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Analytical properties of the two-variables Jacobi matrix polynomials with applications

  • Mohamed Abdalla EMAIL logo and Muajebah Hidan
Published/Copyright: June 15, 2021
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 54 Issue 1

Abstract

In the current study, we introduce the two-variable analogue of Jacobi matrix polynomials. Some properties of these polynomials such as generating matrix functions, a Rodrigue-type formula and recurrence relations are also derived. Furthermore, some relationships and applications are reported.

Keywords: Jacobi matrix polynomials; generating matrix relations; recurrence formulas
MSC 2010: 33C50; 33C45; 33D15; 65N35

1 Introduction

The generating function of the classical Jacobi polynomials is given by (cf., e.g., [1,2])

(1.1) ∑ n = 0 ∞ P n ( α , β ) ( z ) t n = 2 α + β ϱ − 1 ( 1 − t + ϱ ) − α ( 1 + t + ϱ ) − β ,

where

P n ( α , β ) ( z ) = Γ ( α + n + 1 ) n ! Γ ( α + β + n + 1 ) ∑ m = 0 n n m Γ ( α + β + n + m + 1 ) Γ ( α + m + 1 ) z − 1 2 m ,

ϱ = ϱ ( z , t ) = ( 1 − 2 z t + t 2 ) 1 2 and Γ ( . ) is the usual gamma function.

These polynomials are generalizations of several families of orthogonal polynomials like the Legendre, Chebyshev and Gegenbauer (ultraspherical) polynomials. In addition, the classical orthogonal polynomials of Jacobi have played important roles in many different applications of mathematics, physics and engineering sciences (see, e.g., [1,2,3, 4,5,6]).

In contrast, the special functions and polynomials with matrix parameters have many applications in various areas of mathematical analysis, physics, probability theory, statistics and engineering (see, [7,8, 9,10,11, 12,13,14, 15,16]). One particular special matrix polynomial which frequently appears in the recent investigations is the Jacobi matrix polynomial (JMP) that has been introduced in [17,18,19].

The aim of the present work is to study two-variable analogue of Jacobi matrix polynomials (2VAJMP) J n ( E , F , z , w ) and their properties, which have been proposed on the pattern for two-variables Konhauser matrix polynomials [20], two-variable Shivley’s matrix polynomials [21], two-variable Laguerre matrix polynomials [22], two-variable Hermite generalized matrix polynomials [23], two-variable Gegenbauer matrix polynomials [24] and the second kind Chebyshev matrix polynomials with two variables [25]. The current work is assumed to be extensions to the matrix setting of the results of [26].

The paper is organized as follows. In Section 2, we summarize definition and previous results to be used in the following sections. Section 3 contains the definition of the 2VAJMP J n ( E , F , z , w ) , for parameter matrices E and F associated with some generating matrix relations. A Rodrigue-type formula and recurrence relations for 2VAJMP J n ( E , F , z , w ) are archived in Section 4. Finally, we give some relationships and applications in Section 5.

2 Preliminaries

In this section, we recall some definitions and facts whose more detailed accounts and applications can be found in [7,16].

If E is a matrix in the complex space C d × d ( d ∈ N ) , its spectrum σ ( E ) denotes the set of all the eigenvalues of E and μ ( E ) = max { Re ( z ) : z ∈ σ ( E ) } , ν ( E ) = min { Re ( z ) : z ∈ σ ( E ) } . The square matrix E is said to be positive stable if and only if ν ( E ) > 0 . I and 0 stand for the identity matrix and the null matrix in C d × d , respectively.

If Φ ( z ) and Ψ ( z ) are holomorphic functions of the complex variable z , which are defined in an open set Ω of the complex plane, and E is a matrix in C d × d such that σ ( E ) ⊂ Ω , then from the properties of the matrix functional calculus, it follows that

(2.1) Φ ( E ) Ψ ( E ) = Ψ ( E ) Φ ( E ) .

Hence, if F in C d × d is a matrix for which σ ( F ) ⊂ Ω and also if E F = F E , then

(2.2) Φ ( E ) Ψ ( F ) = Ψ ( F ) Φ ( E ) .

By application of the matrix functional calculus, for E in C d × d , then from [7], the Pochhammer symbol or shifted factorial defined by

(2.3) ( E ) n = E ( E + I ) … ( E + ( n − 1 ) I ) = Γ − 1 ( A ) Γ ( E + n I ) , n ≥ 1 , I , n = 0 ,

with the condition

(2.4) E + n I is invertible for all integers n ≥ 0 .

Definition 2.1

[27] If E is a matrix in C d × d , such that Re ( z ) > 0 for all eigenvalues z of E , then Γ ( E ) is well defined as

(2.5) Γ ( E ) = ∫ 0 ∞ τ E − I e − τ d τ ; τ E − I = exp ( ( E − I ) ln τ ) .

Definition 2.2

[7] Suppose that N 1 , N 2 and N 3 are matrices in C d × d , such that N 3 satisfies the condition (2.4). Then, the hypergeometric matrix function F 1 2 ( N 1 , N 2 ; N 3 ; z ) is given by

(2.6) F 1 2 ( N 1 , N 2 ; N 3 ; z ) = ∑ n = 0 ∞ ( N 1 ) n ( N 2 ) n [ ( N 3 ) n ] − 1 z n n ! .

Definition 2.3

[7,17,18] Let E and F be positive stable matrices in C d × d , then the JMP P n ( E , F ) ( z ) is defined by

(2.7) P n ( E , F ) ( z ) = ( E + I ) n n ! F 1 2 − n I , E + F + ( n + 1 ) I E + I ; 1 − z 2 .

Definition 2.4

[7,11] Let N 1 , N 2 , N 3 and N 4 be commutative matrices in C d × d with N 3 + n I and N 4 + n I invertible for all integers n ≥ 0 . Then the four Appell hypergeometric matrix function F 4 N 1 , N 2 N 3 , N 4 ; z , w  is defined in the following form:

(2.8) F 4 N 1 , N 2 N 3 , N 4 ; z , w = ∑ μ , ν = 0 ∞ ( N 1 ) μ + ν ( N 2 ) μ + ν [ ( N 3 ) μ ] − 1 [ ( N 4 ) ν ] − 1 μ ! ν ! z μ w ν , ∣ z ∣ 1 2 + ∣ w ∣ 1 2 < 1 .

In [28], it was shown that JMPs are generated by

(2.9) ∑ n = 0 ∞ P n ( E , F ) ( z ) t n = F 4 I + F , I + E I + E , I + F ; ( z − 1 ) t 2 ; ( z + 1 ) t 2 ,

where E , F ∈ C d × d with E + n I and F + n I invertible for every integer n ≥ 0 and E F = F E .

3 Some generating relations for 2VAJMP

Let E and F be matrices in the complex space C d × d , satisfying the conditions

(3.1) Re ( z ) > − 1 ∀ z ∈ σ ( E ) , Re ( z ) > − 1 ∀ z ∈ σ ( F ) and E F = F E .

In view of (1.1), we define 2VAJMP with matrix generating form:

(3.2) ∑ n = 0 ∞ J n ( E , F , z , w ) t n = 2 E + F ℛ − 1 ( 1 + w t + ℛ ) − F ( 1 − w t + ℛ ) − E ,

where ℛ = ℛ ( z , w , t ) = ( 1 − 2 z t + w t 2 ) 1 2 .

In this section, we obtain power series and matrix generating relations for 2VAJMP J n ( E , F , z , w ) .

Theorem 3.1

Let E and F be positive stable matrices in C d × d such that E + n I and F + n I are invertible for all integers n ≥ 0 . Then J n ( E , F , z , w ) takes the following explicit forms:

(3.3) J n ( E , F , z , w ) = ∑ s = 0 n ( E + I ) n ( F + I ) n s ! ( n − s ) ! [ ( E + I ) s ] − 1 [ ( F + I ) n − s ] − 1 z − w 2 s z + w 2 n − s

or

(3.4) J n ( E , F , z , w ) = ∑ s = 0 n ( E + I ) n ( E + F + I ) n + s s ! ( n − s ) ! [ ( E + I ) s ] − 1 [ ( E + F + I ) n ] − 1 z − w 2 s .

Proof

Using the relation (2.9) with the following result (see [29]):

(3.5) F 4 P , Q Q , P ; − x ( 1 − x ) ( 1 − y ) , − y ( 1 − x ) ( 1 − y ) = ( 1 − x y ) − 1 ( 1 − x ) P ( 1 − y ) Q ,

where F 4 is defined in (2.8) with P and Q positive stable matrices in C d × d .

If we take P = I + F , Q = I + E and

(3.6) − x ( 1 − x ) ( 1 − y ) = t ( z − 1 ) 2 , − y ( 1 − x ) ( 1 − y ) = t ( z + 1 ) 2 .

From (3.2), let us consider

(3.7) ℛ = ( 1 − 2 z t + w t 2 ) 1 2 , x = 1 − 2 1 + w t + ℛ , y = 1 − 2 1 − w t + ℛ .

Using (3.7), it follows that

(3.8) − x ( 1 − x ) ( 1 − y ) = 1 1 − y 1 − 1 1 − x = 1 − w t + ℛ 2 1 − 1 + w t + ℛ 2 = ( 1 − w t + ℛ ) ( 1 − w t − ℛ ) 4 = ( 1 − w t ) 2 − ℛ 2 4 = t ( z − w ) 2 .

Similarly,

(3.9) − y ( 1 − x ) ( 1 − y ) = t ( z + w ) 2 .

Now, according to (2.9), (3.8) and (3.9), we have

(3.10) ∑ n = 0 ∞ J n ( E , F , z , w ) t n = F 4 I + F , I + E I + E , I + F ; t 2 ( z − w ) , t 2 ( z + w ) .

Therefore, equation (3.10) can be rewritten as

(3.11) ∑ n = 0 ∞ J n ( E , F , z , w ) t n = ∑ n , s = 0 ∞ ( E + I ) n + s ( F + I ) n + s s ! n ! [ ( E + I ) s ] − 1 [ ( F + I ) n ] − 1 1 2 ( z − w ) s 1 2 ( z + w ) n t n ,

which coincides with our assertion (3.3).□

Remark 3.1

It may be noted that for w = 1 , (2VAJMP) J n ( E , F , z , w ) are reduced to JMPs of one variable P n ( E , F ) ( z ) (see [17]).

Remark 3.2

From (3.3) and (3.4), we have the matrix representation for (2VAJMP) J n ( E , F , z , w ) in terms of matrix hypergeometric series in the following forms:

(3.12) J n ( E , F , z , w ) = ( A + I ) n n ! z + w 2 n F 1 2 − n I , − ( F + n I ) I + E ; z − w z + w

and

(3.13) J n ( E , F , z , w ) = ( E + I ) n n ! F 1 2 − n I , E + F + I ( n + 1 ) I + E ; w − z 2 .

Remark 3.3

For E = F = 0 , (3.12) reduces to Legendre polynomials of two complex variables P n ( z , w ) in the following form:

(3.14) P n ( z , w ) = z + w 2 n F 1 2 − n I , − n I I ; z − w z + w .

Theorem 3.2

Let E , F ∈ C d × d be positive stable matrices, with ∣ t ∣ < 1 , ( z − w ) ( 1 − t ) 2 < 1 . The generating matrix function of J n ( E , F , z , w ) is as follows:

(3.15) ∑ ν = 0 ∞ ( E + F + I ) ν [ ( E + I ) ν ] − 1 J ν ( E , F , z , w ) t ν = ( 1 − t ) − ( I + E + F ) F 1 2 1 2 ( E + F + I ) , 1 2 ( 2 I + E + F ) I + E ; 2 t ( z − w ) ( 1 − t ) 2 .

Proof

For convenience, suppose that the left-hand side of (3.15) is denoted by Ξ . From relation (3.3) to Ξ , we obtain

Ξ = ∑ ν = 0 ∞ ( E + F + I ) ν [ ( E + I ) ν ] − 1 J ν ( E , F , z , w ) t ν = ∑ ν = 0 ∞ ∑ k = 0 ν ( E + F + I ) ν + k k ! ( ν − k ) ! [ ( E + I ) k ] − 1 z 2 − w 2 k t ν = ∑ ν = 0 ∞ ∑ k = 0 ∞ ( E + F + I ) ν + 2 k 2 k k ! ν ! [ ( E + I ) k ] − 1 ( z − w ) k t ν + k = ∑ k = 0 ∞ ∑ ν = 0 ∞ ( E + F + ( 2 k + 1 ) I ) ν t ν ν ! ( E + F + I ) 2 k ( z − w ) k k ! 2 k [ ( E + I ) k ] − 1 t k = ∑ k = 0 ∞ [ ( E + I ) k ] − 1 ( E + F + I ) 2 k ( z − w ) k t k k ! 2 k ( 1 − t ) − ( E + F + ( 1 + 2 k ) I ) .

Using the identity

( E ) 2 k = 2 2 k E 2 k E + I 2 k ,

evidently leads us to the required result in (3.15).□

Remark 3.4

Setting w = 1 in (3.15), we get the result of [28] given by

∑ n = 0 ∞ ( E + F + I ) n P n ( E , F ) ( z ) [ ( E + I ) n ] − 1 t n = ( 1 − t ) − ( E + F + I ) F 1 2 E + F + I 2 , E + F + 2 I 2 I + E ; 2 t ( z − 1 ) ( 1 − t ) 2 .

Theorem 3.3

Let E and F be positive stable matrices in C d × d such that E and F satisfy the spectral condition (2.4) with t 2 ( z − w ) < 1 and t 2 ( z + w ) < 1 , the following Bateman’s generating matrix function holds true:

(3.16) ∑ ν = 0 ∞ J ν ( E , F , z , w ) [ ( E + I ) ν ] − 1 [ ( F + I ) ν ] − 1 t ν = F 1 0 − E + I ; t 2 ( z − w ) × F 1 0 − F + I ; t 2 ( z + w ) .

Proof

Starting with the explicit series

∑ ν = 0 ∞ J ν ( E , F , z , w ) [ ( E + I ) ν ] − 1 [ ( F + I ) ν ] − 1 t ν = ∑ ν = 0 ∞ ∑ k = 0 ∞ 1 2 ( z − w ) k 1 2 ( z + w ) ν − k k ! ( ν − k ) ! [ ( E + I ) k ] − 1 [ ( F + I ) ν − k ] − 1 t ν = ∑ ν = 0 ∞ 1 2 ( z − w ) ν ν ! [ ( E + I ) ν ] − 1 ∑ ν = 0 ∞ 1 2 ( z + w ) ν ν ! [ ( F + I ) ν ] − 1 = F 1 0 − E + I ; t 2 ( z − w ) × F 1 0 − F + I ; t 2 ( z + w ) ,

which is precisely assertion (3.16) of Theorem 3.3.□

Working on the same lines as in previous theorems, we state matrix version of Brafman’s generating function in the next theorem without proof.

Theorem 3.4

Let D , E and F be commutative matrices in C d × d such that E and F satisfy the spectral condition (2.4) with 1 − w t − ℛ 2 < 1 and 1 + w t − ℛ 2 < 1 , the following Brafman’s generating matrix function holds true:

∑ ν = 0 ∞ ( D ) ν ( E + F − D + I ) ν [ ( E + I ) ν ] − 1 [ ( F + I ) ν ] − 1 J ν ( E , F , z , w ) t ν = F 1 2 D , E + F − D + I E + I ; 1 − w t − ℛ 2 F 1 2 D , E + F − D + I F + I ; 1 + w t − ℛ 2 , ℛ = ( 1 − 2 z t + w t 2 ) − 1 2 .

4 Rodrigues’ formula and recurrence relations

Two more basic properties of the 2VAJMP J n ( E , F , z , w ) are developed in this section, which enjoy a Rodrigues’ formula obtained from Theorem 4.1 and with the help of (3.3). Also, some various recurrence relations for the 2VAJMP are given.

4.1 Rodrigues’ formula

Theorem 4.1

Let E and F be matrices in C d × d that satisfy (3.1). Then the 2VAJMP J n ( E , F , z , w ) defined in (3.3) may be expressed as

(4.1) J n ( E , F , z , w ) = ( z − w ) − E ( z + w ) − F 2 n n ! D n [ ( z − w ) E + n I ( z + w ) F + n I ] , D ≡ d d ( z ± w ) .

Proof

We begin with the explicit expression (3.3) for J n ( E , F , z , w ) :

(4.2) J n ( E , F , z , w ) = ∑ s = 0 n ( E + I ) n ( F + I ) n 2 n s ! ( n − s ) ! [ ( E + I ) s ] − 1 [ ( F + I ) n − s ] − 1 ( z − w ) s ( z + w ) n − s .

We recall that (cf., e.g., [17,18])

(4.3) D μ z m I + E = ( E + I ) m [ ( E + I ) m − μ ] − 1 z ( m − μ ) I + E , D ≡ d d z .

In view of (4.3), we get

(4.4) D k ( z + w ) F + I m = [ ( F + I ) m − k ] − 1 ( F + I ) m ( z + w ) m I − k I + F

and

(4.5) D m − k ( z − w ) F + n I = [ ( F + I ) m ] − 1 ( E + I ) m ( z − w ) E + k I .

Using (4.4) and (4.5) in (4.1), it follows that

J n ( E , F , z , w ) = ( z − w ) − E ( z + w ) − F 2 n n ! ∑ s = 0 ∞ n ! s ! ( n − s ) ! [ D n − s ( z − w ) E + n I ] [ D s ( z + w ) F + n I ]

Applying the Leibnitz rule yields the desired result of Theorem 4.1.□

4.2 Recurrence relations

Following some various matrix recurrence relations satisfied by 2VAJMP J n ( E , F , z , w ) in (3.2) as follows:

First, the 2VAJMP J n ( E , F , z , w ) satisfy the following total differential matrix recurrence relations:

(4.6) z − w w [ ( E + F + n I ) D J n ( E , F , z , w ) + ( E + I n ) D J n − 1 ( E , F , z , w ) ] = ( E + F + n I ) [ n J n ( E , F , z , w ) − ( E + n I ) J n − 1 ( E , F , z , w ) ] , D ≡ d d z w ,

(4.7) z + w w [ ( E + F + n I ) D J n ( E , F , z , w ) + ( F + I n ) D J n − 1 ( F , E , z , w ) ] = ( E + F + n I ) [ n J n ( E , F , z , w ) − ( F + n I ) J n − 1 ( E , F , z , w ) ] ,

(4.8) z − w w D J n ( E , F , z , w ) − n J n − 1 ( E , F , z , w ) = − ( E + I ) n [ ( E + F + I ) n ] − 1 ∑ s = 0 n − 1 ( E + F + I ) s [ ( E + I ) s ] − 1 × ( E + F + n I ) J n ( E , F , z , w ) + 2 z − w w D J n ( E , F , z , w ) ,

(4.9) z − w w D J n ( E , F , z , w ) − n J n − 1 ( E , F , z , w ) = ( E + I ) n [ ( E + F + I ) n ] − 1 ∑ s = 0 n − 1 ( − 1 ) n − s ( E + F + 2 k I ) s ( E + F + I ) s [ ( E + I ) s ] − 1 J s ( E , F , z , w ) ,

(4.10) ( E + F + 2 n I ) z 2 − w w ( E + F + 2 n I ) D J n ( E , F , z , w ) = n ( F − E ) w + z ( E + F + 2 n I ) w J n ( E , F , z , w ) − 2 ( E + n I ) ( F + n I ) J n − 1 ( E , F , z , w ) ,

(4.11) 2 z ( E + F + n I ) D J n ( E , F , z , w ) + [ z ( E − F ) − ( E + F + 2 n I ) w ] D J n − 1 ( E , F , z , w ) = w ( E + F + n I ) [ 2 n J n ( E , F , z , w ) − ( E − F ) J n − 1 ( E , F , z , w ) ] ,

(4.12) 2 w ( E + F + n I ) D J n ( E , F , z , w ) + [ w ( E − F ) − ( E + F + 2 n I ) z ] D J n − 1 ( E , F , z , w ) = w ( E + F + n I ) ( E + F + 2 n I ) J n − 1 ( E , F , z , w )

and

(4.13) D s J n ( E , F , z , w ) = 2 − m ( E + F + ( n + 1 ) I ) s J n − s ( E + s I , F + s I , z , w ) , 0 < s ≤ n ,

where E and F are matrices in C d × d that satisfy conditions (3.1).

Second, the 2VAJMP J n ( E , F , z , w ) give the following pure matrix recurrence relation:

2 n ( E + F + n I ) ( E + F − 2 I ( n − 1 ) ) J n ( E , F , z , w ) = ( E + F + ( 2 n − 1 ) I ) w ( E 2 − F 2 ) + z ( E + F + 2 n I ) ( E + F + 2 I ( n − 1 ) ) w J n − 1 ( E , F , z , w ) − 2 ( E + ( n − 1 ) I ) ( F + ( n − 1 ) I ) ( E + F + 2 n I ) J n − 2 ( E , F , z , w ) ,

where E and F are matrices in C d × d that satisfy conditions (3.1).

Remark 4.1

For w = 1 , we obtain matrix recurrence relations of the Jacobi matrix polynomial P n ( E , F ) ( z ) .

5 Applications

In this section, we obtain some other interesting results and applications involving J n ( E , F , z , w ) by the formalism developed in the above sections.

  1. Following relationships can easily be obtained from (3.2) as follows:

    ∑ n = 0 ∞ J n ( E , F , z , w ) t n = 2 E + F ℛ − 1 ( 1 + w t + ℛ ) − F ( 1 − w t + ℛ ) − E = 2 E + F ( 1 − 2 z t + w t 2 ) 1 2 ( 1 + w t + ℛ ) − F ( 1 − w t + ℛ ) − E = ∑ n = 0 ∞ P n ( z , w ) t n 1 + w t + ℛ 2 − F 1 − w t + ℛ 2 − E ,

    where ℛ = ( 1 − 2 z t + w t 2 ) − 1 2 , thus, we have

    (5.1) J n ( E , F , z , w ) = P n ( z , w ) 1 + w t + ℛ 2 − F 1 − w t + ℛ 2 − E ,

    where P n ( z , w ) is the two-variable Legendre polynomial, putting w = 1 in (5.1), we obtain

    P n E , F ( z ) = P n ( z ) 1 + t + ℛ 2 − F 1 − t + ℛ 2 − E ,

    where P n ( z ) is the Legendre polynomial of one veritable (see [1]).

  2. In Theorem 3.1, if x , y and ℛ are chosen as

    (5.2) ℛ = ℛ ( η , z , w , t ) = ( η − 2 z t + w t 2 ) 1 2 , x = 1 − 2 η η + w t η + ℛ , y = 1 − 2 η η − w t η + ℛ .

    Then, according to (2.9) with the same way as in Theorem 3.1, we get

    ∑ n = 0 ∞ J n ( E , F , z , w ) t n = ( 2 η ) E + F ℛ − 1 ( η + w t + ℛ ) − F ( η − w t + ℛ ) − E .

    Also, we can write

    (5.3) ∑ n = 0 ∞ J n ( E , F , z , w ) t n = F 4 I + F , I + E I + E , I + F ; t 2 ( z − η w ) , t 2 ( z + η w ) .

  3. Taking

    (5.4) ℛ = ( 1 − 2 ( z + η ) t + w t 2 ) 1 2 , x = 1 − 2 1 + w t + ℛ , y = 1 − 2 1 − w t + ℛ .

    Then

    − x ( 1 − x ) ( 1 − y ) = 1 1 − y 1 − 1 1 − x = 1 − w t + ℛ 2 1 − 1 + w t + ℛ 2 .

    Hence,

    − x ( 1 − x ) ( 1 − y ) = t ( ( z + η ) − w ) 2 .

    Similarly,

    (5.5) − y ( 1 − x ) ( 1 − y ) = t ( ( z + η ) + w ) 2 .

    Using (5.4) and (5.5) in (2.9), we have

    (5.6) ∑ n = 0 ∞ J n ( E , F , z + η , w ) t n = ( 2 ) E + F ℛ − 1 ( 1 + w t + ℛ ) − F ( 1 − w t + ℛ ) − E

    or

    (5.7) ∑ n = 0 ∞ J n ( E , F , z + η , w ) t n = F 4 I + F , I + E I + E , I + F ; t 2 ( ( z + η ) − w ) , t 2 ( ( z + η ) + w ) .

  4. Furthermore,

    (5.8) ℛ = ( 1 − 2 z t + ( w + η ) t 2 ) 1 2 , x = 1 − 2 1 + ( w + η ) t + ℛ , y = 1 − 2 1 − ( w + η ) t + ℛ ,

    give us

    (5.9) − x ( 1 − x ) ( 1 − y ) = 1 1 − y 1 − 1 1 − x = 1 − ( w + η ) t + ℛ 2 1 − 1 + ( w + η ) t + ℛ 2 = t ( z − ( w + η ) ) 2 .

    Similarly,

    (5.10) − y ( 1 − x ) ( 1 − y ) = t ( z + ( w + η ) ) 2 .

    Applying (5.9) and (5.10) in (2.9), as follows that

    (5.11) ∑ n = 0 ∞ J n ( E , F , z , ( w + η ) ) t n = ( 2 ) E + F ℛ − 1 ( 1 + ( w + η ) t + ℛ ) − F ( 1 − ( w + η ) t + ℛ ) − E ,

    (5.12) ∑ n = 0 ∞ J n ( E , F , z , ( w + η ) ) t n = F 4 I + F , I + E I + E , I + F ; t 2 ( z − ( w + η ) ) , t 2 ( z + ( w + η ) ) .

  5. Setting ℛ = ( 1 − 2 α z t + β w t 2 ) 1 2 , α , β ∈ C , then from (2.9), we obtain

    (5.13) ∑ n = 0 ∞ J n ( E , F , α z , β w ) t n = ( 2 ) E + F ℛ − 1 ( 1 + β w t + ℛ ) − F ( 1 − β w t + ℛ ) − E ,

    (5.14) ∑ n = 0 ∞ J n ( E , F , α z , β w ) t n = F 4 I + F , I + E I + E , I + F ; t 2 ( α z − β w ) , t 2 ( α z + β w ) .

Acknowledgements

The authors would like to thank anonymous referees for their careful corrections and comments on the original version of this article.

  1. Author contributions: M.A., M.H.: writing-review and editing. All authors have read and agreed to the published version of the paper.

  2. Conflict of interest: Authors state no conflict of interest.

  3. Data availability statement: Data sharing is not applicable to this paper as no new data were created or analyzed in this study.

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Received: 2021-03-11
Revised: 2021-05-03
Accepted: 2021-05-14
Published Online: 2021-06-15

© 2021 Mohamed Abdalla and Muajebah Hidan, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2021-0021
Keywords for this article
Jacobi matrix polynomials; generating matrix relations; recurrence formulas
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Articles in the same Issue

  1. Regular Articles
  2. Graded I-second submodules
  3. Corrigendum to the paper “Equivalence of the existence of best proximity points and best proximity pairs for cyclic and noncyclic nonexpansive mappings”
  4. Solving two-dimensional nonlinear fuzzy Volterra integral equations by homotopy analysis method
  5. Chandrasekhar quadratic and cubic integral equations via Volterra-Stieltjes quadratic integral equation
  6. On q-analogue of Janowski-type starlike functions with respect to symmetric points
  7. Inertial shrinking projection algorithm with self-adaptive step size for split generalized equilibrium and fixed point problems for a countable family of nonexpansive multivalued mappings
  8. On new stability results for composite functional equations in quasi-β-normed spaces
  9. Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1]
  10. Meromorphic solutions of the (2 + 1)- and the (3 + 1)-dimensional BLMP equations and the (2 + 1)-dimensional KMN equation
  11. On the equivalence between weak BMO and the space of derivatives of the Zygmund class
  12. On some fixed point theorems for multivalued F-contractions in partial metric spaces
  13. On graded Jgr-classical 2-absorbing submodules of graded modules over graded commutative rings
  14. On almost e-ℐ-continuous functions
  15. Analytical properties of the two-variables Jacobi matrix polynomials with applications
  16. New soft separation axioms and fixed soft points with respect to total belong and total non-belong relations
  17. Pythagorean harmonic summability of Fourier series
  18. More on μ-semi-Lindelöf sets in μ-spaces
  19. Range-Kernel orthogonality and elementary operators on certain Banach spaces
  20. A Cauchy-type generalization of Flett's theorem
  21. A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaces
  22. Robust numerical method for singularly perturbed differential equations with large delay
  23. Special Issue on Equilibrium Problems: Fixed-Point and Best Proximity-Point Approaches
  24. Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces
  25. Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications
  26. Some aspects of generalized Zbăganu and James constant in Banach spaces
  27. An iterative approximation of common solutions of split generalized vector mixed equilibrium problem and some certain optimization problems
  28. Generalized split null point of sum of monotone operators in Hilbert spaces
  29. Comparison of modified ADM and classical finite difference method for some third-order and fifth-order KdV equations
  30. Solving system of linear equations via bicomplex valued metric space
  31. Special Issue on Computational and Theoretical Studies of free Boundary Problems and their Applications
  32. Dynamical study of Lyapunov exponents for Hide’s coupled dynamo model
  33. A statistical study of COVID-19 pandemic in Egypt
  34. Global existence and dynamic structure of solutions for damped wave equation involving the fractional Laplacian
  35. New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal
  36. Some results on generalized finite operators and range kernel orthogonality in Hilbert spaces
  37. Structures of spinors fiber bundles with special relativity of Dirac operator using the Clifford algebra
  38. A new iteration method for the solution of third-order BVP via Green's function
  39. Numerical treatment of the generalized time-fractional Huxley-Burgers’ equation and its stability examination
  40. L ∞-error estimates of a finite element method for Hamilton-Jacobi-Bellman equations with nonlinear source terms with mixed boundary condition
  41. On shrinkage estimators improving the positive part of James-Stein estimator
  42. A revised model for the effect of nanoparticle mass flux on the thermal instability of a nanofluid layer
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  44. An adjusted Grubbs' and generalized extreme studentized deviation
  45. Existence and uniqueness of the weak solution for Keller-Segel model coupled with Boussinesq equations
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  47. Stability analysis of fractional order SEIR model for malaria disease in Khyber Pakhtunkhwa
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