Theory and Modern Applications
Table 7 Results for Boas–Buck–Bernoulli polynomials \({}_{F}B^{(j)}_{n}(x,y)\)
From: Certain results on a hybrid class of the Boas–Buck polynomials
S. No. | Results | Expressions |
|---|---|---|
I | Series expansion | \({}_{F}B^{(j)}_{n}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} F_{n-k}(x) B^{(j)}_{k}(0,y)\) |
II | Multiplicative operator | \(\hat{M}_{{}_{F}B^{(j)}}=x\partial _{x}C^{\prime }(C^{-1}(\sigma ))\sigma ^{-1} +\frac{A^{\prime }(C^{-1}(\sigma ))}{A(C^{-1}(\sigma ))}+C(\sigma ) +jyC^{1-j}(\sigma )+(1-\exp (C^{-1}(\sigma )))^{-1}\) |
III | Derivative operator | \(\hat{P}_{{}_{F}B^{(j)}}=C^{-1}(\sigma )\) |
IV | Differential equation | \(\begin{array}[t]{l} ( x\partial _{x}C^{\prime }(C^{-1}(\sigma ))\sigma ^{-1}-nC(\sigma ) +\frac{A^{\prime }(C^{-1}(\sigma ))}{A(C^{-1}(\sigma ))}+C(\sigma ) +jyC^{1-j}(\sigma ) \\ \quad {}+(1-\exp (C^{-1}(\sigma )))^{-1} )\,{}_{F}B^{(j)}_{n}(x,y) =0\end{array}\) |