*2.2. Jacobi Polynomials*

**Definition 5.** *The Jacobi polynomials [31]* <sup>J</sup> *<sup>α</sup>*,*<sup>β</sup> <sup>n</sup>* (*z*) *for indices <sup>α</sup>*, *<sup>β</sup>* <sup>&</sup>gt; <sup>−</sup><sup>1</sup> *and degree <sup>n</sup> are the solutions of the Sturm–Liouville problems. These are orthogonal polynomials with respect to the Jacobi weight function <sup>ω</sup>*(*z*)=(<sup>1</sup> <sup>−</sup> *<sup>z</sup>*)*α*(<sup>1</sup> <sup>+</sup> *<sup>z</sup>*)*<sup>β</sup> in interval* [−1, 1]*, defined as follows:*

$$\mathcal{L}\_{n}^{\mathfrak{a},\mathfrak{b}}(z) = \frac{\Gamma(\mathfrak{a} + n + 1)}{n!\Gamma(\mathfrak{a} + \mathfrak{b} + n + 1)} \sum\_{m=0}^{n} \left(\frac{n}{m}\right) \frac{\Gamma(\mathfrak{a} + \mathfrak{b} + n + m + 1)}{\Gamma(\mathfrak{a} + m + 1)} \left(\frac{z - 1}{2}\right)^{m}.\tag{7}$$

The *k*th derivative of Jacobi polynomials defined as

$$\frac{d^k}{dz^k} \mathcal{J}\_n^{a, \beta}(z) = \frac{\Gamma(\alpha + \beta + n + 1 + k)}{2^k \Gamma(\alpha + \beta + n + 1)} \mathcal{J}\_{n-k}^{(a+k, \beta+k)}(z), \ k \in \mathbb{N},\tag{8}$$

satisfy the recurrence relation

$$
\mathcal{J}\_{n+1}^{a,\beta}(z) = (\mathcal{A}\_n z - \mathcal{B}\_n) \mathcal{J}\_n^{a,\beta}(z) - \rho\_n \mathcal{J}\_{n-1}^{a,\beta}(z), \ n \ge 1,\tag{9}
$$

where

$$\mathcal{J}\_0^{a,\beta}(z) = 1,\ \mathcal{J}\_1^{a,\beta}(z) = \frac{1}{2}(a+\beta+z) + \frac{1}{2}(a-\beta),$$

$$\mathcal{A}\_n = \frac{(2n+a+\beta+1)(2n+a+\beta+2)}{2(n+1)(n+a+\beta+1)},\ \mathcal{B}\_n = \frac{(2n+a+\beta+1)(a^2-\beta^2)}{2(n+1)(n+a+\beta+1)(2n+a+\beta)}.$$

and

$$\rho\_n = \frac{(2n+\kappa+\beta+2)(n+\kappa)(n+\kappa)}{(n+1)(n+\kappa+\beta+1)(2n+\kappa+\beta)}.$$

For the transformation of the interval [−1, 1] to [0, 1], we use the relation, *z* = 2*x* − 1. The recurrence relation (9) becomes

$$\mathcal{J}\_{n+1}^{a,\beta}(\mathbf{x}) = (\mathcal{J}\_n \mathbf{x} - \mathcal{K}\_n) \mathcal{J}\_n^{a,\beta}(\mathbf{x}) - \rho\_n \mathcal{J}\_{n-1}^{a,\beta}(\mathbf{x}), \ n \ge 1,\tag{10}$$

where

$$\begin{aligned} \zeta\_{\mathbb{H}} &= \frac{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}{(n+1)(n+\alpha+\beta+1)},\\ \mathcal{K}\_{\mathbb{H}} &= \frac{(2n+\alpha+\beta+1)(2n^2+(1+\beta)(\alpha+\beta)+2n(\alpha+\beta+1))}{(n+1)(n+\alpha+\beta)(2n+\alpha+\beta)}.\end{aligned}$$

It satisfies the orthogonality relation

$$\int\_0^1 \mathcal{J}\_n^{a,\emptyset}(\mathbf{x}) \mathcal{J}\_m^{a,\emptyset}(\mathbf{x}) w\_1^{a,\emptyset}(\mathbf{x}) d\mathbf{x} = \delta\_{n,m} \mathcal{H}\_n^{a,\emptyset}, \text{ a.e. } \emptyset > -1,\tag{11}$$

where *<sup>δ</sup>n*,*<sup>m</sup>* is the Kronecker delta function, *<sup>w</sup>α*,*<sup>β</sup>* <sup>1</sup> (*x*) = *<sup>x</sup>β*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*)*<sup>α</sup>* is the weight function, and

$$\mathcal{H}\_n^{\mathfrak{a},\mathfrak{f}} = \frac{\Gamma(n+1+\beta)\Gamma(n+\mathfrak{a}+1)}{(2n+1+\mathfrak{a}+\beta)n!\Gamma(n+1+\mathfrak{a}+\beta)}.\tag{12}$$

The summation form of the Jacobi polynomials <sup>J</sup> *<sup>α</sup>*,*<sup>β</sup> <sup>n</sup>* (*x*) is written as

$$\mathcal{J}\_{\boldsymbol{n}}^{\boldsymbol{\alpha},\boldsymbol{\beta}}(\boldsymbol{x}) = \sum\_{k=0}^{l} (-1)^{k+j} \frac{\Gamma(k+j+1+\boldsymbol{\alpha}+\boldsymbol{\beta})\Gamma(j+1+\boldsymbol{\beta})}{k!\Gamma(j+1+\boldsymbol{\alpha}+\boldsymbol{\beta})\Gamma(k+1+\boldsymbol{\beta})},\tag{13}$$

and the *p*th derivative of Equation (13) in [0, 1], can be further rewritten in terms of *x* as

$$\frac{d^p}{d\mathbf{x}^p} \mathcal{J}\_n^{a,\beta}(\mathbf{x}) = \frac{\Gamma(a+\beta+n+1+p)}{\Gamma(a+\beta+n+1)} \mathcal{J}\_{n-p}^{(a+p,\beta+p)}(\mathbf{x}), \ p \in \mathbb{N}.\tag{14}$$

The value of the shifted Jacobi polynomials at the end points are given as

$$\mathcal{J}\_{n}^{a,\beta}(0) = (-1)^{n} \frac{\Gamma(1+n+\beta)}{\Gamma(1+\beta)n!}, \; \mathcal{J}\_{n}^{a,\beta}(1) = \frac{\Gamma(1+n+a)}{\Gamma(1+a)n!}.\tag{15}$$

Let *v*(*x*) be a square integrable function defined on [0, 1], then

$$w(\mathbf{x}) = \sum\_{i=0}^{\infty} c\_i \mathcal{J}\_i^{a,\emptyset}(\mathbf{x}),\tag{16}$$

where *ci* are the unknown coefficients (*ci*, *i* = 0, 1, 2 . . .) determined by the relation

$$\mathcal{L}\_{i} = \frac{1}{\mathcal{H}\_{i}^{a,\emptyset}} \int\_{0}^{1} v(\mathbf{x}) w^{a,\emptyset}(\mathbf{x}) \mathcal{J}\_{i}^{a,\emptyset}(\mathbf{x}) d\mathbf{x}, \text{ i } = 0, 1, 2 \dots \dots \dots \tag{17}$$

Truncating the series in Equation (16) up to (*m* + 1) terms, the approximation of *v*(*x*) is given as

$$w\_m(\mathbf{x}) = \sum\_{i=0}^m c\_i \mathcal{J}\_i^{\mathbf{a}, \emptyset}(\mathbf{x}). \tag{18}$$
