MODIFIED BETA APPROXIMATING OPERATORS OF THE FIRST AND SECOND KIND

. By using the modiﬁed beta distributions of the ﬁrst kind (MB1) and of the second kind (MB2) we obtain a general class of modiﬁed beta ﬁrst kind operators and modiﬁed beta second kind operators. We obtain several positive linear operators as a special case of this modiﬁed beta operators.

We first review the definitions of the generalized beta distributions of the first kind (GB1) and the second kind (GB2) and special cases.
The Euler beta distribution of the first kind is defined for p, q > 0 by the following formula (1.1) B(t; p, q) = 1 B(p,q) t p−1 (1 − t) q−1 for t ∈ (0, 1), and zero otherwise, where B(p, q) is the beta function.
The generalized beta distribution of the first kind is defined by the probability density function (pdf) (see [4]) for 0 < y e < d e and zero otherwise, where the parameters d, p, q are positive.The defined kth-order moments of GB1 random variables are given by [4] (1. 3) B(p,q) for p + k/e > 0.
This four-parameter pdf is very flexible and includes the modified beta distribution of the first kind for e = 1.
The standard beta distribution of the first kind (1.1) corresponds to (1.4) with b = 1.
The Euler beta distribution of the second kind is defined for p, q > 0 by the following formula (1.5) B(u; p, q) = 1 B(p,q) • u p−1 (1+u) p+q , u > 0 and zero otherwise.
The generalized beta distribution of the second kind is defined by the pdf (see [4]) ) p+q for v > 0 and zero otherwise.
The defined kth-order moments of the GB2 are given by [4] (1.7) for − p < k/e < q Letting e = 1 in (1.6) gives the modified beta distribution of the second kind (MB2) The standard beta distribution of the second kind (1.5) is obtained by (1.8) for d = 1.

THE MODIFIED BETA FIRST KIND OPERATORS
Let us denote by M [0, ∞) the linear space of functions defined on [0, ∞), bounded and Lebesgue measurable in each interval [c, d], 0 < d < c < ∞.
By using the modified beta distribution of the first kind (MB1), defined by (1.4) we can define the following general transform: B(p+a,q+b) , and we obtain from (2.1) the (a, b)-modified beta operator (see also [9]) If we put in (2.2) b = 0 we obtain the modified beta first kind operator p,q is a linear positive operator.I.If we choose in (2.3) a = 1 we obtain the modified beta first kind operator Theorem 2.1.The moments of order k of the operator B p,q have the following values Proof.The result follows from (1.3) for e = 1.

THE MODIFIED BETA SECOND KIND OPERATORS
We shall define a linear transform by using the modified beta distribution of the second kind (MB2), defined by (1.8) B(p+a,q+b) x and we obtain the (a, b)-modified beta operator (see also [10]) B(p+a,q+b) • u a x (1+u) a+b du 5.For q = p + 2 we obtain by (3.4) the positive linear operator considered by H. Karski in [1], [2], for p = n + 1.