Applied and Computational Mathematics

Submit a Manuscript

Publishing with us to make your research visible to the widest possible audience.

Propose a Special Issue

Building a community of authors and readers to discuss the latest research and develop new ideas.

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

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.

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

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.

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.