{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:32:28Z","timestamp":1760239948630,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,1,29]],"date-time":"2019-01-29T00:00:00Z","timestamp":1548720000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The principal object of this paper is to introduce two variable Shivley\u2019s matrix polynomials and derive their special properties. Generating matrix functions, matrix recurrence relations, summation formula and operational representations for these polynomials are deduced. Finally, Some special cases and consequences of our main results are also considered.<\/jats:p>","DOI":"10.3390\/sym11020151","type":"journal-article","created":{"date-parts":[[2019,1,29]],"date-time":"2019-01-29T11:27:52Z","timestamp":1548761272000},"page":"151","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Two Variables Shivley\u2019s Matrix Polynomials"],"prefix":"10.3390","volume":"11","author":[{"given":"Fuli","family":"He","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ahmed","family":"Bakhet","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Hidan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science for Girls, King Khalid University, Abha 61471, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4686-2143","authenticated-orcid":false,"given":"M.","family":"Abdalla","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science for Girls, King Khalid University, Abha 61471, Saudi Arabia"},{"name":"Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,29]]},"reference":[{"key":"ref_1","unstructured":"Rainville, E.D. (1997). Special Functions, Macmillan."},{"key":"ref_2","unstructured":"Shively, R. (1953). On Pseudo-Laguerre Polynomials. [Ph. D. Thesis, University of Michigan]."},{"key":"ref_3","first-page":"59","article-title":"New matrix formulas for Laguerre matrix polynomials","volume":"3","author":"Altin","year":"2013","journal-title":"J. Class. Anal."},{"key":"ref_4","first-page":"283","article-title":"Laguerre random polynomials: Definition, differential and statistical properties","volume":"98","author":"Company","year":"2015","journal-title":"Util. Math."},{"key":"ref_5","first-page":"37","article-title":"On the Laguerre matrix polynomials","volume":"53","author":"Sastre","year":"1998","journal-title":"Util. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2069","DOI":"10.2298\/FIL1410069K","article-title":"Modified Laguerre matrix polynomials","volume":"28","author":"Kargina","year":"2014","journal-title":"Filomat"},{"key":"ref_7","first-page":"443","article-title":"Some relations on Laguerre matrix polynomials","volume":"9","author":"Shehata","year":"2015","journal-title":"Malays. J. Math. Sci."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Abdalla, M. (2018). Special matrix functions: Characteristics, achievements and future directions. Linear Multilinear Algebra.","DOI":"10.1080\/03081087.2018.1497585"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"235204","DOI":"10.1088\/1751-8113\/43\/23\/235204","article-title":"2-variables Laguerre matrix polynomials and Lie-algebraic techniques","volume":"43","author":"Khan","year":"2010","journal-title":"J. Phys. A"},{"key":"ref_10","first-page":"603","article-title":"Hermite matrix based polynomials of two variables and lie algebraic techniques","volume":"38","author":"Yasmin","year":"2014","journal-title":"Southeast Asian Bull. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"197","DOI":"10.18514\/MMN.2012.356","article-title":"The matrix version for the multivariable Humbert polynomials","volume":"13","year":"2012","journal-title":"Miskolc Math. Notes"},{"key":"ref_12","first-page":"271","article-title":"A note on two variable Laguerre matrix polynomials","volume":"24","year":"2017","journal-title":"J. Associ. Arab Univ. Basic Appl. Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"87","DOI":"10.4236\/alamt.2018.82008","article-title":"Pseudo laguerre matrix polynomials, operational identities and quasi-monomiality","volume":"8","author":"Pathan","year":"2018","journal-title":"Adv. Linear Algebra Matrix Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2059","DOI":"10.2298\/FIL1509059C","article-title":"q-Matrix polynomials in several variables","volume":"29","year":"2015","journal-title":"Filomat"},{"key":"ref_15","first-page":"807","article-title":"A study of a two variables Gegenbauer matrix polynomials and second order matrix partial differential equations","volume":"2","author":"Kahmmash","year":"2008","journal-title":"Int. J. Math. Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1221","DOI":"10.3923\/jas.2008.1221.1227","article-title":"On Hermite matrix polynomials of two variables","volume":"8","author":"Khammash","year":"2008","journal-title":"J. Appl. Sci."},{"key":"ref_17","first-page":"223","article-title":"On a multivariable extension of Laguerre matrix ploynomials","volume":"6","author":"Singh","year":"2018","journal-title":"Electron. J. Math. Anal. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1002\/mma.3020","article-title":"Reverse generalized Bessel matrix differential equation, polynomial solutions, and their properties","volume":"38","author":"Kishka","year":"2015","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_19","first-page":"564287","article-title":"Infinite matrix products and the representation of the gamma matrix function","volume":"2015","author":"Sols","year":"2015","journal-title":"Abstr. Appl. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/S0377-0427(98)00158-7","article-title":"On the hypergeometric matrix function","volume":"99","year":"1998","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","first-page":"81","article-title":"Operational formulae of the multivariable hypergeometric matrix functions and related matrix polynomials","volume":"3","author":"Abdalla","year":"2017","journal-title":"Gen. Lett. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"6385","DOI":"10.1016\/j.amc.2011.12.006","article-title":"A note on Hermite polynomials of several variables","volume":"218","author":"Khan","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1080\/10652469.2018.1543669","article-title":"On the Wright hypergeometric matrix functions and their fractional calculus","volume":"30","author":"Bakhet","year":"2019","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/S0377-0427(98)00149-6","article-title":"Some applications of the Hermite matrix polynomials series expansions","volume":"99","author":"Defez","year":"1998","journal-title":"J. Comput. Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1007\/s40995-017-0183-3","article-title":"Extended Gauss hypergeometric matrix functions","volume":"42","author":"Abdalla","year":"2018","journal-title":"Iran. J. Sci. Technol. Trans. Sci."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/151\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:29:28Z","timestamp":1760185768000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/2\/151"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,29]]},"references-count":25,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["sym11020151"],"URL":"https:\/\/doi.org\/10.3390\/sym11020151","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,1,29]]}}}