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

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]