Introduction

In [1], Patel introduced the sequence of positive linear operators, which connecting to the Wright function as follows

$$\begin{aligned} P_n^{(\beta )}(f;x)=\dfrac{1}{\phi _{1,\beta }(z)}\sum _{k=0}^{\infty }\dfrac{(nx)^k}{k! \Gamma (k+\beta )} f\left( \dfrac{k}{n}\right) , \end{aligned}$$
(1.1)

where f is real valued bounded function provided summation is convergent, and the entire function (of z)

$$\begin{aligned} \phi (\rho ,\beta ;z)=\phi _{\rho ,\beta }(z) = \sum _{k=0}^{\infty } \frac{z^k}{k!\Gamma (\rho k +\beta )}, ~~~(\rho >-1,\beta ,z\in \mathbb {C}) \end{aligned}$$
(1.2)

named after the British mathematician E. M. Wright, has appeared for the first time in the case \(\rho >0\) in connection with his investigations in the asymptotic theory of partitions in [2].

Note that \(\phi (1,1;z)= I_0(2\sqrt{z})\), where \(I_n(z)=i^{-n}J_n(iz)=\dfrac{(z/2)^n}{\Gamma (1+n)} {}_{0}F_{1}\left( -;1+n;\dfrac{z^2}{4}\right) \), n not a negative integer. The function \(I_n(z)\) is called modified Bessel function of the first kind of index n. Here, \(J_n\) is Bessel function and \({}_{0}F_{1}\) is hypergeometric function. The modified function \(I_n\) is related to \(J_n\) in much the same way that the hyperbolic function are related to the trigonometric functions.

Also, note that

$$\begin{aligned} \phi _{1,2}(z)=\dfrac{1}{\sqrt{z}}I_1\left( 2\sqrt{z}\right) ;~~~~~ \phi _{1,m}(z)=\left( \dfrac{1}{\sqrt{z}}\right) ^{m/2} I_{m-1}\left( 2\sqrt{z}\right) . \end{aligned}$$

The Mittag-Leffler function and Wright function are solution of partial fractional differential equations. Researchers from special function have established some important results in [4,5,6] and references within. As these special functions are important and applicable in fractional differential equations, the operators (2.1) applicable in similar manner and use to approximate functions in \(L_p\) space. From time to time some new generalizations of sequence of positive linear operators were frequently introduced (see [7,8,9,10,11,12,13,14]) and studied several approximation properties.

Construction of Operators

Let \(\beta >0\) be fixed. For all \(n\in \mathbb {N}\), we introduce the Wright operators involving gamma function and Mittag-Leffler function as

$$\begin{aligned} W_n^{(\beta )}(f;x)=\dfrac{1}{E_{1,\beta }(nx)}\sum _{k=0}^{\infty }\dfrac{(nx)^k}{k! \Gamma (k+\beta )} \int _0^{\infty } e^{-t}t^{k} f\left( \dfrac{t}{n}\right) dt, \end{aligned}$$
(2.1)

where \( f\in C_B\left( [0,\infty )\right) :=\left\{ f\in C\left[ 0,\infty \right) : f \text { is bounded }\right\} \), and \(E_{\alpha ,\beta }\) is two index Mittag-Leffler function defined by Wiman [3]

$$\begin{aligned} E_{\alpha ,\beta }(z)=\sum _{k=0}^{\infty } \dfrac{z^k}{\Gamma (\alpha k +\beta )}, ~~~(z,\beta \in \mathbb {C}, \Re (\alpha )>0). \end{aligned}$$

Here, \(C\left[ 0,\infty \right) \) denote the space of continuous functions defined on \([0,\infty )\). Recall that the Banach lattice \(C_B\left( [0,\infty )\right) \) is endowed with the norm

$$\begin{aligned} \Vert f\Vert :=\sup _{x\in [0,\infty )} |f(x)|. \end{aligned}$$

One can note that, the operators \(W_n^{(\beta )}\) defined in (2.1) are linear and positive and these operators (2.1) are linking with three special functions namely gamma function, Mittag-Leffler function and Wright function.

Let \(\{b_n\}\) be a sequence of positive real numbers and let \(\beta > 0\) be fixed. For all \(n\in \mathbb {N}\), Ozarslan [15] introduced the following Mittag-Leffler operators as

$$\begin{aligned} L_n^{\beta }(f,x)=\dfrac{1}{E_{1,\beta }\left( \dfrac{nx}{b_n}\right) }\sum _{k=0}^{\infty } f\left( \dfrac{kb_n}{n}\right) \dfrac{(nx)^k}{b_n^k \Gamma (k+\beta )}, \end{aligned}$$
(2.2)

In particular, \(\beta =1\), the operators \(L_n^{1}(f)\) convert into famous modified Szasz-Mirakjan operators considered in [16]. The generalization of above Mittag-Leffle operators was studied in [17, 18].

We need following lemma for further discussion.

Lemma 1

[15] Let \(\phi _x^2\left( t\right) =\left( t-x\right) ^2\), for each \(x\ge 0\) and \(n,m \in \mathbb {N}\), we have

  1. 1.

    \( L_n^{(\beta )}(1;x)=1;\)

  2. 2.

    \(\left| L_n^{(\beta )}(t;x)-x\right| \le \dfrac{|1-\beta |}{n};\)

  3. 3.

    \( \left| L_n^{(\beta )}(t^2;x)-x^2\right| \le \dfrac{(2|1-\beta |+1)x}{n}+\dfrac{\left( 2(1-\beta )^2+|1-\beta |+|1-\beta ||\beta -2|\right) }{n^2}\).

By direct computations, one can state the following lemma;

Lemma 2

Let \(\phi _x^2\left( t\right) =\left( t-x\right) ^2\), for each \(x\ge 0\) and \(n,m \in \mathbb {N}\), we have

  1. 1.

    \( W_n^{(\beta )}(1;x)=1\);

  2. 2.

    \(\left| W_n^{(\beta )}(t;x)-x\right| \le \dfrac{|1-\beta |+1}{n}\);

  3. 3.

    \( \left| W_n^{(\beta )}(t^2;x)-x^2\right| \le \dfrac{2(|1-\beta |+2)}{n}x+ \dfrac{2|1-\beta |^2+|1-\beta |+4|1-\beta ||\beta -2|+2}{n^2}\);

  4. 4.

    \( W_n^{(\beta )}((t-x)^2;x)\le \dfrac{(4|1-\beta |+6)}{n}x+ \dfrac{2|1-\beta |^2+|1-\beta |+4|1-\beta ||\beta -2|+2}{n^2}\).

Proof

We note that

$$\begin{aligned} W_n^{(\beta )}(1;x)&=\dfrac{1}{E_{1,\beta }(nx)}\sum _{k=0}^{\infty }\dfrac{(nx)^k}{k! \Gamma (k+\beta )} \int _0^{\infty } e^{-t}t^{k} dt\\&= \dfrac{1}{E_{1,\beta }(nx)}\sum _{k=0}^{\infty }\dfrac{(nx)^k}{k! \Gamma (k+\beta )}\Gamma (k+1)=1. \end{aligned}$$

Now,

$$\begin{aligned} W_n^{(\beta )}(t;x)&=\dfrac{1}{E_{1,\beta }(nx)}\sum _{k=0}^{\infty }\dfrac{(nx)^k}{k! \Gamma (k+\beta )} \int _0^{\infty } e^{-t}t^{k} \dfrac{t}{n}dt\\&=\dfrac{1}{nE_{1,\beta }(nx)}\sum _{k=1}^{\infty } \dfrac{(nx)^{k}}{\Gamma (k+\beta )}\left[ k+\beta -1+1-\beta \right] +\dfrac{1}{nE_{1,\beta }(nx)}\sum _{k=0}^{\infty } \dfrac{(nx)^{k}}{\Gamma (k+\beta )}\\&=x+ \dfrac{1-\beta }{nE_{1,\beta }(nx)}\sum _{k=1}^{\infty } \dfrac{(nx)^{k}}{\Gamma (k+\beta )}+\dfrac{1}{n} \end{aligned}$$

Hence,

$$\begin{aligned} \left| W_n^{(\beta )}(t;x)-x\right| =\dfrac{|1-\beta |}{nE_{1,\beta }(nx)}\sum _{k=1}^{\infty } \dfrac{(nx)^{k}}{\Gamma (k+\beta )}+\dfrac{1}{n}\le \dfrac{|1-\beta |+1}{n}. \end{aligned}$$

Similarly, by \(k(k-1)=(k+\beta -1)(k+\beta -2)+2(1-\beta )k+(1-\beta )(\beta -2)\) and \(\Gamma (k+\beta )=(k+\beta -1)(k+\beta -2)\Gamma (k+\beta -2)\), we get

$$\begin{aligned} W_n^{(\beta )}(t^2;x)&=\dfrac{1}{E_{1,\beta }(nx)}\sum _{k=0}^{\infty }\dfrac{(nx)^k}{k! \Gamma (k+\beta )} \int _0^{\infty } e^{-t}t^{k} \left( \dfrac{t}{n}\right) ^2dt\\&=\dfrac{1}{n^2E_{1,\beta }(nx)}\sum _{k=0}^{\infty } \dfrac{(nx)^{k}}{k! \Gamma (k+\beta )}\Gamma (k+3)\\&=\dfrac{1}{n^2E_{1,\beta }(nx)}\sum _{k=0}^{\infty } \dfrac{(nx)^{k}}{\Gamma (k+\beta )}(k+2)(k+1)\\&=\dfrac{1}{n^2E_{1,\beta }(nx)}\sum _{k=0}^{\infty } \dfrac{(nx)^{k}}{\Gamma (k+\beta )}k^2 +\dfrac{3}{n^2E_{1,\beta }(nx)}\sum _{k=0}^{\infty } \dfrac{(nx)^{k}}{\Gamma (k+\beta )}k\\&\quad + \dfrac{2}{n^2E_{1,\beta }(nx)}\sum _{k=0}^{\infty } \dfrac{(nx)^{k}}{\Gamma (k+\beta )}\\&=L_n^{[\beta ]}(t^2,x)+\dfrac{3}{n}L_n^{[\beta ]}(t,x)+\dfrac{2}{n^2}L_n^{[\beta ]}(1,x). \end{aligned}$$

Now,

$$\begin{aligned} \left| W_n^{(\beta )}(t^2;x)-x^2 \right|&\le \left| L_n^{[\beta ]}(t^2,x)-x^2 \right| + \dfrac{3}{n}\left| L_n^{[\beta ]}(t,x)\right| +\dfrac{2}{n^2}\left| L_n^{[\beta ]}(1,x)\right| \\&=\dfrac{2(|1-\beta |+2)}{n}x+ \dfrac{2|1-\beta |^2+|1-\beta |+4|1-\beta ||\beta -2|+2}{n^2}. \end{aligned}$$

Finally,

$$\begin{aligned} W_n^{(\beta )}((t-x)^2;x)\le & {} \left| W_n^{(\beta )}(t^2;x)-x^2 \right| +2x\left| W_n^{(\beta )}(t;x)-x \right| +x^2 \left| W_n^{(\beta )}(1;x)-1 \right| \\\le & {} \dfrac{(4|1-\beta |+6)}{n}x+ \dfrac{2|1-\beta |^2+|1-\beta |+4|1-\beta ||\beta -2|+2}{n^2}. \end{aligned}$$

\(\square \)

Lemma 3

Let \(h \in C_B\left( [0,\infty )\right) \). Then, we have

$$\begin{aligned} \Vert W_n^{[\beta ]}(h)\Vert \le \Vert h\Vert . \end{aligned}$$

Proof

Using the definition of the newly introduced Wright operators involving Gamma function and the values obtained above, the following inequality is readily obtained.

$$\begin{aligned} \Vert W_n^{[\beta ]}(h)\Vert&\le e^{-nx}\sum _{k=0}^{\infty }\dfrac{(nx)^k}{k! \Gamma (k+\beta )} \int _0^{\infty } e^{-t}t^{k+\beta -1} \left| h\left( \dfrac{t}{n}\right) \right| dt,\\&\le \Vert h\Vert e^{-nx}\sum _{k=0}^{\infty }\dfrac{(nx)^k}{k! \Gamma (k+\beta )} \int _0^{\infty } e^{-t}t^{k+\beta -1} dt,\\&\le \Vert h\Vert W_n^{[\beta ]}(1,x)=\Vert h\Vert , \end{aligned}$$

which completes the proof. Now, we can present the central moments of the newly constructed operator that will be used in the main theorems of the paper as follows. \(\square \)

Lemma 4

Let \(x\in [0,\infty )\). In the circumstances, we obtained the following equalities for central moments:

  1. 1.

    \(W_n^{[\beta ]}(t-x,x)=\dfrac{\beta }{n}\).

  2. 2.

    \(W_n^{[\beta ]}((t-x)^2,x)= \dfrac{2x}{n}+\dfrac{\beta (\beta +1)}{n^2}\).

  3. 3.

    \(W_n^{[\beta ]}((t-x)^3,x)=\dfrac{6 x (1 + \beta )}{n^2} + \dfrac{\beta (1 + \beta ) (2 + \beta )}{n^3}\).

  4. 4.

    \(W_n^{[\beta ]}((t-x)^4,x)=\dfrac{\beta \left( \beta ^3+6 \beta ^2+11 \beta +6\right) }{n^4}+\dfrac{12 \left( \beta ^2+3 \beta +2\right) x}{n^3}+\dfrac{12 x^2}{n^2}\).

Theorem 1

Let \(h\in C_B\left( [0,\infty )\right) \). Then, we have

$$\begin{aligned} lim_{n\rightarrow \infty } W_n^{[\beta ]}(h,x)=h(x), \end{aligned}$$

for uniformly in each compact subsets of \([0,\infty )\).

Proof

With the aid of Lemma 2, one can easily obtain the following equality:

$$\begin{aligned} \lim _{n\rightarrow \infty } W_n^{[\beta ]}(t^k,x)=x^k, \end{aligned}$$

for uniformly in each compact subset of \([0,\infty )\) for \(k = 0, 1, 2\). Then, according to the result of the Bohmans-Korovkin theorem, we deduce the \(\displaystyle \lim _{n\rightarrow \infty } W_n^{[\beta ]}(h,x)=h(x)\) for uniformly in each compact subset of \([0,\infty )\). This completes the proof of the theorem. \(\square \)

The Asymptotic Formula

One of the fundamental challenges in approximation theory is the calculation of the rate of convergence of positive linear operators to the test functions. For this purpose, we will present and prove the Voronovskaya-type theorem to determine the asymptotic behaviour of newly constructed operators utilising well-recognised Taylor expansion.

Theorem 2

Let h be bounded and integrable on the interval \(x \in [0,\infty )\), \(h'\) and \(h''\) exist at a fixed point \(x\in [0,\infty )\), in this circumstance the following limit holds:

$$\begin{aligned} \lim _{n\rightarrow \infty } n\left[ W_n^{[\beta ]}(h,x)-h(x)\right] =\beta h(x)+x h''(x). \end{aligned}$$

Proof

First, starting with the well-recognised Taylor formula at \(t=x\) of function h, we readily deduce that

$$\begin{aligned} h(t)=h(x)+h'(x)(t-x)+\dfrac{1}{2}(t-x)^2 h''(x)+\chi (t,x)(t-x)^2, \end{aligned}$$
(3.1)

where

$$\begin{aligned} \chi (t,x)=\dfrac{h''(\xi )-h''(x)}{2} \end{aligned}$$

such that \(\xi \) lying between x and t and

$$\begin{aligned} \lim _{t\rightarrow x} \chi (t,x)=0. \end{aligned}$$

If we apply the new operator \(W_n^{[\beta ]}\) to equality 3.1, we easily obtained that,

$$\begin{aligned} W_n^{[\beta ]}(h,x)= & {} h(x)+h'(x)W_n^{[\beta ]}\left( (t-x),x\right) +\dfrac{1}{2}h''(x)W_n^{[\beta ]}\left( (t-x)^2,x\right) \\{} & {} +W_n^{[\beta ]}\left( \chi (t,x)(t-x)^2,x\right) . \end{aligned}$$

Multiplying each side of the equation here by n will result in the following equality:

$$\begin{aligned} n\left[ W_n^{[\beta ]}(h,x)-h(x)\right]= & {} h'(x)n W_n^{[\beta ]}\left( (t-x),x\right) +\dfrac{1}{2}h''(x)nW_n^{[\beta ]}\left( (t-x)^2,x\right) \\{} & {} +nW_n^{[\beta ]}\left( \chi (t,x)(t-x)^2,x\right) . \end{aligned}$$

If one states this expression in the limit case, we deduce that

$$\begin{aligned}&\lim _{n\rightarrow \infty }n\left[ W_n^{[\beta ]}(h,x)-h(x)\right] =h'(x)\lim _{n\rightarrow \infty } n W_n^{[\beta ]}\left( (t-x),x\right) \\&\quad +\dfrac{1}{2}h''(x)\lim _{n\rightarrow \infty }nW_n^{[\beta ]}\left( (t-x)^2,x\right) +\lim _{n\rightarrow \infty }nW_n^{[\beta ]}\left( \chi (t,x)(t-x)^2,x\right) . \end{aligned}$$

As a consequence of our previous calculations in Lemma 2, the following two expressions can be easily obtained:

$$\begin{aligned} \lim _{n\rightarrow \infty }nW_n^{[\beta ]}(t-x,x)=\beta ~~~~~\lim _{n\rightarrow \infty }nW_n^{[\beta ]}((t-x)^2,x)=2x. \end{aligned}$$

Then, the following is obtained when we replace the information we have obtained above

$$\begin{aligned} \lim _{n\rightarrow \infty }n\left[ W_n^{[\beta ]}(h,x)-h(x)\right]&=\beta h'(x)+xh''(x) +\lim _{n\rightarrow \infty }nW_n^{[\beta ]}\left( \chi (t,x)(t-x)^2,x\right) . \end{aligned}$$
(3.2)

Finally, if we show,

$$\begin{aligned} \lim _{n\rightarrow \infty }nW_n^{[\beta ]}\left( \chi (t,x)(t-x)^2,x\right) =0, \end{aligned}$$

we can conclude the proof.

Hence, by applying the well-known Cauchy-Schwarz inequality, we obtain

$$\begin{aligned} nW_n^{[\beta ]}\left( \chi (t,x)(t-x)^2,x\right) \le \sqrt{n^2W_n^{[\beta ]}\left( \chi ^2(t,x),x\right) }\cdot \sqrt{nW_n^{[\beta ]}\left( (t-x)^4,x\right) }. \end{aligned}$$
(3.3)

Then, with the help of the Korovkin theorem, we can deduce that,

$$\begin{aligned} \lim _{n\rightarrow \infty }W_n^{[\beta ]}\left( \chi ^2(t,x),x\right) =\chi ^2(x,x)=0, \end{aligned}$$
(3.4)

since \(\chi ^2(x, x) = 0\) and \(\chi ^2(\cdot , x)\) is continuous at \(t\in [0,\infty )\)] and bounded as \(t\rightarrow \infty \) and as \(W_n^{[\beta ]}\left( (t-x)^4,x\right) = O(n^{-4})\). As a result, by substituting (3.3) and (3.4) into (3.2), the proof is completed. \(\square \)

Weighted Approximation

After the computation of asymptotic formulae of the introduced operator, now we discuss the Korovkin-type theorem for a weighted approximation. For this purpose, we benefit from the results presented by Gadjiev in [19].

Initially, set \(\rho (x) = 1 + x^2\) as a weight function that is continuous on \(\mathbb {R}\) and the \(\lim _{|x|\rightarrow \infty } \rho (x) = \infty \), \(\rho (x)\ge 1\) for all \(x \in [0,\infty )\). Then, we shall denote by \(C([0,\infty ))\) the set of all \([0,\infty )\rightarrow \mathbb {R}\) functions that are continuous. Then let us consider the following weighted spaces. For all \(x \in [0,\infty )\), the weighted space of real-valued functions h described on \(\mathbb {R}\) with the property \(|h(x)| \le M_h\rho (x)\), where \(M_h\) is a constant depending on the function h defined as

$$\begin{aligned} B_{\rho }\left( [0,\infty )\right) =\left\{ h:[0,\infty )\rightarrow \mathbb {R}: |h(x)| \le M_h \rho (x), x\in [0,\infty )\right\} \end{aligned}$$

and

$$\begin{aligned} C_{\rho }\left( [0,\infty )\right) =\left\{ h\in B_{\rho }\left( [0,\infty )\right) : h \text { is continuous on } \mathbb {R}\right\} =C\left( [0,\infty )\right) \cap B_{\rho }\left( [0,\infty )\right) . \end{aligned}$$

These spaces are normed spaces with

$$\begin{aligned} \Vert h\Vert _{\rho }=\sup _{x\in [0,\infty )} \dfrac{|h(x)|}{\rho (x)}. \end{aligned}$$

Since \(\rho \) is a weight function, \(B_{\rho }\left( [0,\infty )\right) \) and \(C_{\rho }\left( [0,\infty )\right) \) spaces are called weighted spaces. Additionally, if we set that \(\kappa _h\) is a constant dependent on the function h, we can define the following subspace:

$$\begin{aligned} C_{\rho }^{\kappa }\left( [0,\infty )\right) =\left\{ h\in C_{\rho }\left( [0,\infty )\right) : \lim _{|x|\rightarrow \infty } \dfrac{h(x)}{\rho (x)}=\kappa _h \text { exists and it is finite } \right\} , \end{aligned}$$

which is a subspace of space \(C_{\rho }\left( [0,\infty )\right) \). Now, we can provide the following lemma for the new operators.

Lemma 5

Let \(h\in C_{\rho }\left( [0,\infty )\right) \). Then, the following inequality holds

$$\begin{aligned} \Vert W_n^{[\beta ]}\left( h\right) \Vert _{\rho }\le C\Vert h\Vert _{\rho }, \end{aligned}$$

for the operator \(W_n^{[\beta ]}\), which means that the sequence of the Wright operators based on Gamma function \(W_n^{[\beta ]}\) is an approximation process from \(C_{\rho }\left( [0,\infty )\right) \) to \(C_{\rho }\left( [0,\infty )\right) \).

Proof

This lemma can be readily proven by using the definition of operators and the results of 2. Thus, the desired result has been obtained.

Now, we can present and prove the main theorem of this section by following Gadjiev’s technique for an unbounded interval. \(\square \)

Theorem 3

Let \(g\in C_{\rho }^{\kappa }\left( [0,\infty )\right) \). Then, for following equality holds:

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert W_n^{[\beta ]}(f)-f\Vert _{\rho }=0, \end{aligned}$$

for the Wright operators involving Gamma function.

Proof

Utilising Gadjiev’s [19] theorem, it suffices to demonstrate that \(\displaystyle \lim \nolimits _{n\rightarrow \infty }\Vert W_n^{[\beta ]}(t^i)-t^i\Vert _{\rho }= 0\) holds for \(i = 0, 1, 2\). It is clear that the equation for \(k = 0\),which is \(W_n^{[\beta ]}(1,x)=1\) is initially provided. Secondly, using the result of Lemma 2 for \(i = 1\), we readily deduce that,

$$\begin{aligned} \Vert W_n^{[\beta ]}(t)-t\Vert _{\rho }&=\sup _{x\in [0,\infty )}\dfrac{|W_n^{[\beta ]}(t,x)-x|}{1+x^2}\\&=\sup _{x\in [0,\infty )}\dfrac{\left| \dfrac{\beta }{n}\right| }{1+x^2}\le \dfrac{\beta }{n}. \end{aligned}$$

If we take the limit of the above findings, one can readily express that \(\displaystyle \lim \nolimits _{n\rightarrow \infty }\Vert W_n^{[\beta ]}(t)-t\Vert _{\rho }=0\) as \(\displaystyle \lim \nolimits _{n\rightarrow \infty }\dfrac{1}{n}=0\). Finally, we need to find an upper bound of \(\lim _{n\rightarrow \infty } \Vert W_n^{[\beta ]}(t^2)-t^2\Vert _{\rho }\). For that, we have

$$\begin{aligned} \Vert W_n^{[\beta ]}(t^2)-t^2\Vert _{\rho }&=\sup _{x\in [0,\infty )}\dfrac{|W_n^{[\beta ]}(t^2,x)-x^2|}{1+x^2}\\&=\sup _{x\in [0,\infty )}\dfrac{\left| \dfrac{2(\beta +1)x}{n}+\dfrac{\beta (\beta +1)}{n^2}\right| }{1+x^2}\le \dfrac{2(\beta +1)}{n}+\dfrac{\beta (\beta +1)}{n^2} \end{aligned}$$

is obtained in a similar way. In the limit case, we have the desired results, which concludes the proof. \(\square \)

Rate of Convergence

In this section, we provide the convergence rate of the Wright Operators involving Gamma function in terms of the modulus of continuity. Here, for the closed interval \([0, x_0]\), \(x_0\ge 0\), we denote the standard modulus of continuity of h by \(\omega _{x_0}(h, \delta )\) and it can be defined as follows:

$$\begin{aligned} \omega _{x_0}(h,\delta )=\sup _{|t-x|\le \delta , x,t\in [0,x_0]} |g(t)-g(x)|. \end{aligned}$$

It is obvious that the modulus of continuity \(\omega _{x_0} (h, \delta ) \rightarrow 0\) as \(\delta \rightarrow 0\) for the function \(g \in C_b[0,\infty )\). Let us show the corresponding rate of convergence theorem for the newly constructed Wright operator involving Gamma function \((W_n^{[\beta ]})_{n\ge 1}\). Now, we can provide the main theorem of this section.

Theorem 4

Let \(\omega _{x_0}(h,\delta )\) be the modulus of continuity on the finite interval \([0,x_0+1]\subset [0,\infty )\). In the circumstances, the following inequality holds:

$$\begin{aligned} \left| W_n^{[\beta ]}(h,x)-h(x)\right|{} & {} \le 3M_h\left( \dfrac{2x_0n+\beta (\beta +1)}{n^2}\right) (1+x_0^2)\\{} & {} \quad \ +2\omega _{x_0+1} \left( h,\sqrt{\dfrac{2x_0n+\beta (\beta +1)}{n^2}}\right) , \end{aligned}$$

where \(M_h\) is fixed just depending on h.

Proof

Now, let \(h\in C_B\left( [0,\infty )\right) \), \(0\le x\le x_0\) and \(t>x_0+1\). Then, we can deduce that

$$\begin{aligned} |h(t)-h(x)|&\le |h(t)|+|h(x)|\\&\le M_h\left( 2+t^2+x^2\right) \\&= M_h\left( (t-x)^2+2x(t-x)+2+2x^2\right) \\&\le M_h\left( (t-x)^2+2x(t-x)^2+2(t-x)^2+2x^2(t-x)^2\right) \\&=M_h (t-x)^2(2x^2+2x+3)\\&\le M_h(t-x)^2\left( 3x_0^2+6x_0+3\right) \\&=3M_h(t-x)^2(1+x_0)^2 \end{aligned}$$

for \(t-x>1\). Then, again let \(h\in C_B\left( [0,\infty )\right) \), \(0\le x\le x_0\). In the circumstances, the following inequality holds:

$$\begin{aligned} |h(t)-h(x)|&\le \omega _{x_0+1} \left( h,|t-x|\right) \\&\le \omega _{x_0+1} \left( h,\delta \right) \left( 1+\dfrac{1}{\delta }|t-x|\right) \end{aligned}$$

for \(t\le x_0 + 1\). As a consequences, from the above inequalities, we deduce that

$$\begin{aligned} |h(t)-h(x)|\le 3M_h(t-x)^2(1+x_0)^2+\omega _{x_0+1} \left( h,\delta \right) \left( 1+\dfrac{1}{\delta }|t-x|\right) , \end{aligned}$$
(5.1)

for \(0\le x\le x_0\) and \(0\le t<\infty \). Applying \(W_n^{[\beta ]}\) and the Cauchy-Schwarz inequality to (5.1), we obtain

$$\begin{aligned} |W_n^{[\beta ]}\left( h,x\right) -h(x)|&\le 3M_hW_n^{[\beta ]}\left( (t-x)^2,x\right) (1+x_0)^2+\omega _{x_0+1} \left( g,\delta \right) \\&\left( 1+\dfrac{1}{\delta }\sqrt{W_n^{[\beta ]}\left( (t-x)^2,x\right) }\right) ,\\&\le 3M_h\left( \dfrac{2x_0n+\beta (\beta +1)}{n^2}\right) (1+x_0^2)\\&\quad \ +2\omega _{x_0+1} \left( h,\sqrt{\dfrac{2x_0n+\beta (\beta +1)}{n^2}}\right) , \end{aligned}$$

by choosing \(\delta =\sqrt{\dfrac{2x_0n+\beta (\beta +1)}{n^2}}\), which completes the proof. \(\square \)

Pointwise Estimate

In this section, let us examine some pointwise estimates of the rates of convergence of the newly defined Wright operators involving Gamma function. First, the local approximation and the relationship between the local smoothness of h are given. For that, let us describe the following. Let \(s\in (0,1]\) and \(Q \subset [0,\infty )\). In the circumstances, a function \(h\in C_B[0,\infty )\) can be said \(Lip_{M_h}(s)\) on Q if the following condition holds:

$$\begin{aligned} \left| h(t)-h(x)\right| \le M_{h,s}|t-x|^s, t\in [0,\infty ) \text { and } x\in Q, \end{aligned}$$

where \(M_{h,s}\) is fixed just depending on h and s.

Theorem 5

Let \(h\in C_B\left( [0,\infty )\right) \cap Lip_{M_g}(s)\) such that \(s\in (0,1]\) and \(Q\subset [0,\infty )\) given as above. In the circumstances, we have the following inequality:

$$\begin{aligned} |W_n^{[\beta ]}\left( h,x\right) -h(x)|\le M_{h,s}\left[ \left( \dfrac{2x_0n+\beta (\beta +1)}{n^2}\right) ^{s/2}+2\left( d(x,Q)\right) ^s\right] , x\in (0,\infty ), \end{aligned}$$

where \(M_{h,s}\) is defined as above and d(x, Q) is the distance between x and Q described as

$$\begin{aligned} d(x,Q)=\inf \{|t-x|: t\in Q\}. \end{aligned}$$

Proof

Let us describe the closure of the set Q as \(\bar{Q}\). Then, one can say that there exists at least one point \(y_0 \in Q\) such that

$$\begin{aligned} d(x,Q)=|x-t_0|. \end{aligned}$$

Then, utilising the monotonicity properties of \(\left( W_n^{[\beta ]}\right) _{n\ge 1}\), we deduce that

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right|&\le W_n^{[\beta ]}\left( |h(t)-h(t_0)|,x\right) +W_n^{[\beta ]}\left( |h(x)-h(t_0)|,x\right) \\&\le M_{h,s} \left[ W_n^{[\beta ]}\left( |t-t_0|^s,x\right) +|x-t_0|^s\right] \\&\le M_{h,s}\left[ W_n^{[\beta ]}\left( |t-t_0|^s,x\right) +2|x-t_0|^s\right] . \end{aligned}$$

In the circumstances,with the help of the Hölder inequality, we obtain the following result:

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right|&\le M_{h,s}\left[ \left( W_n^{[\beta ]}\left( |t-t_0|^2,x\right) \right) ^{s/2}+2\left( d(x,Q)\right) ^s\right] \\&\le M_{h,s}\left[ \left( \dfrac{2xn+\beta (\beta +1)}{n^2}\right) ^{s/2}+2\left( d(x,Q)\right) ^s\right] , \end{aligned}$$

which finalises the proof. \(\square \)

Let us now calculate the local direct estimate of the Wright function involving Gamma function. For this purpose, we need to review the Lipschitz-type maximal function of order s given in [20], that is

$$\begin{aligned} \widetilde{\omega }_s(g,x)=\sup _{0\le t <\infty , t\ne x} \dfrac{|h(t)-h(x)|}{|t-x|^s}, \end{aligned}$$

where \(s\in (0, 1]\) and \(x\in (0,\infty )\). Now, we can present and prove the theorem.

Theorem 6

Let \(h\in C_B[0,\infty )\) and \(s \in (0, 1]\), then the following inequality holds:

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right| \le \widetilde{\omega }_s(h,x)\left( \dfrac{2xn+\beta (\beta +1)}{n^2}\right) ^{s/2}. \end{aligned}$$

Proof

Using the definition of \(\widetilde{\omega }_s(h,x)\) give above and well-recognised Hölder inequality, we obtain that

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right|&\le W_n^{[\beta ]}\left( \left| h(t)-h(x)\right| ,x\right) \\&\le \widetilde{\omega }_s(h,x) W_n^{[\beta ]}\left( |t-x|^s,x\right) \\&\le \widetilde{\omega }_s(h,x)W_n^{[\beta ]}\left( |t-x|^2,x\right) ^{s/2}\\&\le \widetilde{\omega }_s(h,x)\left( \dfrac{2xn+\beta (\beta +1)}{n^2}\right) ^{s/2} \end{aligned}$$

thus, the desired result is obtained.

Finally, let us consider the following Lipschitz-type space with two parameters, \(\alpha ,\beta >0\), such that

$$\begin{aligned} Lip_M^{\alpha ,\beta }(s)=\left\{ h\in C[0,\infty ): |h(t)-h(x)|\le M \dfrac{|t-x|^s}{(ax^2+bx+t)^{s/2}}, x,t\in (0,\infty )\right\} \end{aligned}$$

introduced in [15], where \(s\in (0,1]\) and M is a positive constant. \(\square \)

Theorem 7

Let us consider \(h\in Lip_M^{\alpha ,\beta }(s)\) and \(x\in (0,\infty )\). Then we have

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right| \le M \left[ \dfrac{2xn+\beta (\beta +1)}{n^2(ax^2+bx)}\right] ^{s/2}, \end{aligned}$$

where \(\alpha ,\beta >0\).

Proof

The proof of this inequality is shown in two steps. First, we take \(s = 1\), that is,

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right|&\le W_n^{[\beta ]}\left( \left| h(t)-h(x)\right| ,x\right) \\&\le M W_n^{[\beta ]}\left( \dfrac{|t-x|}{\sqrt{ax^2+bx+t}},x\right) \\&\le \dfrac{M}{\sqrt{ax^2+bx}}W_n^{[\beta ]}\left( \left| t-x\right| ,x\right) , \end{aligned}$$

\(g\in Lip_M^{\alpha ,\beta }(1)\) and \(x\in (0,\infty )\). Here, applying the Chauchy-Schwarz inequality, we deduce that

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right| \le \dfrac{M}{\sqrt{ax^2+bx}}\left[ W_n^{[\beta ]}\left( \left| t-x\right| ^2,x\right) \right] ^{1/2}\le \left[ \dfrac{2xn+\beta (\beta +1)}{n^2(ax^2+bx)}\right] ^{1/2}, \end{aligned}$$

which confirms the proof of the theorem for \(s = 1\). Then, let us consider \(s\in (0,1)\). For \(g\in Lip_M^{\alpha ,\beta }(s)\) and \(x\in (0,\infty )\), we obtain that

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right| \le \dfrac{M}{\left( ax^2+bx\right) ^{s/2}}W_n^{[\beta ]}(|t-x|^s,x). \end{aligned}$$

With the help of Hölder inequalities, we obtain the following inequality:

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right| \le \dfrac{M}{\left( ax^2+bx\right) ^{s/2}}W_n^{[\beta ]}(|t-x|^s,x)\le \dfrac{M}{\left( ax^2+bx\right) ^{s/2}} \left( W_n^{[\beta ]}(|t-x|,x)\right) ^s. \end{aligned}$$

Finally, applying the Cauchy-Schwarz inequality, we have,

$$\begin{aligned} \left| W_n^{[\beta ]}\left( h,x\right) -h(x)\right| \le \dfrac{M}{\left( ax^2+bx\right) ^{s/2}}\left( W_n^{[\beta ]}(|t-x|^2,x)\right) ^{s/2}\le M\left[ \dfrac{2xn+\beta (\beta +1)}{n^2(ax^2+bx)}\right] ^{s/2}, \end{aligned}$$

which completes the proof. \(\square \)

Conclusion

We investigated a novel class of operators incorporating three special functions: the Gamma function, Mittag-Leffler function, and Wright function. Our results include establishing a relationship between the local smoothness of functions and local approximation, determining the local rate of convergence, and analysing the asymptotic behaviour of this new sequence of operators. We demonstrate that this class is effective for approximating continuous signals on unbounded intervals. Additionally, the admissible values of the involved parameters enable us to make optimal choices, resulting in improved estimations.