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

| Peer-Reviewed

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

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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), 2024. Published by Science Publishing Group

Keywords

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

References

[1]	W. A. Al-Salam and L. Carlitz. “Some orthogonal q- polynomials”. In: Math. Nachr. 30 (1965), pp. 47–61.
[2]	N. A. Al-Salam. “Some operational formulas for the q- Laguerre polynomials”. In: Fibonacci Quart. 22 (1984), no. 2, pp. 166–170.
[3]	G. Bengocheaa, L. Verde-Starb and M. Ortigueira. “Solution of equations with q-derivatives and Ward’s derivatives using an operational method”. In: Filomat. 36 (2022), pp. 1487–1498.
[4]	J. Cao. “Notes on Carlitz’s q-operators”. In: Taiwanese J. Math. 14 (2010), pp. 2229–2244.
[5]	J. Cao. “New proofs of generating functions for Rogers– Szegö polynomials”. In: Appl. Math. Comput. 207 (2009), pp. 486–492.
[6]	J. Cao. “Generalizations of certain Carlitz’s trilinear and Srivastava–Agarwal type generating functions”. In: J. Math. Anal. Appl. 396 (2012), no. 1, pp. 351–362.
[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.
[8]	L. Carlitz. “Some polynomials related to theta functions”. In: Ann. Mat. Pura Appl. 41 (1956), pp. 359–373.
[9]	L. Carlitz. “Some polynomials related to theta functions”. In: Duke Math. J. 24 (1957), pp. 521–527.
[10]	L. Carlitz. “Generating functions for certain q-orthogonal polynomials”. In: Collect. Math. 23 (1972), pp. 91–104.
[11]	J. Cigler. “Operatormethoden für q-Identitäten” (German). In: Monatsh. Math. 88 (1979), pp. 87–105.
[12]	J. Cigler, “Operatormethoden für q-Identitäten”. II. q- Laguerre-Polynome (German). In: Monatsh. Math. 91 (1981), no. 2, 105–117.
[13]	W. Y. C. Chen and Z.-G. Liu, “Parameter augmenting for basic hypergeometric series, I.” In: Sagan, B. E., Stanley, R. P. (eds.) Mathematical Essays in Honor of Gian-Carlo Roto, Birkh ”auser, Basel 1998, pp. 111–129.
[14]	G. Gasper and M. Rahman, Basic hypergeometric series. With a foreword by Richard Askey”, Second edition. Encyclopedia of Mathematics and its Applications, 96. Cambridge University Press, Cambridge, 2004.
[15]	R. C. Gunning, Introduction to Holomorphic Functions of Several Variables, Vol. 1: Function Theory, Wadsworth and Brooks/Cole Mathematics Series, Brooks/Cole Publishing Company, Belmont, 1990; Reprinted by CRC Press, Boca Raton, London and New York, 2017.
[16]	B. He. “On a generalized homogeneous Hahn polynomial”. In: Sci. China Math. 66 (2023), pp. 957– 976.
[17]	M. A. Khan. “q-Analogues of certain operational formulae”. In: Houston J. Math. 13 (1987), no. 1, pp. 75–82.
[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.
[21]	Z.-G. Liu. “On a system of partial differential equations and the bivariate Hermite polynomials”. In: J. Math. Anal. Appl. 454 (2017), pp. 1–17.
[22]	Z.-G. Liu. “A q-operational equation and the Rogers- Szegö polynomials”. In: Sci. China Math. 66 (2023), no. 5, 957–976.
[23]	B. Malgrange, Lectures on the Theory of Functions of Several Complex Variables. Springer, Berlin, 1984.
[24]	L. J. Reid. “Operational equations for a class of symmetric q-polynomials”. In: Duke Math. J. 35 (1968), pp. 783–798.
[25]	H. L. Saad and M. A. Abdlhusein. “New application of the Cauchy operator on the homogeneous Rogers–Szegö polynomials”. In: Ramanujan J. 56 (2021), no. 1, pp. 347–367.
[26]	L. J. Slater, Generalized Hypergeometric Functions. Cambridge University Press: Cambridge, UK; London, UK; New York, NY, USA, 1966.
[27]	H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series. Halsted Press (Ellis Horwood Limited), Chichester, UK; John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisbane, Australia; Toronto, Canada, 1985.
[28]	H. M. Srivastava. “Certain q-polynomial expansions for functions of several variables”. I. In: IMA J. Appl. Math. 30 (1983), pp. 315–323.
[29]	H. M. Srivastava. “Certain q-polynomial expansions for functions of several variables”. II. In: IMA J. Appl. Math. 33 (1984), pp. 205–209.

Cite This Article

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

Copy | Download

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

Copy | Download

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

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

Download PDF

Sections

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

Copy | Download

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

Copy | Download

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