In the present paper, we introduce a new sequence of linear positive operators with the help of generating functions. We obtain some Korovkin type approximation properties for these operators and compute rates of convergence by means of the first and second order modulus of continuities and Peetre’s K-functional. | Turk J Math 33 (2009) , 41 – 53. ¨ ITAK ˙ c TUB doi: Korovkin Type Error Estimates for Positive Linear Operators Involving Some Special Functions Og¨ un Do˘gru and Esra Erku¸s-Duman Abstract In the present paper, we introduce a new sequence of linear positive operators with the help of generating functions. We obtain some Korovkin type approximation properties for these operators and compute rates of convergence by means of the first and second order modulus of continuities and Peetre’s K -functional. In order to obtain explicit expressions for the first and second moment of our operators, we obtain a functional differential equation including our operators. Furthermore, we deal with a modification of our operators converging to integral of function f on the interval (0, 1). Key Words: Positive linear operators, Korovkin-Bohman theorem, Bernstein power series, generating function, Pochhammer symbol, hypergeometric function, Peetre’s K -functional, first and second order modulus of continuities, functional differential equation. 1. Introduction The study of the Korovkin-Bohman type approximation theory is a well established area of active research (see, ., [4, 6, 14]). Especially, it has connections not only with classical approximation theory, but also with other branches of mathematics, such as functional analysis, harmonic analysis, measure theory, probability theory. Cheney and Sharma [8], first introduced a new linear positive operators with the help of generating function expansion of Laguerre polynomial. Recently, two different generalizations of linear positive operators involving some generating functions have been introduced, and Korovkin type error estimates and their rates of convergences have been obtained (see [3, 4]). We now turn to introducing our operators used in this paper. Consider a new sequence of linear positive operators for x ∈ [0, a] , a < 1, t ∈ [0, b], b ∈ R+ , ∞ (Ln f)(x, t) = 1 k (n) f( )g (t)xk , Fn (x, t) cn +
