Asymptotic expansions for certain exponential-type operators connected with 2x3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2x^{3/2}$$\end{document}

In the present paper, we consider the complete asymptotic expansion of certain exponential-type operators connected with 2x3/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2x^{3/2}$$\end{document}. Also, a modification of such exponential-type operators is provided, which preserve the function eAx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{e}^{Ax}$$\end{document}.


Introduction
Ismail and May [5, (3.16)] defined the operators where x ∈ (0, ∞) . The kernel is given by where (t) is the Dirac delta function and I 1 denotes the modified Bessel function of first kind defined by The operators (1) are exponential-type operators as the kernels k n (x, t) of the operators T n satisfy the partial differential equation with p(x) = 2x 3∕2 . The operators (1) reproduce constant and linear functions.
Note that the operators (1) can alternatively be written in the form where x ∈ (0, ∞) and We observe that the operators T n are closely related to the well-known Phillips operators given by (1) Obviously, we have , but unlike the operators T n , Phillips operators are not exponential type operators. The operators T n individually have not been studied by researchers much, due to its complicated behaviour. Very recently, Abel and Gupta [1] studied the rate of convergence of these operators on functions of bounded variation. Also, Gupta [2] and Gupta et al. [3] resp. gave some direct results in ordinary and simultaneous approximation. In the present article, we establish the complete asymptotic expansion of the operators T n as n tends to infinity.

Asymptotic expansion
In the following, we use the falling factorial given by z 0 = 1 , and Firstly, we derive a concise representation of the r-th order moment T n e r (x) where e r (t) = t r , r ∈ ℕ 0 , denote the monomials. Obviously, we have T n e 0 (x) = 1.

Lemma 1 For r ∈ ℕ , the r-th order moment of the Ismail-May operators T n has the representation
Proof For r ∈ ℕ , we have Application of the Leibniz rule for differentiation yields which is the desired formula. ◻ Now we turn to the central moments T n s Let C [0, ∞) be the class of continuous functions f on [0, ∞) satisfying the exponential growth condition f (t) = O(e t ) as t → ∞ , for some > 0.
The next theorem is the main result of this section.
Theorem 1 Let q ∈ ℕ and x ∈ (0, ∞) . For each function f ∈ C [0, ∞) , which has a derivative of order 2q at the point x, the operators T n possess the asymptotic expansion as n → ∞.

Proof
We apply a general result by Sikkema [7]. In view of the localization result (Proposition 1), we can assume that the function f has (at most) polynomial growth as the variable tends to infinity. Using Lemma 2, we obtain as n → ∞ . ◻ More explicitly, we have as n → ∞. A direct corollary is the following Voronovskaja-type formula.
For each function f ∈ C [0, ∞) , which has a second derivative at the point x, we have the asymptotic relation Now we prove a localization result which is of use in the proof of Theorem 1. It is of interest in its own. In what follows let exp denote the exponential function. For real , we write exp (t) ∶= e t . This modified form of operators preserves constants as well as the exponential function exp A , but loose to preserve the linear functions. We have the following limits: Also, we have After simple computations, we derive the following representation.

Lemma 4 For real numbers C, it holds
We may observe here that the r-th order moment T n e r is given by In particular, we have k n (a n (x), t)e At dt, lim n→∞ a n (x) = x = lim A→0 a n (x).