Poisson kernels of qq‐3D Hermite polynomials expansion for functions of several variables via generalized qq‐heat equations

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.