Theory and Modern Applications
From: Certain results on a hybrid class of the Boas–Buck polynomials
S. No. | Results | Expressions |
---|---|---|
I | Series expansion | \({}_{F}E_{n}^{(j)}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} F_{n-k}(x) E^{(j)}_{k}(0,y)\) |
II | Multiplicative operator | \(\hat{M}_{{}_{F}E^{(j)}}=x\partial _{x}C^{\prime }(C^{-1}(\sigma ))\sigma ^{-1} +\frac{A^{\prime }(C^{-1}(\sigma ))}{A(C^{-1}(\sigma ))}+jyC^{1-j}(\sigma ) -(1+\exp (C^{-1}(\sigma )))^{-1}\) |
III | Derivative operator | \(\hat{P}_{{}_{F}E^{(j)}}=C^{-1}(\sigma )\) |
IV | Differential equation | \(\begin{array}[t]{l} (x\partial _{x}C^{\prime }(C^{-1}(\sigma ))\sigma ^{-1}-nC(\sigma ) +\frac{A^{\prime }(C^{-1}(\sigma ))}{A(C^{-1}(\sigma ))} +jyC^{1-j}(\sigma ) \\ \quad {}-(1+\exp (C^{-1}(\sigma )))^{-1} ) \,{}_{F}E^{(j)}_{n}(x,y) =0\end{array}\) |