{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T22:14:34Z","timestamp":1777068874254,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,10,23]],"date-time":"2024-10-23T00:00:00Z","timestamp":1729641600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Ha\u2019il","award":["RG-23 206"],"award-info":[{"award-number":["RG-23 206"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper offers a thorough examination of a unified class of Humbert\u2019s polynomials in two variables, extending beyond well-known polynomial families such as Gegenbauer, Humbert, Legendre, Chebyshev, Pincherle, Horadam, Kinnsy, Horadam\u2013Pethe, Djordjevi\u0107, Gould, Milovanovi\u0107, Djordjevi\u0107, Pathan, and Khan polynomials. This study\u2019s motivation stems from exploring polynomials that lack traditional nomenclature. This work presents various expansions for Humbert\u2013Hermite polynomials, including those involving Hermite\u2013Gegenbauer (or ultraspherical) polynomials and Hermite\u2013Chebyshev polynomials. The proofs enhanced our understanding of the properties and interrelationships within this extended class of polynomials, offering valuable insights into their mathematical structure. This research consolidates existing knowledge while expanding the scope of Humbert\u2019s polynomials, laying the groundwork for further investigation and applications in diverse mathematical fields.<\/jats:p>","DOI":"10.3390\/sym16111415","type":"journal-article","created":{"date-parts":[[2024,10,23]],"date-time":"2024-10-23T12:04:22Z","timestamp":1729685062000},"page":"1415","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On a Class of Generalized Multivariate Hermite\u2013Humbert Polynomials via Generalized Fibonacci Polynomials"],"prefix":"10.3390","volume":"16","author":[{"given":"Noor","family":"Alam","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6484-469X","authenticated-orcid":false,"given":"Shahid Ahmad","family":"Wani","sequence":"additional","affiliation":[{"name":"Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune 412115, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-9885","authenticated-orcid":false,"given":"Waseem Ahmad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Prince Mohammad Bin Fahd University, Al Khobar 31952, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2653-3780","authenticated-orcid":false,"given":"Ketan","family":"Kotecha","sequence":"additional","affiliation":[{"name":"Symbiosis Institute of Technology, Pune Campus, Symbiosis International (Deemed University), Pune 412115, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hasan Nihal","family":"Zaidi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fakhredine","family":"Gassem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anas","family":"Altaleb","sequence":"additional","affiliation":[{"name":"Department of Statistics, College of Business Administration, University of Ha\u2019il, Ha\u2019il 2440, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1215\/S0012-7094-65-03275-8","article-title":"Inverse series relation and other expansions involving Humbert polynomials","volume":"32","author":"Gould","year":"1965","journal-title":"Duke Math. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1215\/S0012-7094-62-02907-1","article-title":"Operational formulas connected with two generalizations of Hermite polynomials","volume":"29","author":"Gould","year":"1962","journal-title":"Duke Math. J."},{"key":"ref_3","first-page":"31","article-title":"On certain generalized polynomial system associated with Humbert polynomials","volume":"23","author":"Agarwal","year":"2012","journal-title":"Sci. Ser. A Math. Sci. (N. S.)"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1080\/00150517.1987.12429670","article-title":"A generalization of Fibonacci polynomials and a representation of Gagenbauer polynomials of integer order","volume":"25","author":"Dilcher","year":"1987","journal-title":"Fibonacci Quart."},{"key":"ref_5","unstructured":"Djorjevi\u0107, G.B. (2024). A generalization of Gegenbauer polynomial with two variables. Indian J. Pure Appl. Math., in press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"279","DOI":"10.2298\/FIL0903279D","article-title":"Polynomials related to generalized Chebyshev polynomials","volume":"23","year":"2009","journal-title":"Filomat"},{"key":"ref_7","first-page":"106","article-title":"Convolutions of the generalized Morgan-Voyce polynomials","volume":"259","author":"Djordjevic","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1016\/j.mcm.2004.10.026","article-title":"Incomplete generalized Jacobsthal and Jacobsthal-Lucas numbers","volume":"42","author":"Srivastava","year":"2005","journal-title":"Math. Comput. Model."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Guan, H., Khan, W.A., and Kizilates, C. (2023). On generalized bivariate (p,q)-Bernoulli-Fibonacci and generalized bivariate (p,q)-Bernoulli-Lucas polynomials. Symmetry, 15.","DOI":"10.3390\/sym15040943"},{"key":"ref_10","unstructured":"Andrews, L.C. (1985). Special Functions for Engineers and Mathematicians, Macmillan Co."},{"key":"ref_11","unstructured":"Abramowitz, M., and Stegun, I.A. (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc.. Reprint of the 1972 Edition."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"258","DOI":"10.2307\/1968431","article-title":"Exponential polynomials","volume":"35","author":"Bell","year":"1934","journal-title":"Ann. Math."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Comtet, L. (1974). Advanced Combinatorics, D. Reidel Publishing Co.","DOI":"10.1007\/978-94-010-2196-8"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Zhang, C., Khan, W.A., and Kizilates, C. (2023). On (p,q)-Fibonacci and (p,q)-Lucas polynomials associated with Changhee numbers with their applications. Symmetry, 15.","DOI":"10.3390\/sym15040851"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"709386","DOI":"10.1155\/2009\/709386","article-title":"On sequences of numbers and polynomials defined by linear recurrence relations of order 2","volume":"2009","author":"He","year":"2009","journal-title":"Int. J. Math. Math. Sci."},{"key":"ref_16","first-page":"268096","article-title":"Sequences of non-Gegenbauer-Humbert polynomials meet the generalized Gegenebauer-Humbert polynomials","volume":"2011","author":"He","year":"2011","journal-title":"Int. Sch. Res. Netw. ISRN Algebra"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.30755\/NSJOM.05832","article-title":"On a class of Humbert-Hermite polynomials","volume":"51","author":"Khan","year":"2021","journal-title":"Novi Sad J. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"920","DOI":"10.1016\/j.dam.2008.03.034","article-title":"A generalization of Lucas polynomials sequence","volume":"157","author":"Cheon","year":"2009","journal-title":"Discret. Appl. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"264842","DOI":"10.1155\/2012\/264842","article-title":"Some properties of the (p,q)-Fibonacci and (p,q)-Lucas polynomials","volume":"2012","author":"Lee","year":"2012","journal-title":"J. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"929","DOI":"10.55730\/1300-0098.3133","article-title":"On a class of generalized Humberts-Hermite polynomials via generalized Fibonacci polynomials","volume":"46","author":"Pathan","year":"2022","journal-title":"Turk. J. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1186\/1687-1847-2014-64","article-title":"Some new results for the (p,q)-Fibonacci and Lucas polynomials","volume":"2014","author":"Wang","year":"2014","journal-title":"Adv. Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1080\/00150517.1987.12429681","article-title":"On some properties of Humbert\u2019s polynomials","volume":"25","author":"Djordjevixcx","year":"1987","journal-title":"Fibonacci Quart."},{"key":"ref_23","first-page":"42","article-title":"Another generalization of Gegenbauer polynomials","volume":"2","author":"Dave","year":"1978","journal-title":"J. Indian Acad. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1080\/00150517.2002.12428637","article-title":"Formulas for convolution Fibonacci numbers and polynomials","volume":"40","author":"Liu","year":"2002","journal-title":"Fibonacci Quart."},{"key":"ref_25","first-page":"9297","article-title":"Identities involving generalized Fibonacci-type polynomials","volume":"217","author":"Ma","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_26","first-page":"208","article-title":"On convolved generalized Fibonacci and Lucas polynomials","volume":"229","year":"2014","journal-title":"Appl. Math. Comput."},{"key":"ref_27","first-page":"759641","article-title":"Some properties of convolved k-Fibonacci numbers","volume":"2013","year":"2013","journal-title":"ISRN Combin."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3179","DOI":"10.1016\/j.chaos.2009.04.048","article-title":"On generalized Fibonacci and Lucas polynomials","volume":"42","author":"Nalli","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","first-page":"439","article-title":"On a polynomial associated with Gegenbauer polynomials","volume":"59","author":"Sinha","year":"1989","journal-title":"Proc. Nat. Acad. Sci. India"},{"key":"ref_30","first-page":"1","article-title":"Polynomial associated with Legendre polynomials","volume":"2","author":"Shrestha","year":"1977","journal-title":"Nepali Math. Sci. Rep. Triv. Univ."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1080\/10652469.2015.1007502","article-title":"Some results on convolved (p,q)-Fibonacci polynomials","volume":"26","author":"Wang","year":"2015","journal-title":"Integral Transform. Spec. Funct."},{"key":"ref_32","first-page":"204","article-title":"Generalized Humbert polynomials via generalized Fibonacci polynomials","volume":"307","author":"Wang","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0969-806X(99)00346-1","article-title":"Exponential operators, quasi-monomials and generalized polynomials","volume":"57","author":"Dattoli","year":"2000","journal-title":"Radiat. Phys. Chem."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1006\/jmaa.1996.0399","article-title":"Generalized Hermite polynomials and super-Gaussian forms","volume":"233","author":"Dattoli","year":"1996","journal-title":"J. Math. Anal. Appl."},{"key":"ref_35","first-page":"47","article-title":"Hermite-Chebyshev polynomials with their generalization form","volume":"29","author":"Batahan","year":"2014","journal-title":"J. Math. Sci. Adv. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1415\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:18:59Z","timestamp":1760113139000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1415"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,23]]},"references-count":35,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["sym16111415"],"URL":"https:\/\/doi.org\/10.3390\/sym16111415","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,23]]}}}