A Note on Truncated Exponential-Based Appell Polynomials

Abstract

This article deals with the introduction of truncated exponential-based Appell polynomials and derivation of their properties. The operational correspondence between these polynomials and Appell polynomials is established. Also, an integral representation for these polynomials in terms of a recently introduced family of polynomials is derived. Special emphasis is given to the truncated exponential-based Bernoulli and Euler polynomials and their related numbers.

Appendix

We give the surface plots of the polynomials ${}_{{}_{{{}_{[2]}}^e}}B_{n}(x,y)$, ${}_{{}_{{{}_{[2]}}^e}}E_{n}(x,y)$, ${}_{e}B_{n}(x,y)$, and ${}_{e}E_{n}(x,y)$, for $n=5$. Also, we find the graphs of the polynomials ${}_{{}_{{{}_{[2]}}^e}}B_{n}(x)$, ${}_{{}_{{{}_{[2]}}^e}}E_{n}(x)$, ${}_{e}B_{n}(x)$, and ${}_{e}E_{n}(x)$, for $n=5$. In order to draw the surface plots and graphs of these polynomials, we use the values of the Bernoulli polynomials $B_n(x)$ and Euler polynomials $E_n(x)$, for $n=0,1,2,3,4,5.$ we give the list of first few Bernoulli and Euler polynomials in Table 3.

Table 3 First few Bernoulli and Euler polynomials

Now, using Table 3, we express the polynomials ${}_{{}_{{{}_{[2]}}^e}}B_{5}(x,y)$, ${}_{{}_{{{}_{[2]}}^e}}E_{5}(x,y)$, ${}_{e}B_{5}(x,y)$, and ${}_{e}E_{5}(x,y)$ in powers of x and y and the polynomials ${}_{{}_{{{}_{[2]}}^e}}B_{5}(x)$, ${}_{{}_{{{}_{[2]}}^e}}E_{5}(x)$, ${}_{e}B_{5}(x)$, and ${}_{e}E_{5}(x)$ in powers of x. We present the expressions of these polynomials in Table 4.

Table 4 Expressions of ${}_{{}_{{{}_{[2]}}^e}}B_{5}(x,y)$, ${}_{{}_{{{}_{[2]}}^e}}E_{5}(x,y)$, ${}_{{e}}B_{5}(x,y)$, ${}_{{e}}E_{5}(x,y)$, ${}_{{}_{{{}_{[2]}}^e}}B_{5}(x)$, ${}_{{}_{{{}_{[2]}}^e}}E_{5}(x)$, ${}_{{e}}B_{5}(x)$, and ${}_{{e}}E_{5}(x)$

By making use of expressions given in Table 4, we get the following surface plots and graphs: