{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:06:56Z","timestamp":1760242016161,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2018,12,16]],"date-time":"2018-12-16T00:00:00Z","timestamp":1544918400000},"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>Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Ces\u00e0ro polynomials. Using these, we also obtain q-analogues for some of their generating functions. Our q-generating functions are given in terms of the basic hypergeometric series       4   \u03d5 5     ,       5   \u03d5 5     ,       4   \u03d5 3     ,       3   \u03d5 2     ,       2   \u03d5 1     , and q-Pochhammer symbols. Starting with our q-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials.<\/jats:p>","DOI":"10.3390\/sym10120758","type":"journal-article","created":{"date-parts":[[2018,12,18]],"date-time":"2018-12-18T02:15:59Z","timestamp":1545099359000},"page":"758","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some Generating Functions for q-Polynomials"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9398-455X","authenticated-orcid":false,"given":"Howard S.","family":"Cohl","sequence":"first","affiliation":[{"name":"Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9545-7411","authenticated-orcid":false,"given":"Roberto S.","family":"Costas-Santos","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica y Matem\u00e1ticas, Facultad de Ciencias, Universidad de Alcal\u00e1, Alcal\u00e1 de Henares, 28871 Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2397-4185","authenticated-orcid":false,"given":"Tanay V.","family":"Wakhare","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Maryland, College Park, MD 20742, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Koekoek, R., Lesky, P.A., and Swarttouw, R.F. (2010). Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer-Verlag. Springer Monographs in Mathematics; With a foreword by Tom H. Koornwinder.","DOI":"10.1007\/978-3-642-05014-5"},{"key":"ref_2","unstructured":"Gasper, G., and Rahman, M. (2004). Basic Hypergeometric Series, Cambridge University Press. [2nd ed.]. Encyclopedia of Mathematics and its Applications; With a foreword by Richard Askey."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1215\/S0012-7094-36-00248-X","article-title":"Two systems of polynomials for the solution of Laplace\u2019s integral equation","volume":"2","author":"Bateman","year":"1936","journal-title":"Duke Math. J."},{"key":"ref_4","first-page":"24","article-title":"Sur la valeur moyenne des coefficients dans le d\u00e9veloppement d\u2019un d\u00e9terminant gauche ou sym\u00e9trique d\u2019un ordre infiniment grand et sur les d\u00e9terminants doublement gauches","volume":"89","author":"Sylvester","year":"1879","journal-title":"C. R. de l\u2019Acad\u00e9mie des Sci."},{"key":"ref_5","unstructured":"Erd\u00e9lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1981). Higher Transcendental Functions, Robert E. Krieger Publishing Co. Inc."},{"key":"ref_6","unstructured":"Srivastava, H.M., and Manocha, H.L. (1984). A Treatise on Generating Functions, Ellis Horwood Ltd."},{"key":"ref_7","unstructured":"Rainville, E.D. (1960). Special Functions, The Macmillan Co."},{"key":"ref_8","first-page":"23","article-title":"Some Properties of a certain Set of Polynomials","volume":"37","author":"Bateman","year":"1933","journal-title":"Tohoku Math. J. First Ser."},{"key":"ref_9","first-page":"209","article-title":"A generalization of the polynomial Fn(x)","volume":"28","author":"Pasternack","year":"1939","journal-title":"Lond. Edinb. Dublin Philos. Mag. J. Sci. Ser. 7"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1090\/S0002-9939-96-03190-5","article-title":"On Jacobi and continuous Hahn polynomials","volume":"124","author":"Koelink","year":"1996","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_11","first-page":"395","article-title":"On some new generating functions","volume":"4","author":"Agarwal","year":"1980","journal-title":"Matematicki Vesnik"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/BF02413168","article-title":"The Rodrigues Type Representations for a Certain Class of Special Functions","volume":"119","author":"Srivastava","year":"1979","journal-title":"Annali di Matematica Pura ed Applicata"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1867","DOI":"10.1007\/s11425-014-4821-3","article-title":"q-Bernoulli polynomials and q-umbral calculus","volume":"57","author":"Kim","year":"2014","journal-title":"Sci. China Math."},{"key":"ref_14","unstructured":"Asif, M. (2010). On Some Problems in Special Functions. [Ph.D. Thesis, Aligarh Muslim University]."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/12\/758\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:34:21Z","timestamp":1760196861000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/12\/758"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,16]]},"references-count":14,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2018,12]]}},"alternative-id":["sym10120758"],"URL":"https:\/\/doi.org\/10.3390\/sym10120758","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,12,16]]}}}