Some Generating Functions for q-Polynomials
Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro 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 hyperge...
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| Veröffentlicht in: | Symmetry (Basel) Jg. 10; H. 12; S. 758 |
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| Abstract | Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro 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 ϕ 5 , 5 ϕ 5 , 4 ϕ 3 , 3 ϕ 2 , 2 ϕ 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. |
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| AbstractList | Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro 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 4ϕ5, 5ϕ5, 4ϕ3, 3ϕ2, 2ϕ1, 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. Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro 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 ϕ 5 , 5 ϕ 5 , 4 ϕ 3 , 3 ϕ 2 , 2 ϕ 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. Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro 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 ϕ 5, 5 ϕ 5, 4 ϕ 3, 3 ϕ 2, 2 ϕ 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.Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro 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 ϕ 5, 5 ϕ 5, 4 ϕ 3, 3 ϕ 2, 2 ϕ 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. Demonstrating the striking symmetry between calculus and -calculus, we obtain -analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain -analogues for some of their generating functions. Our -generating functions are given in terms of the basic hypergeometric series , , , , , and -Pochhammer symbols. Starting with our -generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials. |
| Author | Cohl, Howard S. Costas-Santos, Roberto S. Wakhare, Tanay V. |
| AuthorAffiliation | 2 Departamento de Física y Matemáticas, Facultad de Ciencias, Universidad de Alcalá, Alcalá de Henares, 28871 Madrid, Spain 3 Department of Mathematics, University of Maryland, College Park, MD 20742, USA 1 Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA |
| AuthorAffiliation_xml | – name: 2 Departamento de Física y Matemáticas, Facultad de Ciencias, Universidad de Alcalá, Alcalá de Henares, 28871 Madrid, Spain – name: 1 Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA 92694, USA – name: 3 Department of Mathematics, University of Maryland, College Park, MD 20742, USA |
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| Copyright | 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
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| Keywords | q-polynomials generating functions 33D45 basic hypergeometric functions 33C20 |
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| References | Pasternack (ref_9) 1939; 28 Koelink (ref_10) 1996; 124 Kim (ref_13) 2014; 57 ref_14 Bateman (ref_8) 1933; 37 ref_1 ref_2 Srivastava (ref_12) 1979; 119 ref_5 Sylvester (ref_4) 1879; 89 Bateman (ref_3) 1936; 2 ref_7 ref_6 Agarwal (ref_11) 1980; 4 |
| References_xml | – ident: ref_7 – ident: ref_6 – volume: 57 start-page: 1867 year: 2014 ident: ref_13 article-title: q-Bernoulli polynomials and q-umbral calculus publication-title: Sci. China Math. doi: 10.1007/s11425-014-4821-3 – ident: ref_5 – ident: ref_2 – volume: 37 start-page: 23 year: 1933 ident: ref_8 article-title: Some Properties of a certain Set of Polynomials publication-title: Tohoku Math. J. First Ser. – volume: 4 start-page: 395 year: 1980 ident: ref_11 article-title: On some new generating functions publication-title: Matematicki Vesnik – volume: 119 start-page: 9 year: 1979 ident: ref_12 article-title: The Rodrigues Type Representations for a Certain Class of Special Functions publication-title: Annali di Matematica Pura ed Applicata doi: 10.1007/BF02413168 – ident: ref_14 – ident: ref_1 doi: 10.1007/978-3-642-05014-5 – volume: 2 start-page: 569 year: 1936 ident: ref_3 article-title: Two systems of polynomials for the solution of Laplace’s integral equation publication-title: Duke Math. J. doi: 10.1215/S0012-7094-36-00248-X – volume: 124 start-page: 887 year: 1996 ident: ref_10 article-title: On Jacobi and continuous Hahn polynomials publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-96-03190-5 – volume: 89 start-page: 24 year: 1879 ident: ref_4 article-title: Sur la valeur moyenne des coefficients dans le développement d’un déterminant gauche ou symétrique d’un ordre infiniment grand et sur les déterminants doublement gauches publication-title: C. R. de l’Académie des Sci. – volume: 28 start-page: 209 year: 1939 ident: ref_9 article-title: A generalization of the polynomial Fn(x) publication-title: Lond. Edinb. Dublin Philos. Mag. J. Sci. Ser. 7 |
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| Snippet | Demonstrating the striking symmetry between calculus and q-calculus, we obtain q-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using... Demonstrating the striking symmetry between calculus and -calculus, we obtain -analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using... |
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| SubjectTerms | Functions (mathematics) Laplace transforms Mathematical analysis Polynomials Radiation |
| Title | Some Generating Functions for q-Polynomials |
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