A note on two variable Laguerre matrix polynomials

Peer review under responsibility of University of Bahrain.

Abstract

The principal object of this paper is to present a natural further step toward the mathematical properties and presentations concerning the two variable Laguerre matrix polynomials defined in (Bin-Saad, Maged G., Antar, A. Al-Sayaad, 2015. Study of two variable Laguerre polynomials via symbolic operational images. Asian J. of math. and comput. research, 2(1), 42–50). Series expansions, integral transforms and bilinear and bilateral generating matrix functions for these polynomials are established. Some particular cases and consequences of our main results are also considered.

Keywords:

• Laguerre matrix polynomials
• Operational images
• Functional relations
• Differential matrix equations

1 Introduction

The subject of Laguerre polynomials has gained importance during the last two decades mainly due to its applications in various fields of mathematical physics, such as the solving of delay differential equations (Suayip et al., 2014), pantograph-type Volterra integro-differential equations (Suayip, 2014) and fractional differential equations (Bhrawy and Alghamdi, 2012; Bhrawy et al., 2014a,b; Bhrawy et al., 2015b,c). Numerous other works dealing also with the use of Laguerre polynomials and matrices include those by Ahmadian et al. (2015) in the theory of Tau method for numerical solution of fuzzy fractional kinetic model, by Bhrawy and Taha (2012) and Bhrawy et al. (2015a) in the theory of operational matrix of fractional integration of Laguerre polynomials and generalized Laguerre–Gauss–Radau schema for first order hyperbolic equation on semi-finite domain (Abdelkawy and Taha, 2012) and so on (see also Bin-Saad and Antar (2015)). Further, matrix polynomials seen in the study of many area such as statistics, Lie group theory and number theory are well known. Recently, the matrix versions of the classical families orthogonal polynomials such as Laguerre, Jacobi, Hermite, Gegenbauer, Bessel and Humbert polynomials and some other polynomials were introduced by many authors for matrices in and various properties satisfied by them were given from the scalar case, see for example (Aktas et al., 2013; Aktas et al., 2011; Altin and Cekim, 2012a,b, 2013; Bin-Saad and Antar, 2015; Cekim and Erkus-Duman, 2014; Jódar and Cortés, 1998a,b; Jódar and Company, 1996; Jódar et al., 1995; Pathan et al., 2014; Bayram and Altin, 2015).

If , are elements of and , then we calla matrix polynomial of degree n in x. If is invertible for every integer then.(1.1)

In Bin-Saad and Antar (2015), it is shown that an appropriate combination of methods, relevant to operational calculus and to matrix polynomials, can be a very useful tool to establish and treat a new class of two variable Laguerre matrix polynomials in the following form(1.2) where A be a matrix in where is not an eigenvalue of A for every integer and be a complex number whose real part is positive. The authors in Bin-Saad and Antar (2015) explored the formal properties of the operational identities to derive a number of properties of the new class two variable Laguerre matrix polynomials (1.2)(1.2) and discussed the links with various known polynomials. The generating relation for the matrix function , is given by the following formula (Bin-Saad and Antar, 2015):(1.3) where and .

By setting in (1.2)(1.2) , Eq. (1.2)(1.2) immediately yields the following Laguerre matrix polynomials due to Jódar and Cortés (1998a,b):(1.4)

For the purpose of this work, we recall here same definitions.

Definition 1.1

Let A be a positive stable matrix in , then Gamma matrix function is defined by (Jódar and Cortés, 1998a,b)(1.5)

Definition 1.2

Let A, B and be a positive stable matrix in and , then Beta matrix function is defined by (Jódar and Cortés, 1998a,b)(1.6)

Lemma 1.1

For matrix in where , we have (see Srivastava and Manocha (1984)):(1.7) and(1.8)

Motivated by the important role of the Laguerre matrix polynomials in several diverse fields of physics and the contributions in Bayram and Altin (2013) and Jódar et al. (1994) toward the generalization of the Laguerre polynomials, this work aims at investigating several properties for the two variable Laguerre matrix polynomials . We establish some projection series, integral transforms and bilinear and bilateral generating matrix functions. Many earlier (known) results given by Bayram and Altin (2013) are shown to be special cases of our results.

2 Finite and infinite sums

Theorem 2.1

Let A and B be matrices in satisfying spectral condition is not an eigenvalue of A for every integer is not an eigenvalue of B for every integer and . Then(2.1)

Proof

Using (1.8)(1.8) , we can writewhich on using the multinomial formulagives us(2.2) Now, employing (1.3)(1.3) in (2.2)(2.2) and comparing the coefficients of in the resulting expression, we get the desired result. □

Theorem 2.2

Let A be matrix in satisfying spectral condition is not an eigenvalue of A for every integer . Then(2.3)

Proof

According to formula (1.8)(1.8) , we can writewhere , which in view of (1.3)(1.3) , yields the result(2.4) Now, comparing the coefficients of in (2.4)(2.4) , we get the desired result. □

Theorem 2.3

Let A and C be matrices in satisfying spectral condition is not an eigenvalue of A for every integer and . Then(2.5)

Proof

In view of the identitywe can writewhich on employing the Taylor expressionand then using (1.8)(1.8) and (1.3)(1.3) the theorem can be proved. □

Theorem 2.4

Let A be matrix in satisfying spectral condition is not an eigenvalue of A for every integer and . Then(2.6)

Proof

We have(2.7) Now, comparing the coefficients of in (2.7)(2.7) , we get the desired result. □

It is important to note that Eqs. (2.1)(2.1) , (2.3)(2.3) , (2.5)(2.5) and (2.6)(2.6) provide a generalization of known results in Bayram and Altin (2013) and Eqs. (2.1)(2.1) , (2.4)(2.4) , (2.6)(2.6) and (2.7)(2.7) .

3 Integral transforms

In many situations an integral transform of Laguerre function is more convenient to use than its series representation. First of all, we establish an integral transform for involving double series.

Theorem 3.1

Let A be matrix in satisfying spectral condition is not an eigenvalue of A for every integer and . Then(3.1) where is Appllś function (Srivastava and Karlsson, 1985).

Proof

Denote, for convenience, the left-hand side of Eq. (3.1)(3.1) by I,thenwhich according to the definition of Appell’s series in two variables , yields the required result (3.1)(3.1) . □

A special case of the transformation formula (3.1)(3.1) is worthy of note. Indeed, upon letting , (3.1)(3.1) readily yields(3.2)

Theorem 3.2

Let A be matrix in satisfying spectral condition is not an eigenvalue of A for every integer and . Then(3.3) where is Appllś function (see Srivastava and Karlsson (1985)).

Proof

Denote, for convenience, the left-hand side of Eq. (3.1)(3.1) by I,thenwhich according to the definition of Appell’s series in two variables , yields (3.2)(3.2) . □

Theorem 3.3

Let A be matrix in satisfying spectral condition is not an eigenvalue of A for every integer and . Then(3.4)

Proof

Denote, for convenience, the left-hand side of Eq. (3.1)(3.1) by I,thenputting , we getwhich on applying the definition of Beta matrix function (1.6)(1.6) and considering (1.2)(1.2) , we get (3.4)(3.4) .

4 Bilinear and bilateral generating matrix functions

We aim here at presenting a family of bilinear and bilateral generating matrix functions involving multiple series with essentially arbitrary coefficients for the two variable Laguerre matrix polynomials which are generated by (1.3)(1.3) and given explicitly by (1.2)(1.2) . We begin by stating the following theorem.

Theorem 4.1

Corresponding to an identically non-vanishing function of r complex variables , of complex order and i complex variables of complex order , let(4.1) where ; and(4.2) where is a matrix in whose eigenvalues is not eigenvalue of A for every integer . Then(4.3) provided that each member of (4.3)(4.3) exists for and .

Proof

Denote, for convenience, the right-hand side of Eq. (4.3)(4.3) by I. Then from (4.2)(4.2) , we getwhich on letting , we can writewhich completes the proof of Theorem 4.1. □

By appropriate choices of the generating functions in (4.1)(4.1) , we can derive a number of generating functions involving the products of such polynomials as the familiar Laguerre, Gegenbauer, Jacobi and Hermite matrix polynomials.

First of all, in its special case when and using the generating relation (1.3)(1.3) , the assertion (4.3)(4.3) would obviously yield the following bilinear generating function:(4.4)

Secondly, consider the generating function (Ghazi, 2008):(4.5) where is Gegenbauer matrix polynomials of two variables, upon setting andin (4.3)(4.3) , we shall obtain a bilateral generating function in the following form:(4.6)

Next, consider the double generating function of single polynomials (Pathan and Bin-Saad, 1999):(4.7)

On letting , takingand combing (4.7)(4.7) with (4.1)(4.1) , we get(4.8)

Similarly, on considering the generating function [1].(4.9)

we can show that(4.10)

Finally, we recall here a double generating relation for hypergeometric functions of several variables (see Pathan and Bin-Saad (1999))(4.11) where is Lauricella’s function of n-variables (Srivastava and Karlsson, 1985).

Upon setting , lettingtakingin (4.1)(4.1) and proceeding in the manner described above it is not difficult to obtain the following multi-variable bilateral generating relation:(4.12)

5 Conclusion

In this paper, integral operators and series rearrangement technique has been applied to obtain finite and infinite sums, integral transforms and bilinear and bilateral generating matrix functions for Laguerre matrix polynomials of two variables. Also, some interested particular cases and consequences of our results have been discussed.

Conflict of interest

The authors declare that they have no competing interests.

Notes

Peer review under responsibility of University of Bahrain.