Rogers–Szegö polynomials are the basis in the Scheme of basic hypergeometric orthogonal polynomials. By solving a q-operational equation with formal power series, Liu introduced a new q-exponential operational identity and developed a systematic method to prove the identities involving the Rogers–Szegö polynomials. In this paper, motivated by Carlitz’s q-operators and Liu’s q-operational equation, we construct an q-operational equation for Carlitz’s q-operators and give some applications to some generating functions for Rogers–Szegö polynomials and Hahn polynomials, which generalize the method of exponential operator decomposition introduced by Cao and provide a new proof of results of Carlitz and Saad et al.. We chose Mehler’s formula, q-Nielsen’s formula for Rogers–Szegö polynomials and Mehler’s formula for Hahn polynomials as examples to show that the q-series theory can be applied, which takes us quickly the results. One of the main characteristics of this method is that it provides an effective approach to calculate generating functions for some q-polynomials. This method also brings a new research perspective to problems of the sum and integration of q-polynomials.
Published in | Applied and Computational Mathematics (Volume 12, Issue 4) |
DOI | 10.11648/j.acm.20231204.12 |
Page(s) | 92-108 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2023. Published by Science Publishing Group |
q-Operational Equation, Carlitz’s q-operators, Rogers–Szegö Polynomials, Hahn Polynomials, Generating Function
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APA Style
Jian Cao, Cheng Zhang, Sama Arjika. (2023). A q-Operational Equation for Carlitz’s q-Operators with Some Applications. Applied and Computational Mathematics, 12(4), 92-108. https://doi.org/10.11648/j.acm.20231204.12
ACS Style
Jian Cao; Cheng Zhang; Sama Arjika. A q-Operational Equation for Carlitz’s q-Operators with Some Applications. Appl. Comput. Math. 2023, 12(4), 92-108. doi: 10.11648/j.acm.20231204.12
AMA Style
Jian Cao, Cheng Zhang, Sama Arjika. A q-Operational Equation for Carlitz’s q-Operators with Some Applications. Appl Comput Math. 2023;12(4):92-108. doi: 10.11648/j.acm.20231204.12
@article{10.11648/j.acm.20231204.12, author = {Jian Cao and Cheng Zhang and Sama Arjika}, title = {A q-Operational Equation for Carlitz’s q-Operators with Some Applications}, journal = {Applied and Computational Mathematics}, volume = {12}, number = {4}, pages = {92-108}, doi = {10.11648/j.acm.20231204.12}, url = {https://doi.org/10.11648/j.acm.20231204.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231204.12}, abstract = {Rogers–Szegö polynomials are the basis in the Scheme of basic hypergeometric orthogonal polynomials. By solving a q-operational equation with formal power series, Liu introduced a new q-exponential operational identity and developed a systematic method to prove the identities involving the Rogers–Szegö polynomials. In this paper, motivated by Carlitz’s q-operators and Liu’s q-operational equation, we construct an q-operational equation for Carlitz’s q-operators and give some applications to some generating functions for Rogers–Szegö polynomials and Hahn polynomials, which generalize the method of exponential operator decomposition introduced by Cao and provide a new proof of results of Carlitz and Saad et al.. We chose Mehler’s formula, q-Nielsen’s formula for Rogers–Szegö polynomials and Mehler’s formula for Hahn polynomials as examples to show that the q-series theory can be applied, which takes us quickly the results. One of the main characteristics of this method is that it provides an effective approach to calculate generating functions for some q-polynomials. This method also brings a new research perspective to problems of the sum and integration of q-polynomials.}, year = {2023} }
TY - JOUR T1 - A q-Operational Equation for Carlitz’s q-Operators with Some Applications AU - Jian Cao AU - Cheng Zhang AU - Sama Arjika Y1 - 2023/07/21 PY - 2023 N1 - https://doi.org/10.11648/j.acm.20231204.12 DO - 10.11648/j.acm.20231204.12 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 92 EP - 108 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20231204.12 AB - Rogers–Szegö polynomials are the basis in the Scheme of basic hypergeometric orthogonal polynomials. By solving a q-operational equation with formal power series, Liu introduced a new q-exponential operational identity and developed a systematic method to prove the identities involving the Rogers–Szegö polynomials. In this paper, motivated by Carlitz’s q-operators and Liu’s q-operational equation, we construct an q-operational equation for Carlitz’s q-operators and give some applications to some generating functions for Rogers–Szegö polynomials and Hahn polynomials, which generalize the method of exponential operator decomposition introduced by Cao and provide a new proof of results of Carlitz and Saad et al.. We chose Mehler’s formula, q-Nielsen’s formula for Rogers–Szegö polynomials and Mehler’s formula for Hahn polynomials as examples to show that the q-series theory can be applied, which takes us quickly the results. One of the main characteristics of this method is that it provides an effective approach to calculate generating functions for some q-polynomials. This method also brings a new research perspective to problems of the sum and integration of q-polynomials. VL - 12 IS - 4 ER -