of p-Tricomi matrix function is given as

pCA(z) =Γ−1(A+ 1

2I)√π

∫ 1

−1

(1− t2)A− 1

2I p

0F1

(

−;1

2;−zt2

)

dt.

(ii) Let A be a matrix in CN×N satisfy the condition (1.9). The p-Tricomi

matrix function holds the following integral representation:

pCA(z) =1

2π i

∫

c

ess−(A+I) p0F0

(

−;−;−z

s

)

ds.


Acknowledgement. (1) The Author expresses their sincere appreciation to Prof.Dr. Nassar H. Abdel-All (Department of Mathematics, Faculty of Science, As-siut University, Assiut 71516, Egypt), Dr. M. S. Metwally, (Department ofMathematics, Faculty of Science (Suez), Suez Canal University, Egypt) and Dr.Mahmoud Tawfik Mohamed, (Department of Mathematics and Science, Facultyof Education (New Valley), Assiut University, New Valley, EL-Kharga 72111,Egypt) for their kinds interests, encouragements, helps, suggestions, commentsthem and the investigations for this series of papers.

(2) The author would like to thank the anonymous referees for their severaluseful comments and suggestions towards the improvement of this paper.

References

[1] B. Cekim, A. Altin, Matrix analogues of some properties for Bessel functions, In:1st International Eurasian Conference on Mathematical Sciences and Applications(IECMSA), Kosova, 2012.

[2] B. Cekim, A. Altin, Matrix analogues of some properties for Bessel functions,Journal of Mathematical Sciences 22 (2) (2015) 519–530.

[3] B. Cekim, E. Erkus-Duman, Integral represantations for Bessel matrix functions,Gazi University Journal of Science 27 (1) (2014) 663–667.

[4] A.G. Constantine and R.J. Mairhead, Partial differential equations for hyper-geometric functions of two argument matrices, J. Multivariate Anal. 2 (1972)332–338.

[5] N. Dunford and J.T. Schwartz, Linear Operators, Part I, General Theory, Inter-science Publishers, INC. New York, 1957.

[6] T.M. Flett, Differential Analysis, Cambridge University Press, 1980.

[7] I. Higueras and B. Garcia-Celayeta, Logarithmic norms for matrix pencils, SIAMJ. Matrix Anal. 20 (3) (1999) 646–666.

[8] G.S. Khammash and A. Shehata, On Humbert matrix polynomials, Asian J.Current Engineering and Maths. (AJCEM) 1 (5) (2012) 232–240.

[9] Z.M.G. Kishka, M.A. Saleem, S.Z. Radi, M. Abul-Dahab, On the p and q-Appellmatrix function, Southeast Asian Bull. Math. 35 (2011) 807–818.

[10] B.G. Korenev, Bessel Functions and their Applications, London and New York,2002.

[11] A.T. James, Special functions of matrix and single argument in statistics, In:Theory and Application of Special Functions, R.A. Askey (Ed.), Academic Press,New York, 1975.

[12] L. Jodar, R. Company, E. Navarro, Solving explicitly the Bessel matrix differentialequation, without increasing problem dimension, Congressus Numerantium 92(1993) 261–276.

[13] L. Jodar, R. Company, E. Navarro, Bessel matrix functions: explicit solution ofcoupled Bessel type equations, Utilitas Math. 46 (1994) 129–141.

[14] L. Jodar and J.C. Cortes, Some properties of gamma and beta matrix functions,Appl. Math. Lett. 11 (1998) 89–93.

[15] L. Jodar and J.C. Cortes, On the hypergeometric matrix function, J. Computa-tional Appl. Math. 99 (1998) 205–217.

[16] L. Jodar and J.C. Cortes, Closed form general solution of the hypergeometricmatrix differential equation, Mathematical Computer Modelling 32 (2000) 1017–1028.

288 A. Shehata

[17] L. Jodar, M. Legua, A.G. Law, A matrix method of Frobenius, and applicationto a generalized Bessel equation, Congressus Numerantium 86 (1992) 7–17.

[18] L. Jodar and J. Sastre, On Laguerre matrix polynomials, Utilitas Math. 53 (1998)37-48.

[19] Y.L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962.

[20] M.S. Metwally, On p-Kummers matrix function of complex variable under dif-ferential operators and their properties, Southeast Asian Bull. Math. 35 (2011)1–16.

[21] J. Sastre and L. Jodar, Asymptotics of the modified Bessel and incomplete Gammamatrix functions, Appl. Math. Lett. 16 (6) (2003) 815–820.

[22] J. Sastre and L. Jodar, On the asymptotics of Laguerre matrix polynomials, Util-itas Math. 70 (2006) 71–98.

[23] J. Sastre and L. Jodar, On Laguerre matrix polynomials series, Utilitas Math. 71(2006) 109–130.

[24] K.A.M. Sayyed, M.S. Metwally, M.T. Mohamed, Certain hypergeometric matrixfunction, Scientiae Mathematicae Japonicae 69 (2009) 315–321.

[25] A. Shehata, On p and q-Horn’s matrix function of two complex variables, Appl.Math. 2 (2011) 1437–1442.

[26] A. Shehata, On Tricomi and Hermite-Tricomi matrix functions of complex vari-able, Communications Math. Applications 2(2-3) (2011) 97–109.

[27] A. Shehata, A new extension of Gegenbauer matrix polynomials and their prop-erties, Bulletin Inter. Math. Virtual Institute 2 (2012) 29–42.

[28] A. Shehata, On pseudo Legendre matrix polynomials, Inter. J. Mathematical Sci.Eng. Applications (IJMSEA) 6 (VI) (2012) 251–258.

[29] A. Shehata, Certain pl(m,n)-Kummer matrix function of two complex variablesunder differential operator, Applied Math. 4 (1) (2013) 91–96.

[30] A. Shehata, On Rices matrix polynomials, Afrika Matematika 25 (3) (2014) 757–777.

[31] A. Shehata, On Rainville’s matrix polynomials. Sylwan Journal 158 (9) (2014)158–178.

[32] A. Shehata, On a new family of the extended generalized Bessel-type matrixpolynomials, Mitteilungen Klosterneuburg J. 65 (2) (2015) 100–121.

[33] A. Shehata, New kinds of hypergeometric matrix functions, British J. Math. Com-puter Sci. 5 (1) (2015) 92–102.

[34] A. Shehata, Some relations on Humbert matrix polynomials, Mathematica Bo-hemica (to appear).

[35] A. Shehata, Some relations on Konhauser matrix polynomials, Miskolc Mathe-matical Notes (to appear).

[36] A. Shehata and R. Bhukya, Some properties of Hermite matrix polynomials, J.Inter. Math. Virtual Institute 5 (2015) 1–17.

[37] L.M. Upadhyaya and A. Shehata, On Legendre matrix polynomials and its appli-cations, Inter. Trans. Mathematical Sci. Computer (ITMSC) 4 (2) (2011) 291–310.

[38] L.M. Upadhyaya and A. Shehata, A new extension of generalized Hermite matrixpolynomials, Bulletin Malaysian Mathematical Sci. Soc. 38 (1) (2015) 165–179.

[39] S. Varma, B. Cekim, F. Tasdelen Yesildal, On Konhauser matrix polynomials,Ars Comb. 100 (2011) 193–204.

[40] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge UniversityPress, London, 1959.

[41] G. Yasmin and S. Khan, Hermite matrix based polynomials of two variables and

Lie algebraic techniques, South. Asian Bullet. Math. 38 (4) (2014) 603–618.

Documents

A New Extension of Bessel Matrix Functions · SoutheastAsian Bulletinof Mathematics c SEAMS.2016 Southeast Asian Bulletin of Mathematics (2016) 40: 265–288 A New Extension of Bessel