Certain results on a hybrid class of the Boas–Buck polynomials

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 3 Certain members belonging to the 2VBBGP \({}_{F}P_{n}(x,y)\)

S. No.	Name and notation	Generating function and series expansion
I	Brenke general polynomials \({}_{Y}P_{n}(x,y)\)	\(\varPhi (y,t)A(t)B(xt)=\sum_{n=0}^{\infty }\,{}_{Y}P_{n}(x,y)\frac{t^{n}}{n!}\)
I	Brenke general polynomials \({}_{Y}P_{n}(x,y)\)	\({}_{Y}P_{n}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} Y_{n-k}(x) \phi _{k}(y)\)
II	Sheffer general polynomials \({}_{S}P_{n}(x,y)\)	\(\varPhi (y,t)A(t)\exp (xC(t))=\sum_{n=0}^{\infty }\,{}_{S}P_{n}(x,y)\frac{t^{n}}{n!}\)
II	Sheffer general polynomials \({}_{S}P_{n}(x,y)\)	\({}_{S}P_{n}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} S_{n-k}(x) \phi _{k}(y)\)
III	Appell general polynomials \({}_{L}P_{n}(x,y)\)	\(\varPhi (y,t)A(t)\exp (xt)=\sum_{n=0}^{\infty }\,{}_{L}P_{n}(x,y)\frac{t^{n}}{n!}\)
III	Appell general polynomials \({}_{L}P_{n}(x,y)\)	\({}_{L}P_{n}(x,y)=\sum_{k=0}^{n}{}^{n}C_{k} L_{n-k}(x) \phi _{k}(y)\)