Theory and Modern Applications
From: Certain results on a hybrid class of the Boas–Buck polynomials
S. No. | Name and notation | Generating function and series expansion |
---|---|---|
I | Brenke general polynomials \({}_{Y}P_{n}(x,y)\) | \(\varPhi (y,t)A(t)B(xt)=\sum_{n=0}^{\infty }\,{}_{Y}P_{n}(x,y)\frac{t^{n}}{n!}\) |
\({}_{Y}P_{n}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} Y_{n-k}(x) \phi _{k}(y)\) | ||
II | Sheffer general polynomials \({}_{S}P_{n}(x,y)\) | \(\varPhi (y,t)A(t)\exp (xC(t))=\sum_{n=0}^{\infty }\,{}_{S}P_{n}(x,y)\frac{t^{n}}{n!}\) |
\({}_{S}P_{n}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} S_{n-k}(x) \phi _{k}(y)\) | ||
III | Appell general polynomials \({}_{L}P_{n}(x,y)\) | \(\varPhi (y,t)A(t)\exp (xt)=\sum_{n=0}^{\infty }\,{}_{L}P_{n}(x,y)\frac{t^{n}}{n!}\) |
\({}_{L}P_{n}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} L_{n-k}(x) \phi _{k}(y)\) |