**3. Operational Matrices**

**Theorem 3.** *Let φ<sup>n</sup>* = [Ψ0(*x*), Ψ1(*x*),..., Ψ*n*(*x*)]*<sup>T</sup> be a Shifted Jacobi vector and suppose v* > 0*; then*

$$I^{\overline{v}}\Psi\_i(\mathbf{x}) = I^{(\overline{v})}\Phi\_\mathbf{n}(\mathbf{x})$$

*where <sup>I</sup>*(*v*) <sup>=</sup> (*μ*(*i*, *<sup>j</sup>*)) *is an* (*<sup>n</sup>* <sup>+</sup> <sup>1</sup>) <sup>×</sup> (*<sup>n</sup>* <sup>+</sup> <sup>1</sup>) *operational matrix of the fractional integral of order <sup>v</sup> and its* (*i*, *j*)*th entry is given by*

$$\mu(i,j) = \sum\_{k=0}^{i} \sum\_{l=0}^{j} (-1)^{i+j-k-l} \frac{\Gamma(a+1)\Gamma(i+b+1)\Gamma(i+k+a+b+1)\Gamma(j+l+a+b+1)\Gamma(a+k+l+a+b+1)(2j+a+b+1))!}{(i-k)!(j-l)!(l)!\Gamma(k+b+1)\Gamma(i+a+b+1)\Gamma(\nu+k+1)\Gamma(j+a+1)\Gamma(l+b+1)\Gamma(k+l+\nu+a+b+1)},\tag{13}$$

**Proof .** We refer to reference [44] for the proof.

Now, in particular cases, the operational matrix of integration for various polynomials is given as follows.

For Shifted Legendre polynomials (S1), the (*i*, *j*)th entry of the operational matrix of integration is given as

$$\mu(i,j) = \sum\_{k=0}^{i} \sum\_{l=0}^{j} (-1)^{i+j+k+l} \frac{(i+k)!(j+l)!}{(i-k)!(j-l)!(k)!(l!)^2(a+k+l+1)\Gamma(a+k+l)}.\tag{14}$$

For Shifted Chebyshev polynomials of the first kind (S2), the (*i*, *j*)th entry of the operational matrix of integration is given as:

$$\mu(i,j) = \sum\_{k=0}^{i} \sum\_{l=0}^{j} (-1)^{i+j-k-l} \frac{\Gamma\left(\frac{3}{2}\right) \Gamma\left(i+\frac{3}{2}\right) \Gamma(i+k+2) \Gamma(j+l+2) \Gamma\left(a+k+l+\frac{3}{2}\right) (2j+2) \text{j!}}{(i-k)! (j-l)! (l)! \Gamma\left(k+\frac{3}{2}\right) \Gamma(i+2) \Gamma(a+k+1) \Gamma\left(j+\frac{3}{2}\right) \Gamma\left(l+\frac{3}{2}\right) \Gamma(k+l+a+3)},\tag{15}$$

For Shifted Chebyshev polynomials of the third kind (S3), the (*i*, *j*)th entry of the operational matrix of integration is given as

$$\mu(i,j) = \sum\_{k=0}^{i} \sum\_{l=0}^{j} (-1)^{i+j-k-l} \frac{\Gamma\left(\frac{3}{2}\right) \Gamma\left(i+\frac{1}{2}\right) \Gamma(i+k+1) \Gamma(j+l+1) \Gamma\left(a+k+l+\frac{1}{2}\right) (2j+1) \text{j!}}{(i-k)! (j-l)! (l)! \Gamma\left(k+\frac{1}{2}\right) \Gamma(i+1) \Gamma(a+k+1) \Gamma\left(j+\frac{3}{2}\right) \Gamma\left(l+\frac{1}{2}\right) \Gamma(k+l+a+2)}. \tag{16}$$

For Shifted Chebyshev polynomials of the fourth kind (S4), the (*i*, *j*)th entry of the operational matrix of integration is given as

$$\mu(i,j) = \sum\_{k=0}^{i} \sum\_{l=0}^{j} (-1)^{i+j-k-l} \frac{\Gamma\left(\frac{1}{2}\right) \Gamma\left(i+\frac{3}{2}\right) \Gamma(i+k+1) \Gamma(j+l+1) \Gamma\left(a+k+l+\frac{9}{2}\right) (2j+1) \text{j!}}{(i-k)! (j-l)! (l)! \Gamma\left(k+\frac{3}{2}\right) \Gamma(i+1) \Gamma(a+k+1) \Gamma\left(j+\frac{1}{2}\right) \Gamma\left(l+\frac{3}{2}\right) \Gamma(k+l+a+2)}. \tag{17}$$

For Shifted Gegenbauer polynomials (S5), the (*i*, *j*)th entry of the operational matrix of integration is given as

$$\mu(i,j) = \sum\_{k=0}^{i} \sum\_{l=0}^{j} (-1)^{i+j-k-l} \frac{\Gamma\left(i+a+\frac{1}{2}\right) \Gamma(i+k+2a) \Gamma(j+l+2a) \Gamma\left(a+\frac{1}{2}\right) \Gamma\left(a+k+l+a+\frac{1}{2}\right) (2j+2a) j!}{(i-k)! (j-l)! (l)! \Gamma\left(k+a+\frac{1}{2}\right) \Gamma(i+2a) \Gamma(a+k+1) \Gamma\left(j+a+\frac{1}{2}\right) \Gamma(l+a+\frac{1}{2}) \Gamma(2a+k+l+a+1)} . \tag{18}$$
