Matrix-Valued hypergeometric Appell-Type polynomials

Muajebah Hidan; Ahmed Bakhet; Hala Abd-Elmageed; Mohamed Abdalla; Muajebah Hidan; Ahmed Bakhet; Hala Abd-Elmageed; Mohamed Abdalla

doi:10.3934/era.2022150

Electronic Research Archive

2022, Volume 30, Issue 8: 2964-2980. doi: 10.3934/era.2022150

Previous Article Next Article

Research article Special Issues

Matrix-Valued hypergeometric Appell-Type polynomials

1.
Department of Mathematics, College of Science, King Khalid University, Abha 61471, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt
3.
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt

Academic Editor: Arran Fernandez

Received: 03 March 2022 Revised: 18 April 2022 Accepted: 10 May 2022 Published: 31 May 2022

In recent years, much attention has been paid to the role of special matrix polynomials of a real or complex variable in mathematical physics, especially in boundary value problems. In this article, we define a new type of matrix-valued polynomials, called the first Appell matrix polynomial of two complex variables. The properties of the newly definite matrix polynomial involving, generating matrix functions, recurrence relations, Rodrigues' type formula and integral representation are investigated. Further, relevant connections between the first Appell matrix polynomial and various matrix functions are reported. The current study may open the door for further investigations concerning the practical applications of matrix polynomials associated with a system of differential equations.
- matrix-valued polynomials,
- first Appell matrix functions,
- generating matrix relations,
- functional relations
Citation: Muajebah Hidan, Ahmed Bakhet, Hala Abd-Elmageed, Mohamed Abdalla. Matrix-Valued hypergeometric Appell-Type polynomials[J]. Electronic Research Archive, 2022, 30(8): 2964-2980. doi: 10.3934/era.2022150

Related Papers:

Abstract

In recent years, much attention has been paid to the role of special matrix polynomials of a real or complex variable in mathematical physics, especially in boundary value problems. In this article, we define a new type of matrix-valued polynomials, called the first Appell matrix polynomial of two complex variables. The properties of the newly definite matrix polynomial involving, generating matrix functions, recurrence relations, Rodrigues' type formula and integral representation are investigated. Further, relevant connections between the first Appell matrix polynomial and various matrix functions are reported. The current study may open the door for further investigations concerning the practical applications of matrix polynomials associated with a system of differential equations.

References

[1]	P. Appell, Sur une classe de polynômes, Ann. Sci. École. Norm. Sup, 9 (1880), 119–144. https://doi.org/10.24033/asens.186
[2]	M. Anshelevivh, Appell polynomials and their relatives, Int. Math. Res. Notices, 12 (2004). https://doi.org/10.1155/S107379280413345X
[3]	L. Bedratyuk, N. Luno, Some properties of generalized hypergeomtric Appell polynomials, Carpathian Math. Publ., 12 (2020), 129–137. https://doi.org/10.15330/cmp.12.1.129-137 doi: 10.15330/cmp.12.1.129-137
[4]	S. D. Bajpai, M. S. Arora, Semi-orthogonality of a class of the Gauss' hypergeometric polynomials, Ann. Math. Blaise Pascal., 1 (1994), 75–83. https://doi.org/10.5802/ambp.6 doi: 10.5802/ambp.6
[5]	S. D. Bajpai, Generating function and orthogonality property of a class of polynomials occurring in quantum mechanics, Ann. Math. Blaise Pascal., 1 (1994), 21–26. https://doi.org/10.5802/ambp.2 doi: 10.5802/ambp.2
[6]	I. K. Khanna, V. Srinivasa Bhagavan, Lie group theorietic origins of certain generating functions of the generalized hypergeometric polynomials, Integr. Transforms Spec. Funct., 11 (2001), 177–188. https://doi.org/10.1080/10652460108819309 doi: 10.1080/10652460108819309
[7]	L. Bedratyuk, N. Luno, Some properties of generalized hypergeomtric Appell polynomials, Carpathian Math. Publ., 12 (2020), 129–137. https://doi.org/10.15330/cmp.12.1.129-137 doi: 10.15330/cmp.12.1.129-137
[8]	I. A. Khan, On a generalized hypergeometric polynomial, in International Centre for Theoretical Physics, (1993), 1–5.
[9]	L. N. Djordjevic, D. M. Milosevic, G. V. Milovanovic, H. M. Srivastava, Some finite summation formulas involving multivariable hypergeometric polynomials, Integr. Transforms Spec. Funct., 14 (2003), 349–361. https://doi.org/10.1080/1065246031000081643 doi: 10.1080/1065246031000081643
[10]	O. Bihun, F. Calogero, Properties of the zeros of generalized hypergeometric polynomials, J. Math. Anal. Appl., 419 (2014), 1076–1094. https://doi.org/10.1016/j.jmaa.2014.05.023 doi: 10.1016/j.jmaa.2014.05.023
[11]	C. Bracciali, J. Moreno-Balczar, On the zeros of a class of generalized hypergeometric polynomials, Appl. Math. Comput., 253 (2015), 151–158. https://doi.org/10.1016/j.amc.2014.12.083 doi: 10.1016/j.amc.2014.12.083
[12]	H. Gould, A. Hopper, Operational formulas connected with two generalizations of Hermite polynomials, Duke Math. J., 29 (1962), 51–63. https://doi.org/10.1215/S0012-7094-62-02907-1 doi: 10.1215/S0012-7094-62-02907-1
[13]	B. Çekim, R. Aktaş, Multivariable matrix generalization of Gould-Hopper polynomials, Miskolc Math. Notes, 16 (2015), 79–89. https://doi.org/10.18514/MMN.2015.1112 doi: 10.18514/MMN.2015.1112
[14]	T. Nahid, S. Khan, Construction of some hybrid relatives of Laguerre-Appell polynomials associated with Gould-Hopper matrix polynomials, J. Anal., 29 (2021), 927–946. https://doi.org/10.1007/s41478-020-00288-0 doi: 10.1007/s41478-020-00288-0
[15]	T. Nahid, S. Khan, Differential equations for certain hybrid special matrix polynomials, Bol. Soc. Paran. Mat., (2022), 1–10.
[16]	E. Defez, J. Ibáñez, P. Alonso-Jordá, José M. Alonso, J. Peinado, On Bernoulli matrix polynomials and matrix exponential approximation, J. Comput. Appl. Math., 404 (2022), 113207. https://doi.org/10.1016/j.cam.2020.113207 doi: 10.1016/j.cam.2020.113207
[17]	L. Rodman, Orthogonal matrix polynomials, in NATO ASI Series, Kluwer Academic Publishers, (1990), 345–362. https://doi.org/10.1007/978-94-009-0501-6_16
[18]	R. Horn, C. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991. https://doi.org/10.1017/CBO9780511840371
[19]	A. J. Duran, F. A. Grunbaum, A survey on orthogonal matrix polynomials satisfying second-order differential equations, J. Comput. Appl. Math., 178 (2005), 169–190. https://doi.org/10.1016/j.cam.2004.05.023 doi: 10.1016/j.cam.2004.05.023
[20]	E. Koelink, A. de Ríos, P. Román, Matrix-valued Gegenbauer-type polynomials, Constr. Approx., 46 (2017), 459–487. https://doi.org/10.1007/s00365-017-9384-4 doi: 10.1007/s00365-017-9384-4
[21]	M. Ismail, E. Koelink, P. Roman, Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice, Adv. Appl. Math., 110 (2019), 235–269. https://doi.org/10.1016/j.aam.2019.07.002 doi: 10.1016/j.aam.2019.07.002
[22]	M. Abdalla, Special matrix functions: characteristics, achievements and future directions, Linear Multilinear Algebra, 68 (2020), 1–28. https://doi.org/10.1080/03081087.2018.1497585 doi: 10.1080/03081087.2018.1497585
[23]	H. Abd-Elmageed, M. Abdalla, M. Abul-Ez, N. Saad, Some results on the first Appell matrix function $F_{1}(A; B, B';C; z, w)$, Linear Multilinear Algebra, 68 (2020), 278–292. https://doi.org/10.1080/03081087.2018.1502254 doi: 10.1080/03081087.2018.1502254
[24]	A. Altin, B. Çekim, R. Şahin, On the matrix versions of Appell hypergeometric functions, Quaest. Math., 37 (2014), 31–38. https://doi.org/10.2989/16073606.2013.779955 doi: 10.2989/16073606.2013.779955
[25]	M. Abdalla, H. Abd-Elmageed, M. Abul-Ez, M. Zayed, Further investigations on the two variables second Appell hypergeometric matrix function, Quaest. Math., (2022), 118. https://doi.org/10.2989/16073606.2022.2034680
[26]	L. Jódar, J. C. Cortés, On the hypergeometric matrix function, J. Comp. Appl. Math., 99 (1998), 205–217. https://doi.org/10.1016/S0377-0427(98)00158-7 doi: 10.1016/S0377-0427(98)00158-7
[27]	S. Z. Rida, M. Abul-Dahab, M. A. Saleem, M. T. Mohammed, On Humbert matrix function $\Psi_{1}(A, B;C, C';z, w)$ of two complex variables under differential operator, Int. J. Indus. Math. 32 (2010), 167–179.
[28]	H. L. Manocha, On a polynomial of the form $F_{1}$, Riv. Mat. Univ. Parma., 2 (1967), 143–148.
[29]	M. Abdalla, M. Hidan, Analytical properties of the two variables Jacobi matrix polynomials with applications, Demonstr. Math., (2021), 178–188. https://doi.org/10.1515/dema-2021-0021
[30]	S. Khan, N. Raza, 2-variable generalized Hermite matrix polynomials and Lie algebra representation, Rep. Math. Phys., 66 (2010), 159–174. https://doi.org/10.1016/S0034-4877(10)00024-8 doi: 10.1016/S0034-4877(10)00024-8
[31]	F. He, A. Bakhet, M. Hidan, M. Abdalla, Two variables Shivleys matrix polynomials, Symmetry, 11 (2019), 151. https://doi.org/10.3390/sym11020151 doi: 10.3390/sym11020151

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)