Generalization of Sheffer-λ polynomials

The aim of this study is to define a new generalization of Sheffer- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> λ polynomials with the help of Sheffer polynomials and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> λ -polynomials. For this family, explicit form, summation formulas, quasi-monomiality properties, differential equation and determinant representation are obtained. Subfamilies of these polynomials are introduced and similar properties for subfamilies are found. In addition, 3D graphs and the distribution of real roots are plotted for these subfamilies. Similarly, twice iterated Sheffer- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> </m:math> λ polynomials are defined and basic properties for this family and its subfamilies are obtained, and 3D graphs and the distribution of real roots are also investigated for the subfamilies.