A <i>q</i>-Operational Equation for Carlitz’s <i>q</i>-Operators with Some Applications

Jian Cao; Cheng Zhang; Sama Arjika

doi:doi:10.11648/j.acm.20231204.12

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A q-Operational Equation for Carlitz’s q-Operators with Some Applications

Jian Cao, Cheng Zhang, Sama Arjika

Published in Applied and Computational Mathematics (Volume 12, Issue 4)

Received: 11 June 2023 Accepted: 8 July 2023 Published: 21 July 2023

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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.

Published in	Applied and Computational Mathematics (Volume 12, Issue 4)
DOI	10.11648/j.acm.20231204.12
Page(s)	92-108
Creative Commons	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.
Copyright	Copyright © The Author(s), 2023. Published by Science Publishing Group

Keywords

q-Operational Equation, Carlitz’s q-operators, Rogers–Szegö Polynomials, Hahn Polynomials, Generating Function

References

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[5]	J. Cao. “New proofs of generating functions for Rogers– Szegö polynomials”. In: Appl. Math. Comput. 207 (2009), pp. 486–492.
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[7]	J. Cao, H.-L. Zhou and S. Arjika. “Generalized q-difference equations for (q,c)-hypergeometric polynomials and some applications”. In: Ramanujan J. 60 (2023) no. 4, pp. 1033–1067.
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[18]	R. Koekock and R. F. Swarttouw, The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q- Analogue. Report No. 98-17; Delft University of Technology: Delft, The Netherlands, 1998.
[19]	Z.-G.Liu. “Aq-extensionofapartialdifferentialequation and the Hahn polynomials”. In: Ramanujan J. 38 (2015), no. 3, pp. 481–501.
[20]	Z.-G. Liu and J. Zeng. “Two expansion formulas involving the Rogers–Szegö polynomials with applications”. In: Int. J. Number Theory 11 (2015), no. 2, pp. 507–525.
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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

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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

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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

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@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}
}

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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
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EP  - 108
PB  - Science Publishing Group
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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  -

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Author Information

Jian Cao

School of Mathematics, Hangzhou Normal University, Hangzhou, China
Cheng Zhang

School of Mathematics, Hangzhou Normal University, Hangzhou, China
Sama Arjika

Department of Mathematics and Informatics, University of Agadez, Agadez, Niger

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Plain Text BibTeX RIS

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

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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

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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

Copy | Download

@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}
}

Copy | Download

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  -

Copy | Download