Theory and Modern Applications
From: Certain results on a hybrid class of the Boas–Buck polynomials
S. No. | Φ(y,t) | Polynomial | Generating function |
---|---|---|---|
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 |