EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 10, No. 5, 2017, 995-1004
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Fractional orders of the generalized Bessel matrix
polynomials

M. Abdalla1,∗, M. M. Haidan2

1 Department of Mathematics, Faculty of Science, South Valley University, Qena 83523,
Egypt
2 Department of Mathematics, Faculty of Science for Girls, King Khalid University, Abha,
Saudi Arabia

Abstract. This paper presents and investigates generalized Bessel matrix polynomials (GBMPs)
with order α ∈ < (the set of real numbers). The given result is supposed to be an enhanced and
a generalized form of the scalar form to the fractional analysis setting. By using the Liouville-
Caputo operator of fractional analysis and Rodrigues type representation form of fractional order,
the generalized Bessel matrix functions (GBMFs) Yα(t;A,B), t ∈ C, for matrices A and B in
the complex space CN×N are derived and supplied with a matrix hypergeometric representation
that are satisfied by these functions. Subsequently, a fractional matrix recurrence relationship, a
fractional matrix of second-order differential equation and an orthogonal system are then developed
for GBMFs.

2010 Mathematics Subject Classifications: 33C05, 33C45, 34A05.

Key Words and Phrases: Fractional calculus, Generalized Bessel matrix polynomials, Ro-
drigues’ formula

1. Introduction

The generalized Bessel polynomials (GBPs) formula, a class of orthogonal polyno-
mials which is intimately related with the Bessel functions. They emerged in the solution
of differential equation of spherical waves. These polynomials have been studied first by
Bochner [4] who pointed out their connection with Bessel functions. A comprehensive
study on these polynomials was given by Krall and Frink [17]. Several other authors (see,
e.g., [2, 5, 12]) have contributed to the study of the Bessel polynomials. Special matrix
functions latterly show in several fields (see, for example [15, 24, 25]). A new extension
of hypergeomatric, Humbert and Appel matrix functions were introduced and studied
in [19, 20, 21]. In [1, 22] the scalar case of the generalized Bessel and reverse Bessel
polynomials have already been expanded into matrix setting.

∗Corresponding author.

Email addresses: mabdomath85@gmail.com, m.abdallah@sci.svu.edu.eg. (M. Abdalla),
mhedan@kku.edu.sa (M. Haidan)

http://www.ejpam.com 995 c© 2017 EJPAM All rights reserved.



M. Abdalla, M. M. Haidan / Eur. J. Pure Appl. Math, 10 (5) (2017), 995-1004 996

Several articles and books have been written recently in fractional calculus area, of
which we recommend (for instance, [6, 7, 10, 11, 29, 14, 18, 26, 27, 32]).

In recent years, many researchers have studied various special functions associated
with fractional calculus. Laguerre polynomials, Bell polynomials, Legendre polynomials
and generalized ultraspherical or Gegenbauer functions of arbitrary (fractional) orders have
been defined in [8, 9, 30, 31]. In addition, fractional derivatives of various multivariable
functions have been derived (for examples, [3, 23]).

The major purpose of this work is to obtain generalizations of the (GBMPs) by making
use of fractional calculus and Rodrigues type exemplification form of fractional order.
Therefore, the (GBMPs) with fractional order are obtained and some of their properties
such as a fractional matrix recurrence relations, the fractional matrix differential equation
and an orthogonality property are given. Starting, we mention some the fundamental
definitions of the fractional calculus and some properties of the matrix functions used in
the present work.

Definition 1. The fractional integral of order β ∈ <+, being the set of positive real
numbers, of the function f(τ), τ ≥ b is defined by (see [13, 28] and [26] )

Iβb f(τ) =

∫ τ

b

(τ − u)β−1

Γ(β)
f(u) du. (1)

The Liouville-Caputo fractional derivative of order α ∈ (n − 1, n) (n ∈ N := {1, 2, ...}) of
f(τ), τ ≥ a is defined by

Dα
b f(τ) = In−αb Dn f(τ), D =

d

dτ
. (2)

The fractional derivative of the product g(v)f(v) by [26], the Leibniz rule for fractional
differentiation takes the form

Dα[g(v)f(v)] =
∞∑
s=0

(
α
s

) g(s)(v)Dα−sf(v). (3)

Definition 2. (cf. [1, 16]) For all A in the complex space of matrices CN×N , and

A + nI is invertible for all n ∈ N0 := N ∪ {0}, (4)

then the Pochhammer symbol (the shifted factorial) is defined by

(A)n = A(A + I)...(A + (n− 1)I) = Γ(A + nI)Γ−1(A); (A)0 ≡ I. (5)

where I is unite matrix in CN×N .

Definition 3. [1, 16] Suppose that A,B and D are matrices in CN×N , and D satisfy con-
dition (4), then, the matrix power series of the hypergeometric matrix function is defined
in the form

F (A,B;D; z) =
∞∑
m=0

(A)m(B)m[(D)m]−1

m!
zm. (6)



M. Abdalla, M. M. Haidan / Eur. J. Pure Appl. Math, 10 (5) (2017), 995-1004 997

2. Generalized Bessel matrix functions of fractional order

The classical (GBMPs) Yn(z,A,B) are defined by Rodrigues’ type formula (see [1, 22])

Yn(z,A,B) = B−nz2I−Ae
B
z Dn

(
z2(n−1)I+Ae

−B
z

)
, (7)

where n ≥ 0, A and B are parameter matrices. When A = B = 2I, the analogue Rodrigues’
type formula for the (GBMPs) (7) reduces to the analogue Rodrigues’ type formula Bessel
polynomials proper:

yn(z) = 2−ne
2
z Dn

(
z2ne

−2
z

)
. (8)

By taking the the Liouville-Caputo fractional derivative Dα in (7), we introduce functions
which are naturally refereed to as generalized Bessel matrix functions (GBMFs).

Definition 4. Suppose that α ∈ (n− 1, n) (n ∈ N) and A and B are commuting matrices
in CN×N satisfying the spectral condition (4). Then the GBMFs are defined by the formula

Yα(t;A,B) = B−αt2I−Ae
B
t Lα(t); Lα(t) = Dα(tA+(2α−2)Ie−

B
t ). (9)

Using (9), the GBMFs would be represented by the hypergeometric matrix function

1F1(A,B; t) in the following result:

Theorem 1. The GBMFs can be written as

Yα(t;A,B) =(tB−1)αΓ−1(A + (α− 1)I)Γ(A + (2α− 1)I)

×1F1(−αI;−A + 2(1− α)I;
B
t

).
(10)

Proof. From (9) and the relation (3), we find that

Yα(t;A,B) =B−αt2I−Ae
B
t Dα

∞∑
s=0

(−B)s

s!
tA+(2α−2−s)I

=B−αt2I−Ae
B
t

∞∑
s=0

(−B)s

s!
tA+(α−2−s)I

× Γ−1(A + (α− 1− s)I)Γ(A + (2α− 1− s)I)

=(tB−1)αe
B
t

∞∑
s=0

(−B)s

s!
t−s

× Γ(A + (2α− 1− s)I) Γ−1(A + (α− 1− s)I)

=(tB−1)α Γ(A + (2α− 1)I) Γ−1(A + (α− 1)I)

× 1F1(−αI;−A + 2(1− α)I;
B
t

),

which yields the desired result.



M. Abdalla, M. M. Haidan / Eur. J. Pure Appl. Math, 10 (5) (2017), 995-1004 998

3. Recurrence relations and the differential equation

In this section, we shall show some recurrence relations for the matrix functions
Yα(t;A,B) which generalize (interpolate) those of the GBMPs Yn(z,A,B) (see[1, 22]).
In addition, we generalize the GBMFs (9) by solving the following linear homogeneous
fractional matrix differential equation:

t2 Y ′′α(t;A,B) + (tA + B) Y ′α(t;A,B)

= α(A + (α− 1)I) Yα(t;A,B).

The following lemma enables us to establish Theorem 2.

Lemma 1. Suppose that A and B are commuting matrices in CN×N satisfying the condi-
tion (4). For any α ∈ (n− 1, n) (n ∈ N),

(i) Lα+1(t) (A + (α − 1)I)(A + 2(α − 1)I) = Lα(t)
[
(A + 2αI)(A + 2(α − 1)I)t + B(A −

2I)
]
(A + (2α− 1)I) + Lα−1(t) α B2(A + 2αI).

(ii) Lα+1(t) (A + (α− 1)I) = L′α(t) (A + 2αI)t2 + Lα(t)
[
(A + 2αI)(α+ 1)t− B(α+ 1)

]
.

(iii) Lα+1(t) (A+2(α−1)I)t2 =
[
(A+2(α−1)I)(A+(α−2)I)t+B(A+(α−2)I)

]
Lα(t)+

Lα−1(t)B2α.

Proof. (i) Using the Leibniz rule for fractional derivative [26], the fractional derivative
in (9) yields

(A + (α− 1)I)(A + 2(α− 1)I) Lα+1(t)

=(A + (α− 1)I)(A + 2(α− 1)I)(A + 2αI) DαtA+(2α−1)I e
−B
t

+ B(A + (α− 1)I)(A + 2(α− 1)I) Lα(t)

=(A + (α− 1)I)(A + 2(α− 1)I)(A + 2αI)t Lα(t)

+α(A + (α− 1)I)(A + 2(α− 1)I)(A + 2αI)Dα−1tA+2(α−1)I e
−B
t

+B(A + (2α− 1)I)(A− 2I) Lα(t)

+α B(A + 2αI)
[
(A + 2(α− 1)I) Dα−1tA+(2α−3)I e

−B
t + B Lα−1(t)

]
=
[
(A + (α− 1)I)(A + 2(α− 1)I)(A + 2αI)t+ B(A + (2α− 1)I)(A− 2I)

+α(A + 2(α− 1)I)(A + 2αI)t
]
Lα(t) + α B2 (A+ 2αI) Lα−1(t).

Hence,

(A + (α− 1)I)(A + 2(α− 1)I)Lα+1(t)

=
[
(A + 2αI)(A + 2(α− 1)I)t+ B(A− 2I)

]
(A + (2α− 1)I) Lα(t) + αB2(A + 2αI) Lα−1(t).

(11)



M. Abdalla, M. M. Haidan / Eur. J. Pure Appl. Math, 10 (5) (2017), 995-1004 999

(ii) We have

Lα+1(t) = t2 Dα+1tA+2(α−1)I e
−B
t

+ 2(α+ 1)t DαtA+2(α−1)I e
−B
t

+α(α+ 1) Dα−1tA+2(α−1)I e
−B
t

= t2 L′α(t) + 2(α+ 1)t Lα(t)

+α(α+ 1) Dα−1tA+2(α−1)I e
−B
t

(12)

and

Lα+1(t) =(A + 2αI) DαtA+(2α−1)I e
−B
t + B Lα(t)

=
[
(A + 2αI)t+ B

]
Lα(t)

+α(A + 2αI) Dα−1tA+2(α−1)I e
−B
t .

(13)

If we multiply (12) by (A+ 2αI) and (13) by (α+ 1), then subtract we obtain the required
result.

(iii) Multiply both sides of the equation (ii) above by (A + 2(α− 1)) and substitute for
(A + (α − 1)I)(A + 2(α − 1)I)Lα+1(t) from (i) in (ii) and on rearrangement, we obtain
(iii).

To prove the following result:

Theorem 2. Suppose that A and B are commuting matrices in CN×N satisfying the
condition (4). Then the GBMFs satisfy the following recurrence relations:

(A + (α− 1)I)(A + 2(α− 1)I) Yα+1(t;A,B)

=
[
(A + 2αI)(A + 2(α− 1)I)tB−1 + (A− 2I)

]
(A + (2α− 1)I) Yα(t;A,B) + α(A + 2αI) Yα−1(t;A,B).

(14)

(A + 2αI)t2 Y ′α(t;A,B) = B(A + (α− 1)I) Yα+1(t;A,B)

− (A + (α− 1)I)[(A + 2αI)t+ B] Yα(t;A,B).
(15)

(A + 2(α− 1)I)t2 Y ′α(t;A,B) = α B Yα−1(t;A,B)

+ [α (A + 2(α− 1)I)t− α B] Yα(t;A,B).
(16)

Y ′α(t;A,B) [(A + 2(α− 1)I)t+ B] (A + (α− 2)I)

+α B Yα−1(t;A,B) = α (A + (α− 2)I)(A + 2(α− 1)I) Yα(t;A,B).
(17)



M. Abdalla, M. M. Haidan / Eur. J. Pure Appl. Math, 10 (5) (2017), 995-1004 1000

Proof. Using Lemma 1, substitute for

Lα+1(t) = Bα+1tA−2Ie
−B
t Yα+1(t;A,B),

Lα(t) = BαtA−2Ie
−B
t Yα(t;A,B)

and
Lα−1(t) = Bα−1tA−2Ie

−B
t Yα−1(t;A,B),

in (i), (ii) and (iii) respectively, we get (14), (15) and (16). If we multiply (16) by

1

t2

[
(A + (α− 1)I)[(A + 2(α− 1)I)t+ B](A + 2(α− 1)I)−1

]
and multiply (15 ) by

1

t2
[
αB(A + 2(α− 1)I)−1

]
after replace α by α− 1 and add, we obtain Eq.(17).

Other recurrence relations for the GBMFs Yα(t;A,B) may be derived from the rela-
tions in Theorem 2.
Now, the major property developed here is the differential equation for the GBMFs
Yα(t;A,B) which is derived from their recurrence relation established by Theorem 2. By
differentiating equation (16) we find

t2 (A + 2(α− 1)I) Y ′′α(t;A,B)

+2t (A + 2(α− 1)I) Y ′α(t;A,B)

=α[t(A + 2(α− 1)I)− B] Y ′α(t;A,B)

+α(A + 2(α− 1)I) Yα(t;A,B) + α B Y ′α−1(t;A,B).

(18)

From (17) and (18), a straightforward computation shows that

t2 (A + 2(α− 1)I) Y ′′α(t;A,B) + 2t (A + 2(α− 1)I) Y ′α(t;A,B)

=α(A + (α− 1)I)(A + 2(α− 1)I)Yα(t;A,B),

t2 (A + 2(α− 1)I) Y ′′α(t;A,B) + [t(A + 2(α− 1)I)A

+B(A + 2(α− 1)I)] Y ′α(t;A,B)

=α(A + (α− 1)I)(A + 2(α− 1)I)Yα(t;A,B).

Thus,

t2 Y ′′α(t;A,B) + (tA + B) Y ′α(t;A,B)

= α(A + (α− 1)I) Yα(t;A,B).
(19)

Therefore the following theorem is proved.

Theorem 3. Let A and B be commuting matrices in CN×N , satisfying the spectral condi-
tion (4). Then the GBMFs satisfies fractional matrix differential equation in (19).



M. Abdalla, M. M. Haidan / Eur. J. Pure Appl. Math, 10 (5) (2017), 995-1004 1001

4. Orthogonality property

The research subject of an orthogonal system for the GBMFs is discussed in this section
with the weight function %(t) which is defined by (see, [22])

%(t) =
1

2πi

∞∑
s=0

Γ−1(A + (s− 1)I) Γ(A) (
−B
t

)s, (20)

which satisfies the related matrix nonhomogeneous equation

%(t)
′
(t2 = %(t)(At+ B)− [(A− 2I)(A− I)]t

2πi
. (21)

When the relation (19) is multiplied by %(t), we get

Y ′α(t;A,B))
′

(t2%(t)− Y ′α(t;A,B) (t2%(t))
′
+ Y ′α(t;A,B) (At+B)%(t)

=Yα(t;A,B) αI(A + (α− 1)I)%(t),

and using (21), we have

(zt2%(t)Y ′α(t;A,B))
′
+

[(A− I)(A− B)]t

2πi
Y ′α(t;A,B)

=αI(A + (α− 1)I)Yα(t;A,B)%(t).
(22)

Multiplying Yγ(t;A,B) in (22) and and integrating the result around the unit circle, one
gets ∫

C
(t2%(t)Y ′α(t;A,B))

′Yγ(t;A,B) dt

+

∫
C

[(A− I)(A− 2I)]t

2πi
Y ′α(t;A,B) Yγ(t;A,B) dt

= αI(A + (α− 1)I)

∫
C
%(t)Yα(t;A,B)Yγ(t;A,B) dt.

(23)

Consider the straightforward computation integrating, we see that

αI(A + (α− 1)I)

∫
C
%(t)Yα(t;A,B)Yγ(t;A,B) dt

=−
∫
C
t2%(t)Y ′α(t;A,B)Y ′γ(t;A,B) dt.

(24)

Interchanging α and γ, that is

γI(A + (γ − 1)I)

∫
C
%(t)Yα(t;A,B)Yγ(t;A,B) dt

=−
∫
C
t2%(t)Y ′α(t;A,B)Y ′γ(t;A,B) dt



REFERENCES 1002

and subtracting gives

[αI(A + I(α− 1))− γI(A + (γ − 1)I)]

∫
C
%(t)Yα(t;A,B)Yγ(t;A,B) dt = 0.

Finally, for α 6= γ, we get∫
C
%(t)Yα(t;A,B)Yγ(t;A,B) dt = 0. (25)

This result can be expressed as follows:

Theorem 4. For any real numbers α 6= γ and let A and B be commutative matrices in
CN×N , satisfying the condition (4), then expression (25) hold true.

Acknowledgment

The authors are very grateful to the anonymous referees for many valuable comments
and suggestions which helped to improve the paper.

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