{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:02:11Z","timestamp":1760148131699,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T00:00:00Z","timestamp":1680220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Riyadh, Saudi Arabia","award":["RSPD2023R526"],"award-info":[{"award-number":["RSPD2023R526"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The research described in this paper follows the hypothesis that the monomiality principle leads to novel results that are consistent with past knowledge. Thus, in line with prior facts, our aim is to introduce the \u0394h multi-variate Hermite polynomials \u0394hHm(q1,q2,\u22ef,qr;h). We obtain their recurrence relations by using difference operators. Furthermore, symmetric identities satisfied by these polynomials are established. The operational rules are helpful in demonstrating the novel characteristics of the polynomial families, and thus the operational principles satisfied by these polynomials are derived and will prove beneficial for future observations.<\/jats:p>","DOI":"10.3390\/sym15040839","type":"journal-article","created":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T02:30:20Z","timestamp":1680229820000},"page":"839","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Certain Properties of \u0394h Multi-Variate Hermite Polynomials"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6215-5305","authenticated-orcid":false,"given":"Ibtehal","family":"Alazman","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2864-6888","authenticated-orcid":false,"given":"Badr Saad T.","family":"Alkahtani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh 11989, 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":"Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed University) (SIU), Lavale, Pune 412115, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,31]]},"reference":[{"key":"ref_1","first-page":"93","article-title":"Sur un nouveau d\u00e9velopment en s\u00e9ries de functions","volume":"58","author":"Hermite","year":"1864","journal-title":"Compt. Rend. Acad. Sci. Paris"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1007\/s10473-021-0411-y","article-title":"\u0394h-Gould-Hopper Appell polynomials","volume":"41B","author":"Ozarslan","year":"2021","journal-title":"Acta Math. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s11075-012-9619-1","article-title":"\u0394h-Appell sequences and related interpolation problem","volume":"63","author":"Costabile","year":"2013","journal-title":"Numer. Algorithms"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kim, T. (2019). A Note on the Degenerate Type of Complex Appell Polynomials. Symmetry, 11.","DOI":"10.3390\/sym11111339"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1134\/S1061920818010041","article-title":"Degenerate r-Stirling numbers and r-Bell polynomials","volume":"25","author":"Kim","year":"2018","journal-title":"Russ. J. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Kim, D.S., Kim, T., and Lee, H. (2019). A note on degenerate Euler and Bernoulli polynomials of complex variable. Symmetry, 11.","DOI":"10.3390\/sym11091168"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1007\/s40590-019-00239-1","article-title":"Differential and integral equations for the Laguerre-Gould-Hopper based Appell and related polynomials","volume":"26","author":"Wani","year":"2019","journal-title":"Bolet\u00edn De La Soc. Matem\u00e1tica Mex."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1515\/gmj-2019-2028","article-title":"Fractional calculus and generalized forms of special polynomials associated with Appell sequences","volume":"28","author":"Khan","year":"2021","journal-title":"Georgian Math. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1686","DOI":"10.3906\/mat-1710-55","article-title":"Extended Laguerre-Appell polynomials via fractional operators and their determinant forms","volume":"42","author":"Khan","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4432","DOI":"10.3934\/math.2020283","article-title":"Quasi-monomiality and convergence theorem for Boas-Buck-Sheffer polynomials","volume":"5","author":"Wani","year":"2020","journal-title":"Mathematics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"12680","DOI":"10.3934\/math.2021731","article-title":"A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties","volume":"6","author":"Khan","year":"2021","journal-title":"AIMS Math."},{"key":"ref_12","unstructured":"Jordan, C. (1965). Calculus of Finite Differences, Chelsea Publishing Company."},{"key":"ref_13","first-page":"3631","article-title":"Notes on degenerate tangent polynomials","volume":"11","author":"Ryoo","year":"2015","journal-title":"Glob. J. Pure Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Hwang, K.W., and Ryoo, C.S. (2020). Differential equations associated with two variable degenerate Hermite polynomials. Mathematics, 8.","DOI":"10.3390\/math8020228"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1007\/BF02392231","article-title":"The poweriod, an extension of the mathematical notion of power","volume":"73","author":"Steffensen","year":"1941","journal-title":"Acta. Math."},{"key":"ref_16","first-page":"83","article-title":"Hermite-Bessel and Laguerre-Bessel functions: A by-product of the monomiality principle","volume":"1171","author":"Dattoli","year":"2000","journal-title":"Adv. Spec. Funct. Appl."},{"key":"ref_17","unstructured":"Appell, P., and Kamp\u00e9 de F\u00e9riet, J. (1926). Fonctions Hyperg\u00e9om\u00e9triques et Hypersph\u00e9riques: Polyn\u00d4mes d\u2019Hermite, Gauthier-Villars."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"833","DOI":"10.1016\/j.camwa.2003.09.031","article-title":"Laguerre-type exponentials and generalized Appell polynomials","volume":"48","author":"Bretti","year":"2004","journal-title":"Comput. Math. Appl."},{"key":"ref_19","unstructured":"Andrews, L.C. (1985). Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/S0377-0427(00)00283-1","article-title":"Generalized polynomials operational identities and their applications","volume":"118","author":"Dattoli","year":"2000","journal-title":"J. Comput. Appl. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/S0895-7177(03)00080-3","article-title":"Special polynomials and fractional calculas","volume":"37","author":"Dattoli","year":"2003","journal-title":"Math. Comput. Model."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S0377-0427(99)00111-9","article-title":"Generalized polynomials and associated operational identities","volume":"108","author":"Dattoli","year":"1999","journal-title":"J. Comput. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3245","DOI":"10.1016\/j.jnt.2013.03.004","article-title":"Unified presentation of three families of generalized Apostol-type polynomials based upon the theory of the umbral calculus and the umbral algebra","volume":"13","author":"Dere","year":"2013","journal-title":"J. Number Theory"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/839\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:07:55Z","timestamp":1760123275000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/4\/839"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,31]]},"references-count":23,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["sym15040839"],"URL":"https:\/\/doi.org\/10.3390\/sym15040839","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,3,31]]}}}