The main objective of this section is to derive an operational matrix of fractional integration for modified generalized Laguerre polynomials.

Let u(x)\in {L}_{{w}^{(\alpha ,\beta )}}^{2}(\mathrm{\Lambda}), then u(x) may be expressed in terms of modified generalized Laguerre polynomials as

u(x)=\sum _{j=0}^{\mathrm{\infty}}{a}_{j}{L}_{j}^{(\alpha ,\beta )}(x),\phantom{\rule{2em}{0ex}}{a}_{j}=\frac{1}{{h}_{k}}{\int}_{0}^{\mathrm{\infty}}u(x){L}_{j}^{(\alpha ,\beta )}(x){w}^{(\alpha ,\beta )}(x)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}j=0,1,\dots .

(8)

In practice, only the first (N+1)-terms modified generalized Laguerre polynomials are considered. Then we have

{u}_{N}(x)=\sum _{j=0}^{N}{a}_{j}{L}_{j}^{(\alpha ,\beta )}(x)={C}^{T}\varphi (x),

(9)

where the modified generalized Laguerre coefficient vector *C* and the modified generalized Laguerre vector \varphi (x) are given by

{C}^{T}=[{c}_{0},{c}_{1},\dots ,{c}_{N}],\phantom{\rule{2em}{0ex}}\varphi (x)={[{L}_{0}^{(\alpha ,\beta )}(x),{L}_{1}^{(\alpha ,\beta )}(x),\dots ,{L}_{N}^{(\alpha ,\beta )}(x)]}^{T}.

(10)

**Theorem 3.1** *Let* \varphi (x) *be the modified generalized Laguerre vector and* \nu >0, *then*

{J}^{\nu}\varphi (x)\simeq {\mathbf{P}}^{(\nu )}\varphi (x),

(11)

*where* {\mathbf{P}}^{(\nu )} *is the* (N+1)\times (N+1) *operational matrix of fractional integration of order* *ν* *in the Riemann*-*Liouville sense and is defined as follows*:

{\mathbf{P}}^{(\nu )}=\left(\begin{array}{ccccc}{\mathrm{\Psi}}_{\nu}(0,0)& {\mathrm{\Psi}}_{\nu}(0,1)& {\mathrm{\Psi}}_{\nu}(0,2)& \cdots & {\mathrm{\Psi}}_{\nu}(0,N)\\ {\mathrm{\Psi}}_{\nu}(1,0)& {\mathrm{\Psi}}_{\nu}(1,1)& {\mathrm{\Psi}}_{\nu}(1,2)& \cdots & {\mathrm{\Psi}}_{\nu}(1,N)\\ \vdots & \vdots & \vdots & \cdots & \vdots \\ {\mathrm{\Psi}}_{\nu}(i,0)& {\mathrm{\Psi}}_{\nu}(i,1)& {\mathrm{\Psi}}_{\nu}(i,2)& \cdots & {\mathrm{\Psi}}_{\nu}(i,N)\\ \vdots & \vdots & \vdots & \cdots & \vdots \\ {\mathrm{\Psi}}_{\nu}(N,0)& {\mathrm{\Psi}}_{\nu}(N,1)& {\mathrm{\Psi}}_{\nu}(N,2)& \cdots & {\mathrm{\Psi}}_{\nu}(N,N)\end{array}\right),

(12)

*where*

{\mathrm{\Psi}}_{\nu}(i,j)=\sum _{k=0}^{i}\sum _{r=0}^{j}\frac{{(-1)}^{k+r}{\beta}^{(-\nu )}j!\mathrm{\Gamma}(i+\alpha +1)\mathrm{\Gamma}(k+\nu +\alpha +r+1)}{(i-k)!(j-r)!\phantom{\rule{0.25em}{0ex}}r!\mathrm{\Gamma}(k+\nu +1)\mathrm{\Gamma}(k+\alpha +1)\mathrm{\Gamma}(\alpha +r+1)}.

*Proof* Using the analytic form of the modified generalized Laguerre polynomials {L}_{i}^{(\alpha ,\beta )}(x) of degree *i* (6) and (2), then

\begin{array}{rcl}{J}^{\nu}{L}_{i}^{(\alpha ,\beta )}(x)& =& \sum _{k=0}^{i}{(-1)}^{k}\frac{{\beta}^{k}\mathrm{\Gamma}(i+\alpha +1)}{(i-k)!k!\mathrm{\Gamma}(k+\alpha +1)}{J}^{\nu}{x}^{k}\\ =& \sum _{k=0}^{i}{(-1)}^{k}\frac{{\beta}^{k}\mathrm{\Gamma}(i+\alpha +1)}{(i-k)!\mathrm{\Gamma}(k+\nu +1)\mathrm{\Gamma}(k+\alpha +1)}{x}^{k+\nu},\\ i=0,1,\dots ,N.\end{array}

(13)

Now, approximating {x}^{k+\nu} by N+1 terms of modified generalized Laguerre series, we have

{x}^{k+\nu}=\sum _{j=0}^{N}{c}_{j}{L}_{j}^{(\alpha ,\beta )}(x),

(14)

where {c}_{j} is given from (8) with u(x)={x}^{k+\nu}, that is,

{c}_{j}=\sum _{r=0}^{j}{(-1)}^{r}\frac{{\beta}^{-k-\nu}j!\phantom{\rule{0.25em}{0ex}}\mathrm{\Gamma}(k+\nu +\alpha +r+1)}{(j-r)!r!\mathrm{\Gamma}(r+\alpha +1)},\phantom{\rule{1em}{0ex}}j=1,2,\dots ,N.

(15)

In virtue of (13) and (14), we get

{J}^{\nu}{L}_{i}^{(\alpha ,\beta )}(x)=\sum _{j=0}^{N}{\mathrm{\Psi}}_{\nu}(i,j){L}_{j}^{(\alpha ,\beta )}(x),\phantom{\rule{1em}{0ex}}i=0,1,\dots ,N,

(16)

where

\begin{array}{rcl}{\mathrm{\Psi}}_{\nu}(i,j)& =& \sum _{k=0}^{i}\sum _{r=0}^{j}\frac{{(-1)}^{k+r}{\beta}^{(-\nu )}j!\mathrm{\Gamma}(i+\alpha +1)\mathrm{\Gamma}(k+\nu +\alpha +r+1)}{(i-k)!(j-r)!\phantom{\rule{0.25em}{0ex}}r!\mathrm{\Gamma}(k+\nu +1)\mathrm{\Gamma}(k+\alpha +1)\mathrm{\Gamma}(\alpha +r+1)},\\ j=1,2,\dots ,N.\end{array}

Accordingly, Eq. (16) can be written in a vector form as follows:

{J}^{\nu}{L}_{i}^{(\alpha ,\beta )}(x)\simeq [{\mathrm{\Psi}}_{\nu}(i,0),{\mathrm{\Psi}}_{\nu}(i,1),{\mathrm{\Psi}}_{\nu}(i,2),\dots ,{\mathrm{\Psi}}_{\nu}(i,N)]\varphi (x),\phantom{\rule{1em}{0ex}}i=0,1,\dots ,N.

(17)

Eq. (17) leads to the desired result. □

**Corollary 3.2** *If we define the* *q* *times repeated integration of the modified generalized Laguerre vector* \varphi (x) *by* {J}^{q}\varphi (x), *and*

{J}^{q}\varphi (x)\simeq {\mathbf{I}}^{(q)}\varphi (x),

(18)

*then* {\mathbf{I}}^{(q)}={\mathbf{P}}^{(q)} *is the operational matrix of integration of* \varphi (x), *where* *q* *is an integer value*.