Confluent Appell polynomials

Abstract

In this paper, we introduce the confluent Appell polynomials and prove a Sheffer type characterization theorem for them by means of the Stieltjes integral of hypergeometric polynomials. We investigate their several properties such as explicit representation, integral representation and finite summation formulas. Moreover, by proving a pure recurrence relation and deriving the lowering and the raising operators, in terms of differential and shift operators, we obtain the equation satisfied the confluent Appell polynomials by using the factorization method. And then, we define the confluent Bernoulli and Hermite polynomials and exhibit their main properties such as explicit representations, recurrence formulas (involving the corresponding usual Bernoulli and Hermite polynomials), finite summation formulas and equations involving differential and shift operators. Finally, we construct approximation operators by using confluent Appell polynomials which helps to approximate to a function defined on the semi infinite interval in a weighted function space. We call these as the confluent Jakimovski–Leviatan operators which includes the confluent version of the well-known Szász–Mirakyan operators. Also, an illustrative example in order to show convergence efficiency of the confluent Szász–Mirakyan operators is given.