# Approximation by Bicomplex Favard-Szasz-Mirakjan Operators
## 3 Construction of Bicomplex Favard-Szasz-Mirakjan Operators  
In this section, we aim to introduce the approximation properties of Favard-Szasz-Mirakjan operators for bicomplex numbers. We begin by defining bicomplex Szasz-Mirakjan operators and obtaining a quantitative upper estimate for these operators and their derivatives on a compact \(\mathbb{C}_{2}\) disk. Later, we shall prove a qualitative Voronovskaja-type result for these operators. In other words, our main goal is to extend the concept of Szasz-Mirakjan operators to a bicomplex version of themselves, allowing for the approximation of functions defined on a compact interval. Let \(g:\mathbf{B}(0,r_{1})\times\mathbf{B}(0,r_{2})\longrightarrow\mathbb{C}_{2}\) be a bicomplex-valued function; then, the bicomplex Szasz-Mirakjan operator \(K_{n}(. )\) (\(n\) is a positive integer) of \(g\) is defined as  
\[(K_{n}g)(t_{1}+i_{2}t_{2})=e^{-n(t_{1}+i_{2}t_{2})}\sum_{k=0}^{\infty}\frac{ \left(n(t_{1}+i_{2}t_{2})\right)^{k}}{k!}g\left(\frac{k}{n}\right), \tag{10}\]  
where \(k=0,1,2\ldots\) and for \(t_{1}+i_{2}t_{2}\in\mathbb{C}_{2}\). Analysis of Approximation PropertiesLemma 1: _Let \(e_{k}(t_{1}+i_{2}t_{2})=(t_{1}+i_{2}t_{2})^{k}\) and \(T_{n,k}(t_{1}+i_{2}t_{2})=(K_{n},e_{k})(t_{1}+i_{2}t_{2})\) be for \(k=0,1,2\ldots\); then, the recurrence formula is defined as_  
\[T_{n,k+1}(t_{1}+i_{2}t_{2}) = \frac{(t_{1}+i_{2}t_{2})}{n}D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+ i_{2}t_{2}) \tag{11}\] \[+(t_{1}+i_{2}t_{2})T_{n,k}(t_{1}+i_{2}t_{2}),\]  
_for all \(n\in\mathbb{N}\), \((t_{1}+i_{2}t_{2})\in\mathbb{C}_{2}\)._  
Proof of Lemma 1.: Let us write that \(T_{n,0}(t_{1}+i_{2}t_{2})=1\), \(T_{n,1}(t_{1}+i_{2}t_{2})=t_{1}+i_{2}t_{2}\) for \(k=0,1\). Furthermore, for any \(k\), we have  
\[T_{n,k}(t_{1}+i_{2}t_{2}) =K_{n}(e_{k};t_{1}+i_{2}t_{2})\] \[=\bigg{(}e^{-n(t_{1}-i_{1}t_{2})}\sum_{j=0}^{\infty}\frac{(n(t_{1}- i_{1}t_{2}))^{j}}{j!}\bigg{(}\frac{j}{n}\bigg{)}^{k}\bigg{)}e_{1}\] \[+\bigg{(}e^{-n(t_{1}+i_{1}t_{2})}\sum_{j=0}^{\infty}\frac{(n(t_{1} +i_{1}t_{2}))^{j}}{j!}\bigg{(}\frac{j}{n}\bigg{)}^{k}\bigg{)}e_{2}.\]  
Accordingly, differentiating \(T_{n,k}(t_{1}+i_{2}t_{2})\) with respect to \(t_{1}+i_{2}t_{2}\
eq 0\), we have  
\[D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+i_{2}t_{2}) =D_{t_{1}-i_{1}t_{2}}\bigg{[}\bigg{(}e^{-n(t_{1}-i_{1}t_{2})}\sum _{j=0}^{\infty}\frac{(n(t_{1}-i_{1}t_{2}))^{j}}{j!}\bigg{(}\frac{j}{n}\bigg{)} ^{k}\bigg{)}e_{1}\bigg{]}\] \[+D_{t_{1}+i_{1}t_{2}}\bigg{[}\bigg{(}e^{-n(t_{1}+i_{1}t_{2})}\sum _{j=0}^{\infty}\frac{(n(t_{1}+i_{1}t_{2}))^{j}}{j!}\bigg{(}\frac{j}{n}\bigg{)} ^{k}\bigg{)}e_{2}\bigg{]}\]  
for \(k=0,1,\ldots\). Employing the basic calculations, we obtain the following result:  
\[D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+i_{2}t_{2}) =\bigg{(}-ne^{-n(t_{1}-i_{1}t_{2})}\sum_{j=0}^{\infty}\frac{(n(t_ {1}-i_{1}t_{2}))^{j}}{j!}\bigg{(}\frac{j}{n}\bigg{)}^{k}\] \[+e^{-n(t_{1}-i_{1}t_{2})}\sum_{j=0}^{\infty}\frac{jn(n(t_{1}-i_{1 }t_{2}))^{j-1}}{j!}\bigg{(}\frac{j}{n}\bigg{)}^{k}\bigg{)}e_{1}\] \[T_{n,k+1}(t_{1}+i_{2}t_{2}) = \frac{(t_{1}+i_{2}t_{2})}{n}D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+ i_{2}t_{2}) \tag{11}\]0 \[T_{n,k+1}(t_{1}+i_{2}t_{2}) = \frac{(t_{1}+i_{2}t_{2})}{n}D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+ i_{2}t_{2}) \tag{11}\]1  
for \(n\in\mathbb{N}\). As a conclusion, we can write that  
\[T_{n,k+1}(t_{1}+i_{2}t_{2}) = \frac{(t_{1}+i_{2}t_{2})}{n}D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+ i_{2}t_{2}) \tag{11}\]2 \[T_{n,k+1}(t_{1}+i_{2}t_{2}) = \frac{(t_{1}+i_{2}t_{2})}{n}D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+ i_{2}t_{2}) \tag{11}\]3  
which implies  
\[T_{n,k+1}(t_{1}+i_{2}t_{2}) = \frac{(t_{1}+i_{2}t_{2})}{n}D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+ i_{2}t_{2}) \tag{11}\]4 \[T_{n,k+1}(t_{1}+i_{2}t_{2}) = \frac{(t_{1}+i_{2}t_{2})}{n}D_{t_{1}+i_{2}t_{2}}(T_{n,k})(t_{1}+ i_{2}t_{2}) \tag{11}\]5  
for all \((t_{1}+i_{2}t_{2})\in\mathbf{B}(0,R_{1})\times\mathbf{B}(0,R_{2})\) with \(1<R_{1},R_{2}<\infty\), \(k=0,1,\ldots\) and \(n\in\mathbb{N}\). Lemma 2: _Let us define \(e_{k}\) and \(T_{n,k}\) as stated in Lemma 1.