## 2 Artificial Orthogonal Gland (AOG) Mechanism
### Activation Mechanism
#### 2.3.1 Gegenbauer (Ultra Sphere) Polynomials  
The Gegenbauer polynomials represent a generalization of the classic Legendre polynomials that will be denoted with \(C_{r}^{\lambda}\), where r is an integer and \(\lambda\) is a real number. These polynomials represent the exceptional cases of Jacobi polynomials, and the following relation defines them:  
\[\left(1-2 tx+x^{2}\right)^{-\lambda}=\sum_{r=0}^{\infty}C_{r}^{\lambda}(t)x^{r} \tag{6}\]  
By introducing binomial expansion of the left side of Equation (6), the previous equation can be presented as:  
\[\left(1-(2t-x)x\right)^{-\lambda}=\sum_{n=0}^{\infty}\frac{\left(\lambda \right)_{n}x^{n}(2t-x)^{n}}{n!}=\sum_{n=0}^{\infty}\sum_{m=0}^{n}\frac{\left(- 1\right)^{m}\left(\lambda\right)_{n}(2t)^{n-m}x^{n+m}}{m!(n-m)!} \tag{7}\]  
If \(k=m\geq 0\) and \(r-k=n\), then \(n-m=r-2k\) and \(k\) can vary in the range from 0 to \([r]\), where:  
\[[r]=\left\{\begin{array}{l}\frac{r}{r},\,\,\text{if $r$ is even}\\ \frac{r-1}{2},\,\,\text{if $r$ is odd}\end{array}\right. \tag{8}\]  
Hence, it can be further concluded:  
\[\left(1-(2t-x)x\right)^{-\lambda}=\sum_{r=0}^{\infty}\sum_{k=0}^{[r]}\frac{ \left(-1\right)^{k}\left(\lambda\right)_{r-k}(2t)^{r-2k}}{k!(r-2k)!}x^{r} \tag{9}\]  
Finally, by comparing Equation (9) with Equation (6), it can be obtained:  
\[C_{r}^{\lambda}(t)=\sum_{k=0}^{[r]}\frac{\left(-1\right)^{k}\left(\lambda \right)_{r-k}(2t)^{r-2k}}{k!(r-2k)!} \tag{10}\]  
In the second part of proposing a suitable mathematical apparatus, the equality \(\left[t_{0},t_{f}\right]=[-1,1]\) is specified (for the orthogonality interval), in addition to condition \(\lambda>-1/2\) (as a special case of Jacobi polynomials obtained by setting \(\alpha=\beta=\lambda-1/2\)). After a complex mathematical calculation from [27; 28] the following series is obtained:  
\[\sum_{r=0}^{\infty}C_{r}^{\lambda}(t)x^{r}=\sum_{r=0}^{\infty}\sum_{m=0}^{r} \frac{\left(2\lambda\right)_{r+m}}{m!(r-m)!\left(\lambda+\frac{1}{2}\right)_{ m}}\frac{(t-1)^{m}}{2^{m}}x^{r} \tag{11}\]  
where symbol \(\left(2\lambda\right)_{2k}\) denotes \(\left(2\lambda\right)_{2k}=2^{2k}\left(\lambda\right)_{k}\left(\lambda+1/2 \right)_{k}\), i.e., \(\left(\lambda\right)_{k}=\lambda(\lambda+1)(\lambda+2)\cdots\)\(\left(\lambda+k-1\right)\). A recurrence relation can be presented as [28]:  
\[nC_{n}^{\lambda}(x)=2(n+\lambda-1)xC_{n-1}^{\lambda}(x)-(n+2\lambda-2)C_{n-2}^ {\lambda}(x) \tag{12}\]In order to obtain a basis which will represent the functions of activation mechanisms of artificial glands and activation functions of future OENN, the following function is introduced:  
\[\varphi_{j}(x,\lambda)=\sum_{m=0}^{r}\frac{\left(2\lambda\right)_{r+m}}{m!(r-m)! \left(\lambda+\frac{1}{2}\right)_{m}}\frac{\left(x-1\right)^{m}}{2^{m}},j=0,1,2,\ldots \tag{13}\]  
Finally, the Gegenbauer series of the polynomials applicable for implementations within the proposed intelligent structure are:  
\[\begin{array}{l}\varphi_{0}(x,\lambda)=1,\\ \varphi_{1}(x,\lambda)=2\lambda x,\\ \varphi_{2}(x,\lambda)=2\lambda(1+\lambda)x^{2}-\lambda,\\ \varphi_{3}(x,\lambda)=\frac{4}{3}\lambda(1+\lambda)(2+\lambda)x^{3}-2\lambda (1+\lambda)x,\\ \vdots\\ \varphi_{n-1}(x,\lambda)=\frac{2(n+2)}{n-1}x\varphi_{n-2}(x,\lambda)-\frac{n +5}{n-1}\varphi_{n-3}(x,\lambda)\end{array} \tag{14}\]  
For \(n\) glands, the activation mechanism within each gland (1 to \(n\)) will be calculated as follows:  
\[\begin{array}{l}Act_{\text{deg},1}(t)=\varphi_{0}(CM_{out\_1},SM_{out\_1}) \\ Act_{\text{deg},2}(t)=\varphi_{1}(CM_{out\_2},SM_{out\_2})\\ \cdot\\ \cdot\\ Act_{\text{deg},n}(t)=\varphi_{n-1}(CM_{out\_n-1},SM_{out\_n-1})\end{array} \tag{15}\]  
where \(CM_{out\_i}\) and \(SM_{out\_i}\) are appropriate inputs to the \(i\)-th gland.