### Jacobi polynomials

Jacobi polynomials, which are often called hypergeometric polynomials, are denoted by \(\mathtt{J}_{m}^{\kappa ,\omega }(z)\) and can be defined by the following explicit formula:

$$\begin{aligned}& \mathtt{J}_{m}^{\kappa ,\omega }(z)= \frac{\Gamma (\kappa +m+1)}{m!\Gamma (\kappa +\omega +m+1)}\sum _{i=0}^{m} \binom{m}{i} \frac{\Gamma (\kappa +\omega +i+m+1)}{\Gamma (\kappa +i+1)} \biggl( \frac{z-1}{2} \biggr)^{i}. \end{aligned}$$

Some first few Jacobi polynomials are given by

$$\begin{aligned}& \mathtt{J}_{0}^{\kappa ,\omega }(z)=1, \\& \mathtt{J}_{1}^{\kappa ,\omega }(z)=\kappa +1+(\kappa +\omega +2) \frac{z-1}{2}, \\& \begin{aligned} \mathtt{J}_{2}^{\kappa ,\omega }(z)&=\frac{(\kappa +1)(\kappa +2)}{2}+( \kappa +2) ( \kappa +\omega +3)\frac{z-1}{2}\\ &\quad {}+ \frac{(\kappa +\omega +3)(\kappa +\omega +4)}{2} \biggl(\frac{z-1}{2} \biggr)^{2},\qquad \ldots. \end{aligned} \end{aligned}$$

These polynomials are orthogonal on [−1, 1] with respect to the weight \((1-z)^{\kappa }(1-z)^{\omega }\) and satisfy the following properties:

$$\begin{aligned}& \mathtt{J}_{m}^{\kappa ,\omega }(-1)=(-1)^{m}\binom{m+\omega }{m}, \\& \mathtt{J}_{m}^{\kappa ,\omega }(-z)=(-1)^{m}\mathtt{J}_{m}^{\omega ,\kappa }(z), \\& \mathtt{J}_{m}^{\kappa ,\kappa }(z)= \textstyle\begin{cases} \frac{\Gamma (m+\kappa +1)\Gamma (\frac{m}{2}+1)}{\Gamma (\frac{m}{2}+\kappa +1)\Gamma (m+1)} \mathtt{J}_{\frac{m}{2}}^{\kappa ,\frac{-1}{2}}(2z^{2}-1), & \text{if } m \text{ is even}, \\ \frac{\Gamma (m+\kappa +2)\Gamma (\frac{m}{2}+1)}{\Gamma (\frac{m}{2}+\kappa +1)\Gamma (m+2)}z \mathtt{J}_{\frac{m}{2}}^{\kappa ,\frac{1}{2}}(2z^{2}-1), & \text{if } m \text{ is odd}. \end{cases}\displaystyle \\& \int _{-1}^{1}(1-z)^{\kappa }(1+z)^{\omega } \mathtt{J}_{m}^{\kappa , \omega }(z)\mathtt{J}_{n}^{\kappa ,\omega }(z)\,dz= \frac{2^{\kappa +\omega +1}}{2m+\kappa +\omega +1} \frac{\Gamma (m+\kappa +1)\Gamma (m+\omega +1)}{\Gamma (m+\kappa +\omega +1)m!} \delta _{nm}, \end{aligned}$$

where \(\delta _{nm}\) is Kronecker delta.

$$\begin{aligned} &2m(m+\kappa +\omega ) (2m+\kappa +\omega -2)J_{m}^{\kappa ,\omega }(z)\\ &\quad =(2m+ \kappa +\omega -1) \bigl((2m+\kappa +\omega ) (2m+\kappa +\omega -2)z \\ &\qquad {}+{\kappa }^{2}-{\omega }^{2}\bigr)\mathtt{J}_{m-1}^{\kappa ,\omega }(z)-2(m+ \alpha -1) (m+\omega -1) (2m+\kappa +\omega )\mathtt{J}_{m-2}^{\kappa , \omega }(z). \end{aligned}$$

### Jacobi wavelet of shifted Jacobi polynomial

Jacobi wavelet of the shifted Jacobi polynomial defined on six arguments *k*, *n*, *κ*, *ω*, *m*, *z* is denoted by \(\mathscr{J}(k,n,\kappa ,\omega ,m,z)=\mathscr{J}_{n,m}^{\kappa , \omega }(z)\), and can be defined on [0, 1) as follows [1]:

$$\begin{aligned} \mathscr{J}_{n,m}^{\kappa ,\omega }(z)= \textstyle\begin{cases} 2^{\frac{k}{2}}\mu _{m}^{\kappa ,\omega }\mathtt{J}_{m}^{\kappa , \omega }(2^{k}z-2n+1), & \text{if } z \in [\xi _{1}, \xi _{2}), \\ 0, & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$

(2.1)

where \(\xi _{1}=\frac{n-1}{2^{k-1}}\), \(\xi _{2}=\frac{n}{2^{k-1}}\), and \(\mu _{m}^{\kappa ,\omega }=\sqrt{ \frac{(2m+\kappa +\omega +1)\Gamma (2m+\kappa +\omega +1)m!}{2^{\kappa +\omega +1}\Gamma (m+\kappa +1)\Gamma (m+\omega +1)}}\).

Equivalently, for any positive integer *k*, Jacobi wavelet can also be defined as follows:

$$\begin{aligned} \mathscr{J}_{i}^{\kappa ,\omega }(z)= \textstyle\begin{cases} 2^{\frac{k}{2}}\mu _{m}^{\kappa ,\omega }\mathtt{J}_{m}^{\kappa , \omega }(2^{k}z-2n+1), & \text{if } z \in [\xi _{1}, \xi _{2}), \\ 0, & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$

(2.2)

where *i* is wavelet number determined by \(i=n+2^{k-1}m\), where \(n=0,1,2,\ldots \) and \(m=0,1,2,\ldots,M-1\), where *m* is degree of polynomial. *M* can be determined by \(M=\frac{N}{2^{k-1}}\), where \(k=1,2,\ldots \) .

#### Function approximation by Jacobi wavelet

Let \(\{\mathscr{J}_{1,0}^{\kappa ,\omega },\ldots,\mathscr{J}_{1,M-1}^{ \kappa ,\omega },\mathscr{J}_{2,0}^{\kappa ,\omega },\ldots,\mathscr{J}_{2,M-1}^{ \kappa ,\omega },\mathscr{J}_{2^{k-1},0}^{\kappa ,\omega },\ldots, \mathscr{J}_{2^{k-1},M-1}^{\kappa ,\omega }\}\) be a set of Jacobi wavelets.

Any function \({f}(z) \in L^{2}[0,1)\) can be expressed in terms of Jacobi wavelet as follows [1]:

$$\begin{aligned}& f(z) = \sum_{n=1}^{\infty }\sum _{m=0}^{\infty }a_{n,m}\mathscr{J}_{n,m}^{ \kappa ,\omega }(z) = \sum_{i=1}^{\infty }a_{i} \mathscr{J}_{i}^{ \kappa ,\omega }(z). \end{aligned}$$

For approximation, we truncate this series for a natural number *N*, and we get

$$\begin{aligned} f(z) \approx &\sum_{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}a_{n,m}\mathscr{J}_{n,m}^{ \kappa ,\omega }(z)= \sum_{i=1}^{N}a_{i} \mathscr{J}_{i}^{\kappa , \omega }(z) \end{aligned}$$

(2.3)

$$\begin{aligned} =&a^{T}\mathscr{J}(z), \end{aligned}$$

(2.4)

where *a* and \(\mathscr{J}(z)\) are matrices of order \(N\times 1\) given by

$$\begin{aligned}& \begin{aligned}[b] a&=[a_{1,0},a_{1,1},\ldots, a_{1,M-1},a_{2,0},a_{2,1},\ldots,a_{2,M-1}, \ldots,a_{2^{k-1},0},\ldots,a_{2^{k-1},M-1}]^{T} \\ &= [a_{1},a_{2},\ldots,a_{N}]^{T}, \end{aligned} \end{aligned}$$

(2.5)

$$\begin{aligned}& \begin{aligned}[b] \mathscr{J}(z) &=\bigl[\mathscr{J}_{1,0}^{\kappa ,\omega }(z), \ldots, \mathscr{J}_{1,M-1}^{\kappa ,\omega }(z),\mathscr{J}_{2,0}^{\kappa , \omega }(z), \ldots,\mathscr{J}_{2,M-1}^{\kappa ,\omega }(z),\mathscr{J}_{2^{k-1},0}^{ \kappa ,\omega }(z), \ldots,\mathscr{J}_{2^{k-1},M-1}^{\kappa ,\omega }(z)\bigr] \\ &= \bigl[\mathscr{J}_{1}^{\kappa ,\omega }(z),\ldots, \mathscr{J}_{N}^{\kappa , \omega }(z)\bigr],\end{aligned} \end{aligned}$$

(2.6)

where the coefficient \(a_{i}\) can be determined by \(a_{i} = \langle f(z),\mathscr{J}_{i}^{\kappa ,\omega }(z)\rangle = \int _{0}^{1}f(z)\overline{\mathscr{J}_{i}^{\kappa ,\omega }(z)}\,dz\).

#### Integration of Jacobi wavelet

Let \(\mathscr{J}^{1}_{i}(z)\), \(\mathscr{J}^{2}_{i}(z)\), and \(\mathscr{J}^{3}_{i}(z)\) be the first, second, and third integration of Jacobi wavelet from 0 to *z* respectively. These integrations can be determined as follows:

$$\begin{aligned}& \mathscr{J}_{1,i}^{\kappa ,\omega }(z) = \textstyle\begin{cases} 2^{\frac{-k}{2}}\mu _{m}^{\kappa ,\omega } ( \frac{1}{(m+\kappa +\omega )} ) \{{\mathtt{J}}_{m+1}^{\kappa -1, \omega -1}(\hat{z})-{\mathtt{J}}_{m+1}^{\kappa -1,\omega -1}(-1) \}, & \xi _{1}\leq z < \xi _{2}, \\ 2^{\frac{-k}{2}}\mu _{m}^{\kappa ,\omega } ( \frac{1}{(m+\kappa +\omega )} ) \{{\mathtt{J}}_{m+1}^{\kappa -1, \omega -1}(1)-{\mathtt{J}}_{m+1}^{\kappa -1,\omega -1}(-1) \}, & \xi _{2}\leq z \leq 1, \end{cases}\displaystyle \end{aligned}$$

(2.7)

$$\begin{aligned}& \mathscr{J}_{2,i}^{\kappa ,\omega }(z) = \textstyle\begin{cases} 2^{\frac{-3k}{2}}\mu _{m}^{\kappa ,\omega } ( \frac{1}{(m+\kappa +\omega )} ) \{ ( \frac{1}{(m-2+\kappa +\omega )} ) \{{\mathtt{J}}_{m+2}^{\kappa -2, \omega -2}(\hat{z}) \\ \quad {}-{\mathtt{J}}_{m+2}^{\kappa -2,\omega -2}(-1) \}-(1+\hat{z}){ \mathtt{J}}_{m+1}^{\kappa -1,\omega -1}(-1) \}, & \xi _{1}\leq z < \xi _{2}, \\ 2^{\frac{-3k}{2}}\mu _{m}^{\kappa ,\omega } ( \frac{1}{(m+\kappa +\omega )} ) \{ ( \frac{1}{(m-2+\kappa +\omega )} )\{{\mathtt{J}}_{m+2}^{\kappa -2, \omega -2}(1) \\ \quad {}-{\mathtt{ J}}_{m+2}^{\kappa -2,\omega -2}(-1)\}-2{\mathtt{J}}_{m+1}^{ \kappa -1,\omega -1}(-1) \\ \quad {}+(\hat{z}-1)\{{\mathtt{J}}_{m+1}^{\kappa -1,\omega -1}(1)-{\mathtt{J}}_{m+1}^{ \kappa -1,\omega -1}(-1)\} \}, & \xi _{2}\leq z \leq 1, \end{cases}\displaystyle \end{aligned}$$

(2.8)

where \(\hat{z}=2^{k}z-2n+1\).