Some new semi-exponential operators

In the theory of approximation, linear operators play an important role. The exponential-type operators were introduced four decades ago, since then no new exponential-type operator was introduced by researchers, although several generalizations of existing exponential-type operators were proposed and studied. Very recently, the concept of semi-exponential operators was introduced and few semi-exponential operators were captured from the exponential-type operators. It is more difficult to obtain semi-exponential operators than the corresponding exponential-type operators. In this paper, we extend the studies and define semi-exponential Bernstein, semi-exponential Baskakov operators, semi-exponential Ismail–May operators related to 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} or x3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x^{3}$$\end{document}. Furthermore, we present a new derivation for the semi-exponential Post–Widder operators. In some examples, open problems are indicated.

the linear functions. Many generalizations of exponential-type operators are available in the literature. Tyliba and Wachnicki [7] extended the definition of Ismail and May [4] by proposing a more general family of operators. For a non-negative real number β, they introduced the operators L β λ . For β > 0, they are not of exponential type but similar to exponential-type operators. Recently, Herzog [3] further extended the studies and termed such operators as semi-exponential type operators. Actually, an operator of the form is called a semi-exponential operator if its kernel W L β (λ, x, t) satisfies the differential equation In particular, for β > 0, one has L β λ e 1 = e 1 , where e r (t) = t r (r = 0, 1, 2, . . .). In the case β = 0, the operator L β=0 λ is simply the exponential-type operator studied by Ismail and May [4]. A collection of such operators may be found in the recent book [2,Ch. 1].
Choosing different functions p (x) several exponential-type operators were captured in Ismail and May [4]. It is difficult to construct new exponential-type operators or the corresponding semi-exponential operators by just taking different functions p (x). The essential obstacle is to fulfill the normalization condition which means that L β λ preserves constant functions. Tyliba and Wachnicki [7] captured the semi-exponential operators of Weierstrass and Szász-Mirakyan operators, Herzog [3] got success to define the semi-exponential Post-Widder operators. We represent below the tabular form of known semi-exponential type operators available till date: As pointed out earlier, one can obtain the exponential-type operator as the special β = 0 from semi-exponential operators, but the converse is not analogous. Here we capture some more semi-exponential operators viz. semi-exponential Bernstein polynomials, semi-exponential Baskakov operators, etc.

New semi-exponential operators
In this section, we establish some new exponential-type operators. In all listed cases it is possible to solve the differential equation (1) in the form W L β (n, x, t) = A L (n, t, β) y, but it is difficult to find the normalization, i.e., the factor A L (n, t, β) of the solution y such that respectively. Below we list some instances of p (x), which were considered for well-known exponential-type operators.

Semi-exponential Bernstein operators
where the derivative of y is with respect to the variable x. We conclude that In order to have normalization . Then x = z/ (1 + z), and for any positive integer n, the generating function of the sequence is analytic, for |z| < 1, with an essential singularity at z = −1. Hence, it can be developed as a power series in the disk |z| < 1. The series is convergent for all complex z different from −1. It follows that, for |z| < 1, where the binomial coefficient is to be read as n− j Thus the semi-exponential Bernstein polynomials B The latter representation immediately reveals the special case β = 0.

Semi-exponential Baskakov operators
where the derivative of y is with respect to the variable x. We conclude that In order to have normalization Thus, the semi-exponential Baskakov operators can be defined by In special case β = 0 we have j = 0 and = k such that we get the Baskakov operators.

Semi-exponential Ismail-May operators related to 2x 3/2
If we take p (x) = 2x 3/2 , then for a kernel W U β (n, x, t) = A U (n, t, β) y, we have where the derivative of y is with respect to the variable x. We conclude that Our target is to obtain A U (n, t, β) in order to have normalization If we put, for abbreviation, s = n/ √ x, the normalization condition takes the form where δ (t) denotes Dirac's delta function. Hence, the operators are defined by Thus, the semi-exponential operator, related to 2x 3/2 , takes the form In the special case β = 0, the definition reduces to the Ismail-May operator of exponential type where I 1 (x) is modified Bessel function of the first kind. Further results on the operators U β=0 n can be found in [1].

Semi-exponential Post-Widder operators
Although the semi-exponential Post-Widder operators were captured in [3, Eq. (10)], using Laplace transform, we provide an alternative approach that is shorter. We proceed as follows.
If we take p(x) = x 2 , then for a kernel W P β (n, x, t) = A P (n, t, β) y, we have For normalization, we look for a function A P (n, t, β) such that Putting This is equivalent to It follows that α = n − 1 and

Hence, A P (n, t, β) is given by
Thus, semi-exponential Post-Widder operators take the form Observing that A P (n, t, β) = n (nt/β) (n−1)/2 I n−1 2 √ nβt , where I n denotes the modified Bessel function of the first kind, we obtain the alternative representation

Semi-exponential Ismail-May operators related to x (1 + x) 2
If we take p (x) = x (1 + x) 2 , then for a kernel W R β (n, x, k/n) = A R (n, k, β) y, we have If we put y = x/ (1 + x) the normalization condition reads Now we put z = ye 1−y , so we have the inverse y = −W (−z/e), where W denotes the Lambert W function, i.e., the inverse of z → ze z . Hence, A R (n, k, β) are the coefficients of the power series which is an easy consequence of the Lagrange expansion theorem. With w = −W (−z/e) we obtain

It follows
i.e., A R (n, k, β) is the coefficient of z k in the latter power series expansion. The semiexponential operators related to x(1 + x) 2 , take the form In the special case β = 0 we have ) and the operators reduce to Eq. (3.14)]. As Ismail and May remarked, the substitution y = x/ (1 + x) leads to the operators It may be considered as an open problem to find a closed form of the coefficients A R (n, k, β).

Semi-exponential Ismail-May operators related to x 3
If we take p (x) = x 3 , then for a kernel W Q β (n, If we put s = n/ 2x 2 the normalization condition reads which implies where δ (t) denotes Dirac's delta function. It is well known that We will take advantage of the convolution formula Combining Eqs. (2) and (3) Thus, semi-exponential operators related to p (x) = x 3 take the form In the special case β = 0 we have A Q (n, t, β = 0) = n 2π t −3/2 e −n/(2t) and the operators reduce to
The operators related to 1 + x 2 take the form The main target to find a closed expression for A T (n, t, β), for general β > 0, may be considered as an open problem.