Table 1 Certain members belonging to the 2-variable general polynomials \(P_{n}(x,y)\)

I

\(e^{yt^{s}}\)

Gould–Hopper polynomials [2]

\(\exp (xt+yt^{s})=\sum_{n=0}^{\infty }G_{n}^{(s)}(x,y)\frac{t^{n}}{n!}\)

II

\(C_{0}(-yt^{s})\)

2-variable generalized Laguerre polynomials [3]

\(\exp (xt) C_{0}(-yt^{s}) = \sum_{n=0}^{\infty } {{}_{s}L_{n}(y,x)} \frac{t^{n}}{n!}\)

III

\(\frac{1}{1-yt^{r}}\)

2-variable truncated exponential polynomials of order r [4]

\(\frac{e^{xt}}{1-yt^{r}}=\sum_{n=0}^{\infty } e_{n}^{(r)}(x,y)\frac{t^{n}}{n!}\)

IV

\(\frac{te^{yt^{j}}}{e^{t}-1}\)

2-dimensional Bernoulli polynomials [5]

\(\frac{t}{e^{t}-1}e^{xt+yt^{j}}=\sum_{n=0}^{\infty }B_{n}^{(j)}(x,y)\frac{t^{n}}{n!}\), |t|<2π

V

\(\frac{2e^{yt^{j}}}{e^{t}+1}\)

2-dimensional Euler polynomials [6]

\(\frac{2}{e^{t}+1}e^{xt+yt^{j}}=\sum_{n=0}^{\infty }E_{n}^{(j)}(x,y)\frac{t^{n}}{n!}\), |t|<π