Theory and Modern Applications
Table 6 Results for Boas–Buck-truncated exponential polynomials \({}_{F}e^{(r)}_{n}(x,y)\)
From: Certain results on a hybrid class of the Boas–Buck polynomials
S. No. | Results | Expressions |
|---|---|---|
I | Series expansion | \({}_{F}e^{(r)}_{n}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} F_{n-k}(x) e^{(r)}_{k}(0,y)\) |
II | Multiplicative operator | \(\hat{M}_{{}_{F}e^{(r)}}=x\partial _{x}C^{\prime }(C^{-1}(\sigma ))\sigma ^{-1} +\frac{A^{\prime }(C^{-1}(\sigma ))}{A(C^{-1}(\sigma ))} +ry\partial _{y} yC^{1-r}(\sigma )\) |
III | Derivative operator | \(\hat{P}_{{}_{F}e^{(r)}}=C^{-1}(\sigma )\) |
IV | Differential equation | \(( x\partial _{x}C^{\prime }(C^{-1}(\sigma ))\sigma ^{-1}-nC(\sigma ) +\frac{A^{\prime }(C^{-1}(\sigma ))}{A(C^{-1}(\sigma ))} +ry\partial _{y} yC^{1-r}(\sigma ) ) \,{}_{F}e^{(r)}_{n}(x,y) =0\) |