On generalization of beta operators

Abstract EN

In this present paper, we introduce a generalization definition to Beta weight functions depend on a non-negative integer called −Beta.

This definition restricts to the classical Lupas and to the classical Beta weight functions whenever =0,1 respectively.

In addition, we used these weight functions to define two operators of summation and of summation-integral types.

Surly, these operators restrict to the classical Lupas operators and to the classical Beta operators of both summation and summation-integral types whenever =0,1 respectively.

In addition, we can get the mixed operators of Lupas-Beta and Beta-Lupas for a suitable chose of integer values.

Furthermore, we derive a Voronovaskaja-type asymptotic formula for the new operators from which we can get the similar formulas for many operators of summation and summation-integral types of mixed Lupas-Beta (or Beta-Lupas) weight functions and more.