Abstract The representation of an analytic function as a series involving ‐polynomials is a fundamental problem in classical analysis and approximation theory Ramanujan J. 19 (2009), no. 3, 281–303. In this paper, our investigation is focusing on ‐analog complex Hermite polynomials, which were motivated by Ismail and Zhang Adv. Appl. Math. 80 (2016), 70–92; Trans. Amer. Math. Soc. 369 (2017), 6779–6821. We give a new pair of ‐3D Hermite polynomials and their corresponding ‐heat equations. In addition, we generalize ‐derivative operator of Zhang Adv. Appl. Math. 121 (2020), 102081, 23pp. and ‐derivative operator of Yang Ramanujan J. 60 (2023), 1127–1149. and give some applications. Moreover, we define the generalized homogeneous Rogers–Szegö polynomial and Stieltjes–Wigert polynomial involving two parameters in the binomial coefficient and find their corresponding ‐partial differential equations. Finally, we define generalized ‐3D Hermite polynomials with double binomial coefficients, find their corresponding ‐partial differential equations, and generalize some results of Ismail and Zhang.
Jian Cao (Tue,) studied this question.