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Table 1 Certain members of 2VGP family \(\mathbb{G}_{n}(x,y)\)

From: Construction of a new family of Fubini-type polynomials and its applications

S. no. φ(y,t) Generating functions Polynomials
I. \(e^{yt^{r}}\) \(e^{xt+yt^{r}}=\sum_{n=0}^{\infty }\mathcal{H}^{(r)}_{n}(x,y) \frac{t^{n}}{n!}\) The Gould–Hopper polynomials \(\mathcal{H}^{(r)}_{n}(x,y)\) [4]
II. \(C_{0}(yt)\) \(e^{xt} C_{0}(yt)=\sum_{n=0}^{\infty }\mathtt{L}_{n} (y,x)\frac{t^{n}}{n!}\) The 2-variable Laguerre polynomials \(\mathtt{L}_{n} (y,x)\) [1]
III. \(\frac{1}{1-yt^{s}}\) \(\frac{1}{1-yt^{s}} e^{xt}=\sum_{n=0}^{\infty }e^{(s)}_{n}(x,y) \frac{t^{n}}{n!}\), The 2-variable truncated exponential of order s \(e^{(s)}_{n}(x,y)\) [2]
VI. \(A(t) e^{yt^{2}}\) \(A(t) e^{xt+yt^{2}}=\sum_{n=0}^{\infty }{}_{\mathcal{H}}A_{n}(x,y) \frac{t^{n}}{n!}\), The Hermite–Appell polynomials \({}_{\mathcal{H}}A_{n}(x,y)\) [10]