New Results of Generalized Jacobsthal–Lucas Polynomials with Some Integral Applications

Abstract

We study a generalized class of Jacobsthal–Lucas polynomials that depends on two parameters. First, we introduce essential formulas for these polynomials, involving their series representation, inverse formula, and moment formula. These formulas allow us to investigate this generalized class of polynomials further and to develop novel formulations. The essential standard linearization problem of these polynomials is solved, and the linearization coefficients are given in simple forms. In addition, some mixed linearization formulas with other classes of polynomials are presented. The derivative formulas of these polynomials, expressed as combinations of different polynomials, are given. By employing symbolic algebra methods—most notably Zeilberger’s algorithm and other well-known identities from the literature—many hypergeometric functions appearing in the coefficients can be reduced, resulting in simpler expressions. In addition, some definite integrals are evaluated using the newly introduced formulas.

1. Introduction

Special functions and their modified and generalized forms are crucial in many applied sciences. They are used as fundamental tools in modern applications. In physics, especially in quantum mechanics, special functions appear in closed-form solutions of differential equations describing fields, potentials, or scattering integrals; see, for example [1,2]. In addition, special functions play important roles in epidemiology and biological modeling, where fractional operators involving Mittag–Leffler and Wright functions capture memory and hereditary effects in disease dynamics; see, for example [3]. In statistics and probability, special functions are also important because they form the mathematical foundation for many common and advanced distributions; see, for example [4]. For additional applications of various special functions, the reader may consult [5,6,7].

The study of special polynomials, as well as their modified and generalized polynomials, continues to attract considerable attention across theoretical and applied mathematics. We present several theoretical contributions to the study of many polynomial sequences. A class of Sobolev orthogonal polynomials on the unit circle was introduced [8]. The theory of orthogonality in two variables was extended through establishing a class of bivariate Koornwinder–Sobolev orthogonal polynomials [9]. Exceptional orthogonal systems were analyzed [10]. In [11], finite exceptional orthogonal polynomial sequences that arise from rational Darboux transforms of Romanovski–Jacobi polynomials were developed. The connection between classical and generalized polynomial sequences was investigated [12]. Authors [13] developed some new formulas regarding specific classes of Jacobi polynomials, and other authors [14] introduced new polynomials that combine the Fibonacci and Lucas polynomials. Some new formulas of the convolved Fibonacci polynomials were established [15]. A study [16] introduced probabilistic Bernoulli and Euler polynomials, and an extension to multi-poly-Bernoulli polynomials was developed [17]. Probabilistic degenerate Bernoulli and Euler polynomials were formulated [18]. In addition, degenerate hypergeometric Bernoulli–Euler polynomials with new recurrence and generating function identities were investigated [19].

Among the essential sequences of polynomials are the Jacobsthal and Jacobsthal–Lucas polynomials. Many contributions have addressed these types of polynomials. In [20], some new results for specific Jacobsthal polynomials were derived. Authors [21] introduced Gaussian–Jacobsthal and Jacobsthal–Lucas polynomials and established several new formulas for them, including their generating functions and recurrence relations. In [22], the authors established generalized Jacobsthal and Jacobsthal–Lucas polynomial and developed some operational identities. In [23], the authors investigated certain third-order Jacobsthal and Jacobsthal–Lucas sequences. In [24], the authors proposed higher-order Jacobsthal–Lucas quaternions, extending the polynomial concept to quaternionic structures and established some algebraic and geometric interpretations. Other authors [25] introduced bivariate Jacobsthal and Jacobsthal–Lucas polynomial sequences. In [26], generalized Jacobsthal polynomials and certain associated formulas are presented, while [27] developed some formulas regarding Jacobsthal-type polynomials. In [28], the authors studied Jacobsthal representation polynomials and established connections with Lucas and Fibonacci frameworks. In [29], the authors investigated third-order Jacobsthal polynomials. Other modified Jacobsthal polynomials were proposed and investigated [30]. In [31], some applications of Jacobsthal and Jacobsthal–Lucas numbers in coding theory were given. Other generalized Jacobsthal polynomials were investigated [32]. Finally, in [33], the authors introduced a Jacobsthal-like sequence associated with k–Jacobsthal–Lucas numbers and presented some of their properties with potential cryptographic and computational applications.

Many formulas related to the study of sequences of polynomials are crucial in numerical analysis, particularly within the framework of spectral methods for the numerical solutions to various types of differential equations. The derivation of the derivatives of various sequences of polynomials in terms of their original polynomials is useful when applying the collocation method, as they help construct the operational matrices of the derivatives that are used to reformulate the governing equation and its associated conditions as a system of equations suitable for numerical treatment; see, for example [34,35]. In addition, the different derivative formulas are useful for computing the required inner products when employing the tau and Galerkin spectral methods; see, for example [36].

Hypergeometric functions are crucial in special functions and their applications. They can represent almost all important functions and polynomials. In addition, linearization and connection problems, which are important problems in this area, can be solved using these functions. Linearization and connection coefficients can often be expressed using the hypergeometric functions of different arguments; see, for example [37,38,39,40,41].

Here, we list the main contributions and the novelty of this work in the following points:

• We introduce a generalized class of Jacobsthal–Lucas polynomials depending on two parameters, which extends some of the special sequences.
• We establish some fundamental formulas of the generalized polynomials, such as the series form, the inversion formula, and the moment formulas. These formulas are derived in closed forms and, to the best of our knowledge, have not been reported in earlier works. In addition, they provide a fundamental basis for further investigation of such generalized polynomials.
• We derive new linearization formulas for these polynomials. More precisely, both standard and some mixed linearization formulas are obtained in closed forms.
• We establish new derivative formulas of these polynomials in terms of various polynomials. The inverse derivative formulas are also established.
• We establish new connection formulas between GLPs and other orthogonal and non-orthogonal polynomials.
• We obtain some new expressions of definite integrals involving the generalized Jacobsthal–Lucas polynomials, which further demonstrates the usefulness of the obtained results.

Here, we discuss the main contributions of related works and highlight the differences between the main results of these works and our results.

• The work in [22] focused mainly on generalized Jacobsthal and Jacobsthal–Lucas polynomials and some of their basic properties, including recurrence relations, generating functions, classical identities, and matrix representations.
• In [26], the authors studied generalized Jacobsthal and Jacobsthal–Lucas polynomials that were defined via higher-order recurrence relations depending on a parameter m.
• In [27], the authors derived some derivative sequences of generalized Jacobsthal and Jacobsthal–Lucas polynomials based on using the generating functions.
• In [32], the authors introduced $h\left(x\right)$-Jacobsthal-type polynomials and investigated their properties, including generating functions, Binet formulas, summation formulas, and combinatorial representations.
• In [33], the authors introduced a Jacobsthal-like sequence associated with the k–Jacobsthal–Lucas numbers, studied algebraic identities, and presented applications in computational and cryptographic contexts.
• From the above discussion, we note that none of the previous works considered the formulas developed in this paper, such as the linearization formulas or the connections of these generalized polynomials with different orthogonal and non-orthogonal polynomials, which may be useful for further analytical and computational applications. This motivated the development of such formulas.

The rest of this manuscript is structured as follows: Section 2 presents an account of some generalized Horadam polynomials, including Jacobsthal–Lucas polynomials as well as an account of some other polynomials. In Section 3, some fundamental formulas regarding GJLPs, such as their series form, their inversion formula, and the moment formula, are developed. Section 4 derives new linearization formulas for GJLPs, including the standard linearization formula of these polynomials. Section 5 derives some derivative formulas of GJLPs as combinations of different sequences of polynomials. This section also presents the inverse derivative formulas. Some new definite integrals involving GJLPs are developed in Section 6 based on the formulas obtained in the earlier sections. Finally, this paper concludes with remarks given in Section 7.

2. An Account of Some Generalized Polynomials and Symbolic Computation

We present an overview of some generalized sequences. Furthermore, we introduce a generalized sequence of polynomials, namely, generalized Jacobsthal–Lucas polynomials (GJLPs), which extend the well-known Jacobsthal–Lucas polynomials. Some other classes of polynomials are also considered. Additionally, an overview of the application of some useful symbolic algorithms is provided.

In [42], Horadam established two important classes of polynomials that can be generated using the following recurrence relations:

$$\begin{array}{cc}\hfill W_n\left(x\right)& =r\left(x\right)W_{j-1}\left(x\right)+s\left(x\right)W_{j-2}\left(x\right),W_0\left(x\right)=0,W_1\left(x\right)=1,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill w_n\left(x\right)& =\overline{r}\left(x\right)w_{j-1}\left(x\right)+\overline{s}\left(x\right)w_{j-2}\left(x\right),w_0\left(x\right)=2,w_1\left(x\right)=\overline{r}\left(x\right).\hfill \end{array}$$

(2)

They have the following Binet formulas:

$$\begin{array}{cc}\hfill W_j\left(x\right)& ={\displaystyle \frac{{\left[r\left(x\right)+\sqrt{r^2\left(x\right)+4s\left(x\right)}\right]}^j-{\left[r\left(x\right)-\sqrt{r^2\left(x\right)+4s\left(x\right)}\right]}^j}{2^j\sqrt{r^2\left(x\right)+4s\left(x\right)}}},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill w_j\left(x\right)& ={\displaystyle \frac{{\left[\overline{r}\left(x\right)+\sqrt{{\overline{r}}^2\left(x\right)+4\overline{s}\left(x\right)}\right]}^j+{\left[\overline{r}\left(x\right)-\sqrt{{\overline{r}}^2\left(x\right)+4\overline{s}\left(x\right)}\right]}^j}{2^j}}.\hfill \end{array}$$

(4)

Certain choices of $r\left(x\right),s\left(x\right),\overline{r}\left(x\right)$, and $\overline{s}\left(x\right)$ yield some important generalizations of Fibonacci and Lucas polynomials. The two generalized classes that are constructed by the following recurrence relations:

$$\begin{array}{cc}\hfill F_k^{a,b}\left(x\right)=& ax{F}_{k-1}^{a,b}\left(x\right)+b{F}_{k-2}^{a,b}\left(x\right),F_0^{a,b}\left(x\right)=1,F_1^{a,b}\left(x\right)=ax,k\ge 2,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill L_k^{c,d}\left(x\right)=& cx{L}_{k-1}^{c,d}\left(x\right)+d{L}_{k-2}^{c,d}\left(x\right),L_0^{c,d}\left(x\right)=2,L_1^{c,d}\left(x\right)=cx,k\ge 2,\hfill \end{array}$$

(6)

are, respectively, particular classes of (3) and (4).

Remark 1.

Jacobsthal–Lucas polynomials are particular cases of (4). They can be constructed with the aid of the following recurrence relation [43]:

$$J{L}_n^{A,B}\left(x\right)=J{L}_{n-1}^{A,B}\left(x\right)+2xJ{L}_{n-2}^{A,B}\left(x\right),J_0^{A,B}\left(x\right)=2,J{L}_1^{A,B}\left(x\right)=1.$$

(7)

Remark 2.

In this paper, we investigate a class of polynomials that generalizes the Jacobsthal–Lucas class of polynomials. More explicitly, we focus on the generalized class of polynomials ${\left\{J{L}_n^{A,B}\left(x\right)\right\}}_{n\ge 0}$ generated by the recurrence formula given below:

$$J{L}_n^{A,B}\left(x\right)=AJ{L}_{n-1}^{A,B}\left(x\right)+BxJ{L}_{n-2}^{A,B}\left(x\right),J{L}_0^{A,B}\left(x\right)=2,J{L}_1^{A,B}\left(x\right)=A.$$

(8)

The first few polynomials of $J{L}_n^{A,B}\left(x\right)$ are

$$\begin{array}{cccc}\hfill J{L}_0^{A,B}\left(x\right)& =2,\hfill & \hfill J{L}_1^{A,B}\left(x\right)& =A,\hfill \\ \hfill J{L}_2^{A,B}\left(x\right)& =A^2+2Bx,\hfill & \hfill J{L}_3^{A,B}\left(x\right)& =A^3+3ABx,\hfill \\ \hfill J{L}_4^{A,B}\left(x\right)& =A^4+4A^{2}Bx+2B^2{x}^2,\hfill & \hfill J{L}_5^{A,B}\left(x\right)& =A^5+5A^{3}Bx+5A{B}^2{x}^2.\hfill \end{array}$$

2.1. An Overview of Some Polynomials

Some orthogonal and non-orthogonal polynomials are described. The emphasis is on two groups of polynomials: symmetric and non-symmetric. Here we denote them ${\varphi }_m\left(x\right)$ and ${\psi }_m\left(x\right)$. They can be expressed as

$$\begin{array}{cc}\hfill {\varphi }_n\left(x\right)=& \sum _{r=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{H}_{r,n}{x}^{n-2r},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\psi }_n\left(x\right)=& \sum _{r=0}^n{F}_{r,n}{x}^{n-r},\hfill \end{array}$$

(10)

where $H_{r,n}$ and $F_{r,n}$ are known coefficients.

The inversion formulas corresponding to (9) and (10) can be assumed to have the following forms:

$$\begin{array}{cc}\hfill x^n=& \sum _{r=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{\overline{H}}_{r,n}{\varphi }_{n-2r}\left(x\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill x^n=& \sum _{r=0}^n{\overline{F}}_{r,n}{\psi }_{n-r}\left(x\right),\hfill \end{array}$$

(12)

and ${\overline{H}}_{r,n}$ and ${\overline{F}}_{r,n}$ stand for known coefficients.

We provide some famous polynomials. The normalized Jacobi polynomials can be expressed as [14]

$$R_r^{(\rho ,\gamma )}\left(x\right)={}_{2}F_1\left.\left(\begin{array}{c}-r,r+\rho +\gamma +1\\ \rho +1\end{array}\right|{\displaystyle \frac{1-x}{2}}\right).$$

(13)

Note that the ultraspherical (normalized Gegenbauer polynomials) are defined as

$$C_r^{\left(\rho \right)}\left(x\right)=R_r^{(\rho -{\displaystyle \frac{1}{2}},\rho -{\displaystyle \frac{1}{2}})}\left(x\right).$$

(14)

The shifted normalized Jacobi polynomials on $[0,1]$ are defined as

$${\tilde{R}}_m^{(\rho ,\gamma )}\left(x\right)=R_m^{(\rho ,\gamma )}(2x-1),$$

(15)

which can be expressed as [14]:

$${\displaystyle {\tilde{R}}_m^{(\rho ,\gamma )}\left(x\right)=\sum _{\ell =0}^m\frac{{(-1)}^{\ell }m!\mathrm{\Gamma }(\rho +1){(\gamma +1)}_m{(\rho +\gamma +1)}_{2m-\ell }}{\ell !(m-\ell )!\mathrm{\Gamma }(m+\rho +1){(\rho +\gamma +1)}_m{(\gamma +1)}_{m-\ell }}{x}^{m-\ell },}$$

(16)

and its inverse formula is

$${\displaystyle x^j=\sum _{r=0}^j\frac{\left(\genfrac{}{}{0pt}{}{j}{r}\right){(\rho +1)}_{j-r}{(j-r+\gamma +1)}_r}{{(2j-2r+\rho +\gamma +2)}_r{(j-r+\rho +\gamma +1)}_{j-r}}{\tilde{R}}_{j-r}^{(\rho ,\gamma )}\left(x\right).}$$

(17)

The set $\left\{{\tilde{R}}_{\ell }^{(\rho ,\gamma )}\left(x\right)\right\}$ forms an orthogonal system on $[0,1]$ in the following sense:

$${\int }_0^1{(1-x)}^{\rho }{x}^{\gamma }{\tilde{R}}_{\ell }^{(\rho ,\gamma )}\left(x\right){\tilde{R}}_k^{(\rho ,\gamma )}\left(x\right)dx=\left\{\begin{array}{cc}0,\hfill & k\ne \ell ,\hfill \\ {\tilde{h}}_{\ell }^{\rho ,\gamma },\hfill & k=\ell ,\hfill \end{array}\right.$$

(18)

with

$${\tilde{h}}_{\ell }^{\rho ,\gamma }={\displaystyle \frac{k!\mathrm{\Gamma }{(\rho +1)}^2\mathrm{\Gamma }(k+\gamma +1)}{(2k+\rho +\gamma +1)\mathrm{\Gamma }(k+\rho +1)\mathrm{\Gamma }(k+\rho +\gamma +1)}}.$$

(19)

Remark 3.

It is worth noting that the four kinds of Chebyshev polynomials are special cases of the classical Jacobi polynomials and admit trigonometric expressions. If $\theta ={cos}^{-1}x$, we can write [5]

$$\begin{array}{cccc}{T}_n(cos\theta )=\hfill & cos\left(n\theta \right),\hfill & \hfill & {\displaystyle U_n(cos\theta )=\frac{sin\left((n+1)\theta \right)}{sin\theta },}\hfill \end{array}$$

(20)

$$\begin{array}{cccc}\hfill V_n(cos\theta )=& {\displaystyle \frac{cos\left(\left(n+\frac{1}{2}\right)\theta \right)}{cos\left(\frac{\theta }{2}\right)},}\hfill & \hfill & {\displaystyle W_n(cos\theta )=\frac{sin\left(\left(n+\frac{1}{2}\right)\theta \right)}{sin\left(\frac{\theta }{2}\right)}.}\hfill \end{array}$$

(21)

Remark 4.

The following unified recurrence relation can be used to construct the four kinds of Chebyshev polynomials:

$$C_n\left(x\right)=2x{C}_{n-1}\left(x\right)-C_{n-2}\left(x\right),$$

(22)

but with different starting values.

Remark 5.

Note that the repeated use of the recurrence relation (22) yields the following moment formula:

$${\displaystyle x^r{C}_n\left(x\right)=\frac{1}{2^r}\sum _{s=0}^r\left(\genfrac{}{}{0pt}{}{r}{s}\right)C_{r+n-2s}\left(x\right).}$$

(23)

Remark 6.

The four shifted Chebyshev polynomials are included in the particular forms of ${\tilde{R}}_n^{(\rho ,\gamma )}\left(x\right)$. We have

$$\begin{array}{cccc}\hfill & T_n^{*}\left(x\right)={\tilde{R}}_n^{(-{\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}})}\left(x\right),\hfill & \hfill & U_n^{*}\left(x\right)=(n+1){\tilde{R}}_n^{({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})}\left(x\right),\hfill \\ \hfill & V_n^{*}\left(x\right)={\tilde{R}}_n^{(-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}})}\left(x\right),\hfill & \hfill & W_n^{*}\left(x\right)=(2n+1){\tilde{R}}_n^{({\displaystyle \frac{1}{2}},-{\displaystyle \frac{1}{2}})}\left(x\right).\hfill \end{array}$$

2.2. An Account of Some Symbolic Computation Including Zeilberger’s Algorithm

In some parts of this paper, we use symbolic computation to derive formulas. The application of Zeilberger’s algorithm is useful for obtaining closed-form expressions for sums and terminating hypergeometric functions in the context of special functions. We present the steps for using this algorithm and other symbolic algorithms to obtain the desired closed forms.

• First, a recurrence relation satisfied by the sum or by the terminating hypergeometric function is derived. This step is carried out using Zeilberger’s algorithm [44,45], which can be implemented via computer algebra programs such as Maple (e.g., using the sumrecursion command).
• The second step is solving the resulting recurrence relation analytically. This can also be performed through suitable symbolic algorithms, such as Petkovšek’s algorithm or van Hoeij’s method [46]. Alternatively, dedicated tools such as the Maple (version 17) package LREtools[hypergeomsols] can be used to obtain closed-form solutions.
• Using the two steps above, many sums and terminating hypergeometric functions can be expressed in closed form, thereby yielding reduced forms.

3. Some New Key Formulas of GJLPs

This section aims to develop new fundamental formulas for GJLPs. The series form, its inversion formula, and the moment formula will be developed. These formulas will help with further investigating GJLPs.

3.1. Power Form and Inversion Formula of GJLPs

Theorem 1.

For every positive integer j, the following expression is valid:

$$J{L}_j^{A,B}\left(x\right)=j\sum _{r=0}^{⌊j/2⌋}{\displaystyle \frac{A^{j-2r}{B}^r}{j-r}}\left({\displaystyle \genfrac{}{}{0pt}{}{j-r}{r}}\right)x^r,$$

(24)

where $\lfloor ·\rfloor$ denotes the well-known floor function.

Proof. 

First, consider the polynomial:

$${\theta }_j^{A,B}\left(x\right)=j\sum _{r=0}^{⌊j/2⌋}{\displaystyle \frac{A^{j-2r}{B}^r}{j-r}}\left({\displaystyle \genfrac{}{}{0pt}{}{j-r}{r}}\right)x^r.$$

(25)

We note that ${\theta }_0^{A,B}\left(x\right)=2,{\theta }_1^{A,B}\left(x\right)=A$. In addition, some algebraic manipulations lead to the following identity:

$${\theta }_n^{A,B}\left(x\right)=A{\theta }_{n-1}^{A,B}\left(x\right)+Bx{\theta }_{n-2}^{A,B}\left(x\right),$$

(26)

and therefore ${\theta }_n^{A,B}=J{L}_n^{A,B}\left(x\right)$ for all $n\ge 0$. □

The following two expressions can be deduced from the expression in (24).

Corollary 1.

For every positive integer j, we have the following two expressions:

$$\begin{array}{c}J{L}_{2j}^{A,B}\left(x\right)=j\sum _{r=0}^j{\displaystyle \frac{2A^{2r}{B}^{j-r}(j+r-1)!}{\left(2r\right)!(j-r)!}}{x}^{j-r},\end{array}$$

(27)

$$\begin{array}{c}J{L}_{2j+1}^{A,B}\left(x\right)=(2j+1)\sum _{r=0}^j{\displaystyle \frac{A^{2r+1}{B}^{j-r}(j+r)!}{(2r+1)!(j-r)!}}{x}^{j-r}.\end{array}$$

(28)

Proof. 

This is directly obtained from Formula (24). □

Theorem 2.

The following inversion formula follows for any non-negative integer k:

$$\begin{array}{cc}\hfill x^k& ={\displaystyle \frac{1}{2}}{B}^{-k}\left(\sum _{m=0}^k{\displaystyle \frac{1}{\left(2m\right)!}}{A}^{2m}{(k-2m+1)}_{2m}J{L}_{2k-2m}^{A,B}\left(x\right)\right.\hfill \\ \hfill & \left.+\sum _{m=0}^{k-1}{\displaystyle \frac{1}{(2m+1)!}}{A}^{2m+1}{(k-2m)}_{2m+1}J{L}_{2k-2m-1}^{A,B}\left(x\right)\right).\hfill \end{array}$$

(29)

Proof. 

We use mathematical induction. The formula is valid for $k=0$. Let (29) be valid. The following identity remains to be proven:

$$\begin{array}{cc}\hfill x^{k+1}& ={\displaystyle \frac{1}{2}}{B}^{-k}\left(\sum _{m=0}^k{\displaystyle \frac{1}{\left(2m\right)!}}{A}^{2m}{(k-2m+1)}_{2m}xJ{L}_{2k-2m}^{A,B}\left(x\right)\right.\hfill \\ \hfill & \left.-\sum _{m=0}^{k-1}{\displaystyle \frac{1}{(2m+1)!}}{A}^{2m+1}{(k-2m)}_{2m+1}xJ{L}_{2k-2m-1}^{A,B}\left(x\right)\right).\hfill \end{array}$$

(30)

Given the recursive Formula (8) written as

$$xJ{L}_k\left(x\right)={\displaystyle \frac{1}{B}}\left(J{L}_{k+2}^{A,B}\left(x\right)-AJ{L}_{k+1}^{A,B}\left(x\right)\right),$$

(31)

we can convert (30) into the following formula:

$$\begin{array}{cc}\hfill x^{k+1}& ={\displaystyle \frac{1}{2}}{B}^{-k-1}\left(\sum _{m=0}^k{\displaystyle \frac{1}{\left(2m\right)!}}{A}^{2m}{(k-2m+1)}_{2m}\left(J{L}_{2k-2m+2}^{A,B}\left(x\right)-AJ{L}_{2k-2m+1}^{A,B}\left(x\right)\right)\right.\hfill \\ \hfill & \left.-\sum _{m=0}^{k-1}{\displaystyle \frac{1}{(2m+1)!}}{A}^{2m+1}{(k-2m)}_{2m+1}\left(J{L}_{2k-2m+1}^{A,B}\left(x\right)-AJ{L}_{2k-2m}^{A,B}\left(x\right)\right)\right),\hfill \end{array}$$

(32)

which can be simplified as

$$\begin{array}{cc}\hfill x^{k+1}& ={\displaystyle \frac{1}{2}}{B}^{-k}\left(\sum _{m=0}^k{\displaystyle \frac{1}{\left(2m\right)!}}{A}^{2m}{(k-2m+1)}_{2m}xJ{L}_{2k-2m}^{A,B}\left(x\right)\right.\hfill \\ \hfill & \left.-\sum _{m=0}^{k-1}{\displaystyle \frac{1}{(2m+1)!}}{A}^{2m+1}{(k-2m)}_{2m+1}xJ{L}_{2k-2m-1}^{A,B}\left(x\right)\right).\hfill \end{array}$$

(33)

This proves the theorem. □

3.2. The Moment Formula of GJLPs

The following theorem derives the moment formula of GJLPs.

Theorem 3.

Let $m,i$ be non-negative integers (NNIs). The following moment formula follows:

$$x^{m}J{L}_i^{A,B}\left(x\right)=\sum _{r=0}^m{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m}{r}\right){(-A)}^{m-r}}{B^m}}J{L}_{i+m+r}^{A,B}\left(x\right).$$

(34)

Proof. 

We provide the proof by induction on m. For $m=0$, the identity holds. Let (34) be valid, and we will show that

$$x^{m+1}J{L}_i^{A,B}\left(x\right)=\sum _{r=0}^{m+1}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m+1}{r}\right){(-A)}^{m-r+1}}{B^{m+1}}}J{L}_{i+m+r+1}^{A,B}\left(x\right).$$

(35)

Multiplying (34) by x and using (31), we get

$$x^{m+1}J{L}_i^{A,B}\left(x\right)=\sum _{r=0}^m{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m}{r}\right){(-A)}^{m-r}}{B^{m+1}}}\left(J{L}_{i+m+r+2}^{A,B}\left(x\right)-AJ{L}_{i+m+r+1}^{A,B}\left(x\right)\right),$$

(36)

which can be converted into

$$x^{m+1}J{L}_i^{A,B}\left(x\right)=\sum _{r=0}^{m+1}\left(\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r-1}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r}}\right)\right)\left({\displaystyle \frac{{(-A)}^{m-r+1}}{B^{m+1}}}J{L}_{i+m+r+1}^{A,B}\left(x\right)\right).$$

(37)

Based on the following combinatorial identity:

$$\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r-1}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r}}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{m+1}{r}}\right),0\le r\le m+1,$$

(38)

we can write

$$x^{m+1}J{L}_i^{A,B}\left(x\right)=\sum _{r=0}^{m+1}{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m+1}{r}\right){(-A)}^{m-r+1}}{B^{m+1}}}J{L}_{i+m+r+1}^{A,B}\left(x\right),$$

(39)

and this completes the proof. □

4. Linearization Formulas of GJLPs

In this section, we develop novel linearization formulas of GJLPs. First, we derive the standard linearization formulas for the GJLPs; thereafter, we derive additional mixed linearization formulas using different polynomials.

4.1. Standard Linearization Formulas of GJLPs

In this part, we develop new linearization formulas for the GJLPs.

Theorem 4.

The following two linearization formulas hold for all positive integers m and n:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J{L}_{2m}^{A,B}\left(x\right)J{L}_n^{A,B}\left(x\right)& =\left(2m\right)!\sum _{s=0}^m{\displaystyle \frac{A^{2s}}{c_s(2m-2s)!\left(2s\right)!}}J{L}_{n+2m-2s}^{A,B}\left(x\right)\hfill \\ \hfill & -2\left(2m\right)!\sum _{s=0}^{m-1}{\displaystyle \frac{A^{2s+1}(m-s)}{(2s+1)(2m-2s)!\left(2s\right)!}}J{L}_{n+2m-2s-1}^{A,B}\left(x\right),\hfill \end{array}\end{array}$$

(40)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J{L}_{2m+1}^{A,B}\left(x\right)J{L}_n^{A,B}\left(x\right)& =(2m+1)!\sum _{s=0}^m{\displaystyle \frac{A^{2s+1}}{(2m-2s)!(2s+1)!}}J{L}_{n+2m-2s}^{A,B}\left(x\right)\hfill \\ \hfill & -(2m+1)!\sum _{s=0}^{m-1}{\displaystyle \frac{A^{2s+2}}{(2m-2s-1)!(2s+2)!}}J{L}_{n+2m-2s-1}^{A,B}\left(x\right),\hfill \end{array}\end{array}$$

(41)

and $c_s$ is defined as

$$c_s=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & s=0,\hfill \\ 1,\hfill & s>0.\hfill \end{array}\right.$$

(42)

Proof. 

Using the series form in (27), we can write

$$J{L}_{2m}^{A,B}\left(x\right)J{L}_n^{A,B}\left(x\right)=\sum _{r=0}^m{\displaystyle \frac{2A^{2r}{B}^{m-r}m(m+r-1)!}{(m-r)!\left(2r\right)!}}{x}^{m-r}J{L}_n^{A,B}\left(x\right).$$

(43)

Given the moment formula in (34), the last formula becomes

$$J{L}_{2m}^{A,B}\left(x\right)J{L}_n^{A,B}\left(x\right)=2m\sum _{r=0}^m{\displaystyle \frac{(m+r-1)!}{(m-r)!\left(2r\right)!}}\sum _{q=0}^{m-r}{(-1)}^{m-q-r}{A}^{m-q+r}\left({\displaystyle \genfrac{}{}{0pt}{}{m-r}{q}}\right)J{L}_{n+q+m-r}^{A,B}\left(x\right).$$

(44)

Rearranging and expanding the above formula enables one to write (44) in the following form:

$$J{L}_{2m}^{A,B}\left(x\right)J{L}_n^{A,B}\left(x\right)=\sum _{s=0}^m{V}_{s,m}J{L}_{n+2m-2s}^{A,B}\left(x\right)+\sum _{s=0}^{m-1}{\overline{V}}_{s,m}J{L}_{n+2m-2s-1}^{A,B}\left(x\right),$$

(45)

where

$$\begin{array}{cc}\hfill V_{s,m}& =\sum _{r=0}^s{\displaystyle \frac{2A^{2r}{B}^{m-r}m{(-A)}^{-2r+2s}{B}^{-m+r}\left(\genfrac{}{}{0pt}{}{m-r}{m+r-2s}\right)(m+r-1)!}{(m-r)!\left(2r\right)!}},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\overline{V}}_{s,m}& =\sum _{r=0}^s{\displaystyle \frac{2A^{2r}{B}^{m-r}m{(-A)}^{1-2r+2s}{B}^{-m+r}\left(\genfrac{}{}{0pt}{}{m-r}{m+r-2s-1}\right)(m+r-1)!}{(m-r)!\left(2r\right)!}}.\hfill \end{array}$$

(47)

The coefficients $V_{s,m}$ and ${\overline{V}}_{s,m}$ can be represented in hypergeometric form as

$$\begin{array}{cc}\hfill V_{s,m}& ={\displaystyle \frac{2A^{2s}\sqrt{\pi }m!{}_3\tilde{F}_2\left(\begin{array}{c}m,\frac{1}{2}-s,-s\\ \frac{1}{2},1+m-2s\end{array}|1\right)}{\left(2s\right)!}},\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\overline{V}}_{s,m}& ={\displaystyle \frac{-2A^{2s+1}\sqrt{\pi }m!{}_3\tilde{F}_2\left(\begin{array}{c}m,-\frac{1}{2}-s,-s\\ \frac{1}{2},m-2s\end{array}|1\right)}{(2s+1)!}},\hfill \end{array}$$

(49)

where the general regularized generalized hypergeometric function that appears in (48) and (49) is defined by

$${}_p\tilde{F}_q\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}|z\right)={\displaystyle \frac{{}_{p}F_q\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}|z\right)}{\mathrm{\Gamma }\left(b_1\right)\dots \mathrm{\Gamma }\left(b_q\right)}},$$

and the generalized hypergeometric function is defined by

$${}_{p}F_q\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}|z\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(a_1\right)}_k\dots {\left(a_p\right)}_k}{{\left(b_1\right)}_k\dots {\left(b_q\right)}_k}}{\displaystyle \frac{z^k}{k!}},$$

where ${\left(a\right)}_k$ denotes the Pochhammer symbol.

With the aid of Zeilberger’s algorithm [44], the coefficients in (48) and (49) can be reduced to give

$$\begin{array}{cc}\hfill V_{s,m}& ={\displaystyle \frac{A^{2s}\left(2m\right)!}{c_s(2m-2s)!\left(2s\right)!}},\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {\overline{V}}_{s,m}& ={\displaystyle \frac{-2A^{2s+1}\left(2m\right)!(m-s)}{(2s+1)(2m-2s)!\left(2s\right)!}}.\hfill \end{array}$$

(51)

Based on the two reduction Formulas in (50) and (51), Formula (45) can now be reduced to give

$$\begin{array}{cc}\hfill J{L}_{2m}^{A,B}\left(x\right)J{L}_n^{A,B}\left(x\right)& =\left(2m\right)!\sum _{s=0}^m{\displaystyle \frac{A^{2s}}{c_s(2m-2s)!\left(2s\right)!}}J{L}_{n+2m-2s}^{A,B}\left(x\right)\hfill \\ & -2\left(2m\right)!\sum _{s=0}^{m-1}{\displaystyle \frac{A^{2s+1}(m-s)}{(2s+1)(2m-2s)!\left(2s\right)!}}J{L}_{n+2m-2s-1}^{A,B}\left(x\right).\hfill \end{array}$$

This proves (40). Formula (41) can be similarly proven. □

4.2. Some Other Linearization Formulas Involving GJLPs

In this part, we give a general theorem for linearizing the products of the GJLPs with non-symmetric polynomials.

Theorem 5.

Let $m,n$ be NNIs. The following linearization formula (LF) follows for all non-symmetric polynomials ${\psi }_m\left(x\right)$ that are expressed in (10):

$$\begin{array}{cc}\hfill {\psi }_m\left(x\right)J{L}_n^{A,B}\left(x\right)=& \sum _{s=0}^m\sum _{L=0}^s{F}_{L,m}{A}^{2s-2L}{B}^{L-m}\left({\displaystyle \genfrac{}{}{0pt}{}{m-L}{L+m-2s}}\right)J{L}_{n+2m-2s}^{A,B}\left(x\right)\hfill \\ \hfill & -\sum _{s=0}^{m-1}\sum _{L=0}^s{F}_{L,m}{A}^{2s-2L+1}{B}^{L-m}\left({\displaystyle \genfrac{}{}{0pt}{}{m-L}{-1+L+m-2s}}\right)J{L}_{n+2m-2s-1}^{A,B}\left(x\right),\hfill \end{array}$$

(52)

where $F_{L,m}$ are the coefficients that appear in (10).

Proof. 

Using the series form of ${\psi }_m\left(x\right)$ in (10), we can write

$${\psi }_m\left(x\right)J{L}_n^{A,B}\left(x\right)=\sum _{r=0}^m{F}_{r,m}{x}^{m-r}J{L}_n^{A,B}\left(x\right),$$

(53)

where $F_{r,m}$ are the coefficients that appear in (10).

We apply the inversion formula of the GJLPs in (29) in the last formula to obtain the following LF:

$${\psi }_m\left(x\right)J{L}_n^{A,B}\left(x\right)=\sum _{r=0}^m{F}_{r,m}{B}^{r-m}\sum _{L=0}^{m-r}{(-A)}^{-L+m-r}\left({\displaystyle \genfrac{}{}{0pt}{}{m-r}{L}}\right)J{L}_{n+L+m-r}^{A,B}\left(x\right),$$

(54)

which, upon expanding and rearranging the terms, results in the following expression:

$$\begin{array}{cc}\hfill {\psi }_m\left(x\right)J{L}_n^{A,B}\left(x\right)=& \sum _{s=0}^m\sum _{L=0}^s{F}_{L,m}{A}^{2s-2L}{B}^{L-m}\left({\displaystyle \genfrac{}{}{0pt}{}{m-L}{L+m-2s}}\right)J{L}_{n+2m-2s}^{A,B}\left(x\right)\hfill \\ & -\sum _{s=0}^{m-1}\sum _{L=0}^s{F}_{L,m}{A}^{2s-2L+1}{B}^{L-m}\left({\displaystyle \genfrac{}{}{0pt}{}{m-L}{-1+L+m-2s}}\right)J{L}_{n+2m-2s-1}^{A,B}\left(x\right).\hfill \end{array}$$

This ends the proof. □

Remark 7.

The general LF in (52) yields several linearization formulas for different non-symmetric polynomials and GJLPs.

Corollary 2.

Let $m,n$ be two NNIs. The below LF follows:

$$\begin{array}{cc}\hfill {\tilde{R}}_m^{(\gamma ,\delta )}J{L}_n^{A,B}\left(x\right)=& -{\displaystyle \frac{B^{-m}m!\mathrm{\Gamma }(1+\gamma )\mathrm{\Gamma }(1-2m-\gamma -\delta )\mathrm{\Gamma }(2m+\gamma +\delta )}{\mathrm{\Gamma }(1+m+\gamma )\mathrm{\Gamma }(1+m+\gamma +\delta )}}\hfill \\ \hfill & \times \sum _{s=0}^m{\displaystyle \frac{A^{2s}}{\left(2s\right)!}}{}_{3}F_2\left(\begin{array}{c}-s,\frac{1}{2}-s,-m-\delta \\ 1+m-2s,-2m-\gamma -\delta \end{array}|-\frac{4B}{A^2}\right)J{L}_{2m+n-2s}^{A,B}\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{B^{-m}m!\mathrm{\Gamma }(1+\gamma )\mathrm{\Gamma }(1-2m-\gamma -\delta )\mathrm{\Gamma }(2m+\gamma +\delta )}{\mathrm{\Gamma }(1+m+\gamma )\mathrm{\Gamma }(1+m+\gamma +\delta )}}\hfill \\ \hfill & \times \sum _{s=0}^{m-1}{\displaystyle \frac{A^{2s+1}}{(2s+1)!}}{}_{3}F_2\left(\begin{array}{c}-s,-\frac{1}{2}-s,-m-\delta \\ m-2s,-2m-\gamma -\delta \end{array}|-\frac{4B}{A^2}\right)J{L}_{2m+n-2s-1}^{A,B}\left(x\right).\hfill \end{array}$$

(55)

Proof. 

Apply (52), and use the coefficients of the inversion formula of the shifted Jacobi polynomials in (17). □

Remark 8.

It is worth mentioning here that the LF in (55) can be reduced for specific choices of the involved parameters. The following corollary presents simplified linearization formulas for particular classes. From now on and due to the importance of the specific case corresponding to the choice $B={\displaystyle \frac{-A^2}{4}}$, we denote

$$J{L}_n^A\left(x\right)=J{L}_n^{A,{\displaystyle \frac{-A^2}{4}}}\left(x\right).$$

(56)

Corollary 3.

Let $m,n$ be NNIs. For $B={\displaystyle \frac{-A^2}{4}}$ and $\gamma =-{\displaystyle \frac{1}{2}}$, the following LF follows:

$$\begin{array}{cc}\hfill {\tilde{R}}_m^{({\displaystyle \frac{-1}{2}},\delta )}J{L}_n^A\left(x\right)=& {\displaystyle \frac{{(-1)}^{m}m!}{\mathrm{\Gamma }\left(\frac{1}{2}+m+\delta \right)}}[\sum _{s=0}^m{\displaystyle \frac{2^{4m-2s}{A}^{2s-2m}\mathrm{\Gamma }\left(\frac{1}{2}+2m-s+\delta \right){(1+2m-2s+\delta )}_s}{(2m-2s)!\left(2s\right)!}}J{L}_{n+2m-2s}^A\left(x\right)\hfill \\ \hfill & -\sum _{s=0}^{m-1}{\displaystyle \frac{2^{-1+4m-2s}{A}^{2s-2m+1}\mathrm{\Gamma }\left(\frac{1}{2}+2m-s+\delta \right){(2m-2s+\delta )}_s}{(2m-2s-1)!(2s+1)!}}J{L}_{n+2m-2s-1}^A\left(x\right)].\hfill \end{array}$$

(57)

Proof. 

The substitution with $B={\displaystyle \frac{-A^2}{4}}$ and $\gamma =-{\displaystyle \frac{1}{2}}$ in (55) yields the following formula:

$$\begin{array}{cc}\hfill {\tilde{R}}_m^{({\displaystyle \frac{-1}{2}},\delta )}{L}_n^A\left(x\right)=& {\displaystyle \frac{{(-1)}^{m+1}{4}^{2m}{A}^{-2m}{(m!)}^2\mathrm{\Gamma }\left(\frac{3}{2}-2m-\delta \right)\mathrm{\Gamma }\left(-\frac{1}{2}+2m+\delta \right)}{\left(2m\right)!\mathrm{\Gamma }\left(\frac{1}{2}+m+\delta \right)}}\hfill \\ \hfill & \times \sum _{s=0}^m{\displaystyle \frac{A^{2s}}{\left(2s\right)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-s,\frac{1}{2}-s,-m-\delta \\ 1+m-2s,\frac{1}{2}-2m-\delta \end{array}|1\right)J{L}_{2m+n-2s}^A\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{B^{-m}\sqrt{\pi }m!\mathrm{\Gamma }\left(\frac{3}{2}-2m-\delta \right)\mathrm{\Gamma }\left(-\frac{1}{2}+2m+\delta \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+m\right)\mathrm{\Gamma }\left(\frac{1}{2}+m+\delta \right)}}\hfill \\ \hfill & \times \sum _{s=0}^{m-1}{\displaystyle \frac{A^{2s+1}}{(2s+1)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-s,-\frac{1}{2}-s,-m-\delta \\ m-2s,-2m-\delta +\frac{1}{2}\end{array}|1\right)J{L}_{2m+n-2s-1}^A\left(x\right).\hfill \end{array}$$

(58)

Now, the two ${}_3\tilde{F}_2\left(1\right)$ that appear in (58) can be summed using Zeilberger’s algorithm [44] in two closed forms to give

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_3\tilde{F}_2\left(\begin{array}{c}\frac{1}{2}-s,-s,-m-\delta \\ 1+m-2s,\frac{1}{2}-2m-\delta \end{array}|1\right)=& {\displaystyle \frac{{(-1)}^s(2m-s+\delta ){\left(\frac{1}{2}+m-s\right)}_s{\left(2m-2s+\delta \right)}_s}{(2m-2s+\delta )(m-s)!\mathrm{\Gamma }\left(\frac{1}{2}-2m+s-\delta \right)}},\hfill \end{array}\end{array}$$

(59)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {}_3\tilde{F}_2\left(\begin{array}{c}-\frac{1}{2}-s,-s,-m-\delta \\ m-2s,-2m-\delta +\frac{1}{2}\end{array}|1\right)=& {\displaystyle \frac{{(-1)}^s{\left(\frac{1}{2}+m-s\right)}_s{\left(2m-2s+\delta \right)}_s}{(m-s-1)!\mathrm{\Gamma }\left(\frac{1}{2}-2m+s-\delta \right)}},\hfill \end{array}\end{array}$$

(60)

and, thus, Formula (58) can be simplified to give

$$\begin{array}{cc}\hfill {\tilde{R}}_m^{({\displaystyle \frac{-1}{2}},\delta )}J{L}_n^A\left(x\right)=& {\displaystyle \frac{{(-1)}^{m}m!}{\mathrm{\Gamma }\left(\frac{1}{2}+m+\delta \right)}}[\sum _{s=0}^m{\displaystyle \frac{2^{4m-2s}{A}^{2s-2m}\mathrm{\Gamma }\left(\frac{1}{2}+2m-s+\delta \right){(1+2m-2s+\delta )}_s}{(2m-2s)!\left(2s\right)!}}J{L}_{n+2m-2s}^A\left(x\right)\hfill \\ \hfill & -\sum _{s=0}^{m-1}{\displaystyle \frac{2^{-1+4m-2s}{A}^{2s-2m+1}\mathrm{\Gamma }\left(\frac{1}{2}+2m-s+\delta \right){(2m-2s+\delta )}_s}{(2m-2s-1)!(2s+1)!}}J{L}_{n+2m-2s-1}^A\left(x\right)].\hfill \end{array}$$

This ends the proof. □

Corollary 4.

For all positive integers $m,n$, the below LF follows:

$$\begin{array}{cc}\hfill T_m^{*}\left(x\right)J{L}_n^A\left(x\right)=& {(-1)}^{m}m[\sum _{s=0}^m{\displaystyle \frac{2^{1+4m-4s}{A}^{2s-2m}(4m-2s-1)!}{\left(2s\right)!(4m-4s)!}}J{L}_{n+2m-2s}^A\left(x\right)\hfill \\ \hfill & -\sum _{s=0}^{m-1}{\displaystyle \frac{2^{-1+4m-4s}{A}^{2s-2m+1}(4m-2s-2)!}{(2s+1)!(4m-4s-2)!}}J{L}_{n+2m-2s-1}^A\left(x\right)].\hfill \end{array}$$

(61)

Proof. 

Formula (61) follows immediately from (57), setting $\delta =-{\displaystyle \frac{1}{2}}$. □

Corollary 5.

For all positive integers $m,n$, the below LF follows:

$$\begin{array}{cc}\hfill V_m^{*}\left(x\right)J{L}_n^A\left(x\right)=& {(-1)}^m[\sum _{s=0}^m{\displaystyle \frac{4^{2m-2s}{A}^{2s-2m}(4m-2s+1)!}{\left(2s\right)!(4m-4s+1)!}}J{L}_{n+2m-2s}^A\left(x\right)\hfill \\ \hfill & -\sum _{s=0}^{m-1}{\displaystyle \frac{4^{-1+2m-2s}{A}^{2s-2m+1}(4m-2s)!}{(2s+1)!(4m-4s-1)!}}J{L}_{n+2m-2s-1}^A\left(x\right)].\hfill \end{array}$$

(62)

Proof. 

Formula (62) follows immediately from (57), setting $\delta ={\displaystyle \frac{1}{2}}$. □

Corollary 6.

Consider $m,n$ as NNIs, and let $C_i\left(x\right)$ be the Chebyshev polynomials that are generated by (22). The following LFs hold:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & J{L}_{2m}^{A,B}{C}_n\left(x\right)=2^{1-m}{B}^m\sum _{r=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r}}\right){}_{4}F_3\left(\begin{array}{c}-r,\frac{1}{2}+\frac{m}{2},\frac{m}{2},-m+r\\ \frac{1}{4},\frac{1}{2},\frac{3}{4}\end{array}|\frac{A^4}{16B^2}\right)C_{m+n-2r}\left(x\right)\hfill \\ \hfill & +2^{1-m}{A}^2{B}^{m-1}{m}^2\sum _{r=0}^{m-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{r}}\right){}_{4}F_3\left(\begin{array}{c}-r,\frac{1}{2}+\frac{m}{2},1+\frac{m}{2},1-m+r\\ \frac{3}{4},\frac{5}{4},\frac{3}{2}\end{array}|\frac{A^4}{16B^2}\right)C_{m+n-2r-1}\left(x\right),\hfill \end{array}\end{array}$$

(63)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & J{L}_{2m+1}^{A,B}{C}_n\left(x\right)=2^{-m}A{B}^m(1+2m)\sum _{r=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r}}\right){}_{4}F_3\left(\begin{array}{c}-r,\frac{1}{2}+\frac{m}{2},1+\frac{m}{2},-m+r\\ \frac{1}{2},\frac{3}{4},\frac{5}{4}\end{array}|\frac{A^4}{16B^2}\right)C_{m+n-2r}\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{2^{-m}{A}^3{B}^{m-1}(2m+1)(m+1)m}{3}}\sum _{r=0}^{m-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{r}}\right){}_{4}F_3\left(\begin{array}{c}-r,1+\frac{m}{2},\frac{3}{2}+\frac{m}{2},1-m+r\\ \frac{5}{4},\frac{3}{2},\frac{7}{4}\end{array}|\frac{A^4}{16B^2}\right)C_{m+n-2r-1}\left(x\right).\hfill \end{array}\end{array}$$

(64)

Proof. 

Using the series form in (27) yields

$$J{L}_{2m}^{A,B}{C}_n\left(x\right)=2m\sum _{L=0}^m{\displaystyle \frac{A^{2L}{B}^{m-L}(-1+L+m)!}{\left(2L\right)!(m-L)!}}{x}^{m-L}{C}_n\left(x\right).$$

(65)

Inserting the moment formula in (23) allows one to obtain the following formula:

$$J{L}_{2m}^{A,B}{C}_n\left(x\right)=2m\sum _{L=0}^m{\displaystyle \frac{A^{2L}{B}^{m-L}(-1+L+m)!2^{L-m}}{\left(2L\right)!(m-L)!}}\sum _{s=0}^{m-L}\left({\displaystyle \genfrac{}{}{0pt}{}{m-L}{s}}\right)C_{m+n-L-2s}\left(x\right),$$

(66)

which can be alternatively written as

$$\begin{array}{cc}\hfill J{L}_{2m}^{A,B}{C}_n\left(x\right)& =\sum _{r=0}^m\sum _{L=0}^r{\displaystyle \frac{2^{1+2L-m}{A}^{4L}{B}^{m-2L}m\left(\genfrac{}{}{0pt}{}{m-2L}{r-L}\right)(2L+m-1)!}{\left(4L\right)!(m-2L)!}}{C}_{m+n-2r}\left(x\right)\hfill \\ \hfill & +\sum _{r=0}^{m-1}\sum _{L=0}^r{\displaystyle \frac{2^{2+2L-m}{A}^{2+4L}{B}^{m-1-2L}m\left(\genfrac{}{}{0pt}{}{m-1-2L}{r-L}\right)(2L+m)!}{(4L+2)!(m-2L-1)!}}{C}_{m+n-2r-1}\left(x\right).\hfill \end{array}$$

(67)

which can also be represented as

$$\begin{array}{cc}\hfill J{L}_{2m}^{A,B}{C}_n\left(x\right)& =2^{1-m}{B}^m\sum _{r=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r}}\right){}_{4}F_3\left(\begin{array}{c}\frac{1}{2}+\frac{m}{2},\frac{m}{2},-r,-m+r\\ \frac{1}{4},\frac{1}{2},\frac{3}{4}\end{array}|\frac{A^4}{16B^2}\right)C_{m+n-2r}\left(x\right)\hfill \\ \hfill & +2^{1-m}{A}^2{B}^{-1+m}{m}^2\sum _{r=0}^{m-1}\left({\displaystyle \genfrac{}{}{0pt}{}{m-1}{r}}\right){}_{4}F_3\left(\begin{array}{c}\frac{1}{2}+\frac{m}{2},1+\frac{m}{2},-r,1-m+r\\ \frac{3}{4},\frac{5}{4},\frac{3}{2}\end{array}|\frac{A^4}{16B^2}\right)C_{m+n-2r-1}\left(x\right).\hfill \end{array}$$

(68)

This proves (63). Formula (64) can also be derived using similar steps. □

5. Some New Derivative Formulas of GJLPs and Their Inverse Formulas

In this section, we give several expressions of the derivatives of GJLPs in terms of different polynomials. We provide closed-form expressions for the derivatives of the GJLPs as combinations of their original expressions. In addition, other expressions of the derivatives of the GJLPs are given in terms of other polynomials. Some inverse formulas are also given.

5.1. Derivatives of GJLPs in Terms of Their Original Polynomials

The following theorem presents the two expressions of the general derivative formulas of the GJLPs in terms of their original polynomials.

Theorem 6.

Let $m,r$ be two NNIs with $r\ge m$. The following derivative formulas hold:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^{m}J{L}_{2r}^{A,B}\left(x\right)& =B^m\sqrt{\pi }r![\sum _{s=0}^{r-m}{\displaystyle \frac{A^{2s}}{\left(2s\right)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-s,r,\frac{1}{2}-s\\ \frac{1}{2},1+r-2s-m\end{array}|1\right)J{L}_{2r-2m-2s}^{A,B}\left(x\right)\hfill \\ \hfill & -\sum _{s=0}^{r-m-1}{\displaystyle \frac{A^{2s+1}}{(2s+1)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-s,r,-\frac{1}{2}-s\\ \frac{1}{2},r-2s-m\end{array}|1\right)J{L}_{2r-2m-2s-1}^{A,B}\left(x\right)],\hfill \end{array}\end{array}$$

(69)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^{m}J{L}_{2r+1}^{A,B}\left(x\right)& ={\displaystyle \frac{B^m\sqrt{\pi }(2r+1)r!}{4}}[\sum _{s=0}^{r-m}{\displaystyle \frac{A^{2s+1}}{\left(2s\right)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-s,1+r,\frac{1}{2}-s\\ \frac{3}{2},1+r-2s-m\end{array}|1\right)J{L}_{2r-2m-2s}^{A,B}\left(x\right)\hfill \\ \hfill & -\sum _{s=0}^{r-m-1}{\displaystyle \frac{A^{2s+2}}{(2s+1)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-s,1+r,-\frac{1}{2}-s\\ \frac{3}{2},r-2s-m\end{array}|1\right)J{L}_{2r-2m-2s-1}^{A,B}\left(x\right)].\hfill \end{array}\end{array}$$

(70)

Proof. 

The two formulas can be proved in a similar manner. Here, we prove (69). Using the series form in (27), we get

$${\mathrm{D}}^{m}J{L}_{2r}^{A,B}\left(x\right)=2r\sum _{L=0}^{r-m}{\displaystyle \frac{A^{2L}{B}^{-L+r}(L+r-1)!}{\left(2L\right)!(r-m-L)!}}{x}^{r-L-m}.$$

(71)

Applying the inversion formula in (29) transforms the previous formula into

$$\begin{array}{c}\hfill {\mathrm{D}}^{m}J{L}_{2r}^{A,B}\left(x\right)=2r\sum _{L=0}^{r-m}{\displaystyle \frac{A^{2L}{B}^m(L+r-1)!}{\left(2L\right)!(r-m-L)!}}\left[\sum _{s=0}^{r-L-m}{\displaystyle \frac{A^{2s}{\left(1-L-m+r-2s\right)}_{2s}}{2\left(2s\right)!}}J{L}_{2r-2L-2m-2s}^{A,B}\left(x\right)\right.\\ \hfill \left.-\sum _{s=0}^{r-L-m-1}{\displaystyle \frac{A^{2s+1}{\left(-L-m+r-2s\right)}_{2s+1}}{2(2s+1)!}}J{L}_{2r-2L-2m-2s-1}^{A,B}\left(x\right)\right],\end{array}$$

(72)

which can be converted again into the following formula after some algebraic manipulations:

$$\begin{array}{cc}\hfill {\mathrm{D}}^{m}J{L}_{2r}^{A,B}\left(x\right)& =B^{m}r\left(\sum _{s=0}^{r-m}{A}^{2s}\sum _{L=0}^s{\displaystyle \frac{(L+r-1)!}{\left(2L\right)!(2s-2L)!(L-m+r-2s)!}}J{L}_{2r-2m-2s}^{A,B}\left(x\right)\right)\hfill \\ \hfill & -\sum _{s=0}^{r-m-1}\left(\sum _{L=0}^s{\displaystyle \frac{A^{2s+1}(L+r-1)!}{\left(2L\right)!(L-m+r-2s-1)!(2s+1-2L)!}}\right)J{L}_{2r-2m-2s-1}^{A,B}\left(x\right).\hfill \end{array}$$

(73)

Finally, in hypergeometric form, we can write

$$\begin{array}{cc}\hfill D^{m}J{L}_{2r}^{A,B}\left(x\right)& =B^m\sqrt{\pi }r![\sum _{s=0}^{r-m}{\displaystyle \frac{A^{2s}}{\left(2s\right)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-s,r,\frac{1}{2}-s\\ \frac{1}{2},1+r-2s-m\end{array}|1\right)J{L}_{2r-2m-2s}^{A,B}\left(x\right)\hfill \\ \hfill & -\sum _{s=0}^{r-m-1}{\displaystyle \frac{A^{2s+1}}{(2s+1)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-s,r,-\frac{1}{2}-s\\ \frac{1}{2},r-2s-m\end{array}|1\right)J{L}_{2r-2m-2s-1}^{A,B}\left(x\right)].\hfill \end{array}$$

This proves Formula (69). Formula (70) can be proved in a similar manner. □

5.2. Derivatives of GJLPs in Terms of Non-Symmetric Polynomials

This part is confined to presenting a general derivative formula for the GJLPs in terms of any non-symmetric polynomial. Additionally, some connection formulas are derived as special cases.

Theorem 7.

For two NNIs $r,m$ with $r\ge m$, the following two derivative formulas hold:

$$\begin{array}{cc}\hfill D^{m}J{L}_{2r}^{A,B}\left(x\right)& =2r\sum _{s=0}^{r-m}\left(\sum _{L=0}^s{\displaystyle \frac{A^{2L}{B}^{r-L}(L+r-1)!}{\left(2L\right)!(r-m-L)!}}{\overline{F}}_{s-L,r-L-m}\right){\psi }_{r-m-s}\left(x\right),\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill D^{m}J{L}_{2r+1}^{A,B}\left(x\right)& =(2r+1)\sum _{s=0}^{r-m}\left(\sum _{L=0}^s{\displaystyle \frac{A^{1+2L}{B}^{r-L}(L+r)!}{(2L+1)!(r-m-L)!}}{\overline{F}}_{s-L,r-L-m}\right){\psi }_{r-m-s}\left(x\right),\hfill \end{array}$$

(75)

where ${\overline{F}}_{r,n}$ are the coefficients that appear in (12).

Proof. 

The analytic form of the GJLPs in (27) allows one to write

$$D^{m}J{L}_{2r}^{A,B}\left(x\right)=2r\sum _{L=0}^{r-m}{\displaystyle \frac{A^{2L}{B}^{r-L}(L+r-1)!}{\left(2L\right)!(r-m-L)!}}{x}^{r-L-m}.$$

(76)

The inversion formula in (12) converts the last formula into the following formula:

$$D^{m}J{L}_{2r}^{A,B}\left(x\right)=2r\sum _{L=0}^{r-m}{\displaystyle \frac{A^{2L}{B}^{r-L}(L+r-1)!}{\left(2L\right)!(r-m-L)!}}\sum _{t=0}^{r-L-m}{\overline{F}}_{t,r-L-m}{\psi }_{r-L-m-t}\left(x\right).$$

(77)

The arrangement of the previous formula leads to

$$D^{m}J{L}_{2r}^{A,B}\left(x\right)=2r\sum _{s=0}^{r-m}\left(\sum _{L=0}^s{\displaystyle \frac{A^{2L}{B}^{r-L}(L+r-1)!}{\left(2L\right)!(r-m-L)!}}{\overline{F}}_{s-L,r-L-m}\right){\psi }_{r-m-s}\left(x\right).$$

(78)

This proves (74). Formula (75) can be similarly proved. □

Remark 9.

Many formulas regarding the derivatives of GJLPs in terms of various non-symmetric polynomials can be obtained. The following corollaries present some of these formulas.

Corollary 7.

Let r and m be NNIs with $r\ge m$. In combinations of the shifted Jacobi polynomials, the derivatives of the GJLPs are represented as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^{m}J{L}_{2r}^{A,B}\left(x\right)& ={\displaystyle \frac{2B^{r}r!\mathrm{\Gamma }(1-m+r+\delta )}{\mathrm{\Gamma }(1+\gamma )}}\hfill \\ \hfill & \times \sum _{s=0}^{r-m}{\displaystyle \frac{(1-2m+2r-2s+\gamma +\delta )\mathrm{\Gamma }(1-m+r-s+\gamma )\mathrm{\Gamma }(1-m+r-s+\gamma +\delta )}{(r-m-s)!s!\mathrm{\Gamma }(1-m+r-s+\delta )\mathrm{\Gamma }(2-2m+2r-s+\gamma +\delta )}}\hfill \\ \hfill & \times {}_{3}F_2\left(\begin{array}{c}-s,r,-1+2m-2r+s-\gamma -\delta \\ \frac{1}{2},m-r-\delta \end{array}|-{\displaystyle \frac{A^2}{4B}}\right){\tilde{R}}_{r-m-s}^{(\gamma ,\delta )}\left(x\right),\hfill \end{array}\end{array}$$

(79)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^{m}J{L}_{2r+1}^{A,B}\left(x\right)& ={\displaystyle \frac{A{B}^r(2r+1)r!\mathrm{\Gamma }(1-m+r+\delta )}{\mathrm{\Gamma }(1+\gamma )}}\hfill \\ \hfill & \times \sum _{s=0}^{r-m}{\displaystyle \frac{(1-2m+2r-2s+\gamma +\delta )\mathrm{\Gamma }(1-m+r-s+\gamma )\mathrm{\Gamma }(1-m+r-s+\gamma +\delta )}{(r-m-s)!s!\mathrm{\Gamma }(1-m+r-s+\delta )\mathrm{\Gamma }(2-2m+2r-s+\gamma +\delta )}}\hfill \\ \hfill & \times {}_{3}F_2\left(\begin{array}{c}-s,r+1,-1-2r+s+2m-\gamma -\delta \\ \frac{3}{2},m-r-\delta \end{array}|-{\displaystyle \frac{A^2}{4B}}\right){\tilde{R}}_{r-m-s}^{(\gamma ,\delta )}\left(x\right).\hfill \end{array}\end{array}$$

(80)

Proof. 

The preceding formulas follow from Theorem 7 by employing the inversion coefficients of ${\tilde{R}}_i^{(\gamma ,\delta )}\left(x\right)$ given in (17). □

Corollary 8.

Let r and m be NNIs with $r\ge m$. In combinations of Bernoulli polynomials, the derivatives of the GJLPs are represented as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^{m}J{L}_{2r}^{A,B}\left(x\right)& =2r\sum _{s=0}^{r-m}{\displaystyle \frac{B^{r-s-1}\left[-A^{2s+2}(s+1)!(r+s)!+B^{1+s}(r-1)!(2s+2)!{}_{2}F_1\left(\begin{array}{c}r,-1-s\\ \frac{1}{2}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)\right]}{(r-m-s)!(s+1)!(2s+2)!}}\hfill \\ \hfill & \times B_{r-m-s}\left(x\right),\hfill \end{array}\end{array}$$

(81)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^{m}J{L}_{2r+1}^{A,B}\left(x\right)& =(2r+1)\sum _{s=0}^{r-m}{\displaystyle \frac{\left[-{\displaystyle \frac{A^{3+2s}{B}^{r-s-1}(r+s+1)!}{(2s+3)!}}+{\displaystyle \frac{A{B}^{r}r!}{(s+1)!}}{}_{2}F_1\left(\begin{array}{c}1+r,-1-s\\ \frac{3}{2}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)\right]}{(r-s-m)!}}{B}_{r-m-s}\left(x\right).\hfill \end{array}\end{array}$$

(82)

Proof. 

The above results can be deduced from Theorem 7 using the following inversion formula for Bernoulli polynomials:

$$x^i=\sum _{r=0}^j{\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{1+j}{j-r}\right)}{1+j}}{B}_{j-r}\left(x\right).$$

(83)

□

Remark 10.

The derivative formulas in this section provide connection formulas for the GJLPs, with non-symmetric polynomials as special cases. The subsequent corollaries illustrate some of these formulas, and the next section introduces additional ones.

5.3. Some New Connection Formulas

This section introduces new connection formulas for the GJLPs, linking them directly as special cases of the derivative formulas to well-known polynomials. In addition, symbolic algebra helps give some connection formulas in simple reduced formulas.

Corollary 9.

The Jacobsthal–Lucas-shifted Jacobi connection formula can be written in the following form:

$$\begin{array}{cc}\hfill J{L}_{2r}^{A,B}\left(x\right)=& {\displaystyle \frac{2B^{r}r!\mathrm{\Gamma }(1+r+\delta )}{\mathrm{\Gamma }(1+\gamma )}}\sum _{s=0}^r{\displaystyle \frac{(1+2r-2s+\gamma +\delta )\mathrm{\Gamma }(1+r-s+\gamma )\mathrm{\Gamma }(1+r-s+\gamma +\delta )}{(r-s)!s!\mathrm{\Gamma }(1+r-s+\delta )\mathrm{\Gamma }(2+2r-s+\gamma +\delta )}}\hfill \\ \hfill & \times {}_{3}F_2\left(\begin{array}{c}r,-s,-1-2r+s-\gamma -\delta \\ \frac{1}{2},-r-\delta \end{array}|-\frac{A^2}{4B}\right){\tilde{R}}_{r-s}^{(\gamma ,\delta )}\left(x\right),\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill J{L}_{2r+1}^{A,B}\left(x\right)=& {\displaystyle \frac{A{B}^r(2r+1)r!\mathrm{\Gamma }(1+r+\delta )}{\mathrm{\Gamma }(1+\gamma )}}\sum _{s=0}^r{\displaystyle \frac{(1+2r-2s+\gamma +\delta )\mathrm{\Gamma }(1+r-s+\gamma )\mathrm{\Gamma }(1+r-s+\gamma +\delta )}{(r-s)!s!\mathrm{\Gamma }(1+r-s+\delta )\mathrm{\Gamma }(2+2r-s+\gamma +\delta )}}\hfill \\ \hfill & \times {}_{3}F_2\left(\begin{array}{c}r+1,-s,-1-2r+s-\gamma -\delta \\ \frac{3}{2},-r-\delta \end{array}|-\frac{A^2}{4B}\right){\tilde{R}}_{r-s}^{(\gamma ,\delta )}\left(x\right).\hfill \end{array}$$

(85)

Proof. 

The two formulas are direct consequences of Formulas (79) and (80), achieved by setting $m=0$. □

Remark 11.

Some simplified connection formulas for specific classes of the shifted Jacobi polynomials with specific classes of the GJLPs can be obtained. The following corollaries exhibit these formulas.

Corollary 10.

For every positive integer r, the below connection formulas follows:

$$\begin{array}{cc}\hfill J{L}_{2r}^A\left(x\right)=& {\displaystyle \frac{{(-1)}^r{2}^{1+2\delta }{A}^{2r}\left(2r\right)!\mathrm{\Gamma }(1+2r+\delta )}{\sqrt{\pi }}}\sum _{s=0}^r{\displaystyle \frac{{(-1)}^s(1+4r-4s+2\delta )\mathrm{\Gamma }\left(\frac{1}{2}+r-s+\delta \right)}{(r-s)!\left(2s\right)!\mathrm{\Gamma }(2+4r-2s+2\delta )}}\hfill \\ \hfill & \times {\tilde{R}}_{r-s}^{({\displaystyle \frac{-1}{2}},\delta )}\left(x\right),\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill J{L}_{2r+1}^A\left(x\right)=& {\displaystyle \frac{{(-1)}^r{2}^{2+2\delta }{A}^{2r+1}(2r+1)!\mathrm{\Gamma }(2+2r+\delta )}{\sqrt{\pi }}}\sum _{s=0}^r{\displaystyle \frac{{(-1)}^s\left(\frac{1}{2}+2r-2s+\delta \right)\mathrm{\Gamma }\left(\frac{1}{2}+r-s+\delta \right)}{(r-s)!(2s+1)!\mathrm{\Gamma }(3+4r-2s+2\delta )}}\hfill \\ \hfill & \times {\tilde{R}}_{r-s}^{({\displaystyle \frac{-1}{2}},\delta )}\left(x\right).\hfill \end{array}$$

(87)

Proof. 

We prove (86). Setting $\gamma =-{\displaystyle \frac{1}{2}},$ and $B=-{\displaystyle \frac{A^2}{4}}$ in (84) yields the following formula:

$$\begin{array}{cc}\hfill J{L}_{2r}^A\left(x\right)& ={\displaystyle \frac{{(-1)}^r{2}^{1-2r}{A}^{2r}r!\mathrm{\Gamma }(1+r+\delta )}{\mathrm{\Gamma }(1+\gamma )}}\sum _{s=0}^r{\displaystyle \frac{\left(\frac{1}{2}+2r-2s+\delta \right)\mathrm{\Gamma }\left(\frac{1}{2}+r-s\right)\mathrm{\Gamma }\left(\frac{1}{2}+r-s+\delta \right)}{(r-s)!s!\mathrm{\Gamma }(1+r-s+\delta )\mathrm{\Gamma }\left(\frac{3}{2}+2r-s+\delta \right)}}\hfill \\ \hfill & \times {}_{3}F_2\left(\begin{array}{c}-s,r,-\frac{1}{2}-2r+s-\delta \\ \frac{1}{2},-r-\delta \end{array}|1\right){\tilde{R}}_{r-s}^{({\displaystyle \frac{-1}{2}},\delta )}\left(x\right).\hfill \end{array}$$

(88)

Based on the Pfaff–Saalschütz identity [47], the above can be summed to give

$${}_{3}F_2\left(\begin{array}{c}-s,r,-\frac{1}{2}-2r+s-\delta \\ \frac{1}{2},-r-\delta \end{array}|1\right)={\displaystyle \frac{{(-1)}^s{\left(\frac{1}{2}+r-s\right)}_s{\left(1+2r-s+\delta \right)}_s}{{\left(\frac{1}{2}\right)}_s{\left(1+r-s+\delta \right)}_s}},$$

(89)

and Formula (88) can be reduced to give

$$\begin{array}{cc}\hfill J{L}_{2r}^A\left(x\right)& ={\displaystyle \frac{{(-1)}^r{2}^{1+2\delta }{A}^{2r}\left(2r\right)!\mathrm{\Gamma }(1+2r+\delta )}{\sqrt{\pi }}}\sum _{s=0}^r{\displaystyle \frac{{(-1)}^s(1+4r-4s+2\delta )\mathrm{\Gamma }\left(\frac{1}{2}+r-s+\delta \right)}{(r-s)!\left(2s\right)!\mathrm{\Gamma }(2+4r-2s+2\delta )}}\hfill \\ \hfill & \times {\tilde{R}}_{r-s}^{\left(-\frac{1}{2},\delta \right)}\left(x\right).\hfill \end{array}$$

This proves (86). Formula (87) can be similarly proved. □

Corollary 11.

Let r be any non-negative integer, and let $T_r^{*}\left(x\right)$ be the shifted Chebyshev polynomials of the first kind. The below connection formula follows:

$$\begin{array}{cc}\hfill J{L}_{2r}^A\left(x\right)& ={(-1)}^r{4}^{1-2r}{A}^{2r}\left(4r\right)!\sum _{s=0}^r{c}_{r-s}{\displaystyle \frac{{(-1)}^s}{(4r-2s)!\left(2s\right)!}}{T}_{r-s}^{*}\left(x\right),\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill J{L}_{2r+1}^A\left(x\right)& ={(-1)}^r{2}^{-4r}{A}^{2r+1}(4r+2)!\sum _{s=0}^r{c}_{r-s}{\displaystyle \frac{{(-1)}^s}{(4r-2s+1)!(2s+1)!}}{T}_{r-s}^{*}\left(x\right),\hfill \end{array}$$

(91)

where

$$c_r=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & r=0,\hfill \\ 1,\hfill & r>0.\hfill \end{array}\right.$$

Proof. 

They are particular formulas of (86) and (87) corresponding to the choice $\delta ={\displaystyle \frac{-1}{2}}$. □

Corollary 12.

Let r be any non-negative integer, and let $V_r^{*}\left(x\right)$ be the shifted Chebyshev polynomials of the first kind. The below connection formula follows:

$$\begin{array}{cc}\hfill J{L}_{2r}^A\left(x\right)& ={(-1)}^r{2}^{-1-8r}{A}^{2r}(1+4r)!\sum _{s=0}^r{\displaystyle \frac{{(-1)}^s{2}^{3+4r}(1+2r-2s)}{(4r-2s+2)!\left(2s\right)!}}{V}_{r-s}^{*}\left(x\right),\hfill \end{array}$$

(92)

$$\begin{array}{cc}\hfill J{L}_{2r+1}^A\left(x\right)& ={\displaystyle \frac{8{(-1)}^r{A}^{2r+1}(2r+1)!\mathrm{\Gamma }\left(\frac{5}{2}+2r\right)}{\sqrt{\pi }}}\sum _{s=0}^r{\displaystyle \frac{{(-1)}^s(1+2r-2s)}{(4r-2s+3)!(2s+1)!}}{V}_{r-s}^{*}\left(x\right).\hfill \end{array}$$

(93)

Proof. 

They are particular formulas of (86) and (87) corresponding to the choice $\delta ={\displaystyle \frac{1}{2}}$. □

Remark 12.

The connection formulas in (90) and (91) enable one to find a new trigonometric expression for the GJLPs for the case corresponding to $B={\displaystyle \frac{-A^2}{4}}$. The following corollary presents these representations.

Corollary 13.

The below trigonometric identity follows:

$$\begin{array}{cc}\hfill & J{L}_r^A\left(\frac{1+cos\theta }{2}\right)=\hfill \\ \hfill & \left\{\begin{array}{cc}\begin{array}{cc}\hfill {(-1)}^{\frac{r}{2}}{4}^{1-r}{A}^r\left(2r\right)!& {\displaystyle \sum _{s=0}^{⌊r/2⌋}\frac{{(-1)}^s{c}_{r-2s}}{(2r-2s)!\left(2s\right)!}cos\left((r-2s)\theta \right),}\hfill \end{array}\hfill & \mathit{if}r\mathit{is}\mathit{even},\hfill \\ \begin{array}{cc}\hfill {(-1)}^{\frac{r-1}{2}}{2}^{-2(r-1)}{A}^r\left(2r\right)!& {\displaystyle \sum _{s=0}^{⌊(r-1)/2⌋}\frac{{(-1)}^s{c}_{r-2s-1}}{(2s+1)!(2r-2s-1)!}cos\left((r-2s-1)\theta \right),}\hfill \end{array}\hfill & \mathit{if}r\mathit{is}\mathit{odd}.\hfill \end{array}\right.\hfill \end{array}$$

(94)

Proof. 

The two connection Formulas (90) and (91) yield the following expressions based on the trigonometric expression of the Chebyshev polynomials of the first kind:

$$\begin{array}{cc}\hfill J{L}_{2r}^A\left(\frac{1+cos\theta }{2}\right)& ={(-1)}^r{4}^{1-2r}{A}^{2r}\left(4r\right)!\sum _{s=0}^r{c}_{r-s}{\displaystyle \frac{{(-1)}^s}{(4r-2s)!\left(2s\right)!}}cos\left((r-s)\theta \right),\hfill \end{array}$$

(95)

$$\begin{array}{cc}\hfill J{L}_{2r+1}^A\left(\frac{1+cos\theta }{2}\right)& ={(-1)}^r{2}^{-4r}{A}^{2r+1}(4r+2)!\sum _{s=0}^r{c}_{r-s}{\displaystyle \frac{{(-1)}^s}{(4r-2s+1)!(2s+1)!}}cos\left((r-s)\theta \right).\hfill \end{array}$$

(96)

□

The above two formulas can be unified to give Formula (94).

Among the important connection formulas are the connection formulas for the GJLPs with Bernoulli polynomials. The following corollary presents these formulas.

Corollary 14.

The GJLP–Bernoulli connection formulas are given in the following:

$$\begin{array}{cc}\hfill J{L}_{2r}^{A,B}\left(x\right)& =2r\sum _{s=0}^r{\displaystyle \frac{B^{r-s-1}\left(-A^{2s+2}(1+s)!(r+s)!+B^{s+1}(r-1)!(2s+2)!{}_{2}F_1\left(\begin{array}{c}r,-1-s\\ \frac{1}{2}\end{array}|-\frac{A^2}{4B}\right)\right)}{(r-s)!(1+s)!(2s+2)!}}{B}_{r-s}\left(x\right),\hfill \end{array}$$

(97)

$$\begin{array}{cc}\hfill J{L}_{2r+1}^{A,B}\left(x\right)& =(2r+1)\sum _{s=0}^r{\displaystyle \frac{\left(-{\displaystyle \frac{A^{2s+3}{B}^{r-s-1}(r+s+1)!}{(2s+3)!}}+{\displaystyle \frac{A{B}^{r}r!}{(s+1)!}}{}_{2}F_1\left(\begin{array}{c}r+1,-1-s\\ \frac{3}{2}\end{array}|-\frac{A^2}{4B}\right)\right)}{(r-s)!}}{B}_{r-s}\left(x\right).\hfill \end{array}$$

(98)

Proof. 

Direct special cases of Formulas (81) and (82) are obtained by setting $m=0$. □

5.4. Derivatives of the GJLPs in Terms of Symmetric Polynomials

In this section, we develop general formulas for the derivatives of the GLPs in terms of symmetric polynomials.

Theorem 8.

Let m and r be NNIs with $r\ge m$. In addition, let ${\varphi }_r\left(x\right)$ be the symmetric polynomials that are expressed in (9). The following derivative formulas hold:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d^{m}J{L}_{2r}^{A,B}\left(x\right)}{d{x}^m}}=& 2r\sum _{s=0}^{\lfloor {\displaystyle \frac{r-m}{2}}\rfloor }\sum _{L=0}^s{\displaystyle \frac{A^{4L}{B}^{-2L+r}(2L+r-1)!}{\left(4L\right)!(-2L-m+r)!}}{\overline{H}}_{s-L,r-2L-m}{\varphi }_{r-m-2s}\left(x\right)\hfill \\ \hfill & +2r\sum _{s=0}^{\lfloor {\displaystyle \frac{r-m-1}{2}}\rfloor }\sum _{L=0}^s{\displaystyle \frac{A^{2+4L}{B}^{-1-2L+r}(2L+r)!}{(4L+2)!(-2L-m+r-1)!}}{\overline{H}}_{s-L,r-2L-m-1}{\varphi }_{r-m-2s-1}\left(x\right),\hfill \end{array}\end{array}$$

(99)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d^{m}J{L}_{2r+1}^{A,B}\left(x\right)}{d{x}^m}}=& (2r+1)\sum _{s=0}^{\lfloor {\displaystyle \frac{r-m}{2}}\rfloor }\sum _{L=0}^s{\displaystyle \frac{A^{1+4L}{B}^{-2L+r}(2L+r)!}{(4L+1)!(-2L-m+r)!}}{\overline{H}}_{s-L,r-2L-m}{\varphi }_{r-m-2s}\left(x\right)\hfill \\ \hfill & +(2r+1)\sum _{s=0}^{\lfloor {\displaystyle \frac{r-m-1}{2}}\rfloor }\sum _{L=0}^s{\displaystyle \frac{A^{3+4L}{B}^{-1-2L+r}(2L+r+1)!}{(4L+3)!(-2L-m+r-1)!}}{\overline{H}}_{s-L,r-2L-m-1}{\varphi }_{r-m-2s-1}\left(x\right),\hfill \end{array}\end{array}$$

(100)

where ${\overline{H}}_{r,n}$ are the coefficients in (11).

Proof. 

Differentiating the power series representation in (27) yields the following formula:

$${\displaystyle \frac{d^{m}J{L}_{2r}^{A,B}\left(x\right)}{d{x}^m}}=2r\sum _{L=0}^{r-m}{\displaystyle \frac{A^{2L}{B}^{-L+r}(L+r-1)!}{\left(2L\right)!(-L-m+r)!}}{x}^{r-L-m}.$$

(101)

Inserting the inversion Formula (11) in the last formula gives

$${\displaystyle \frac{d^{m}J{L}_{2r}^{A,B}\left(x\right)}{d{x}^m}}=2r\sum _{L=0}^{r-m}{\displaystyle \frac{A^{2L}{B}^{-L+r}(L+r-1)!}{\left(2L\right)!(-L-m+r)!}}\sum _{t=0}^{\lfloor {\displaystyle \frac{r-L-m}{2}}\rfloor }{\overline{H}}_{t,r-L-m}{\varphi }_{r-L-m-2t}\left(x\right),$$

(102)

which can be arranged to give

$$\begin{array}{cc}\hfill {\displaystyle \frac{d^{m}J{L}_{2r}^{A,B}\left(x\right)}{d{x}^m}}=& 2r\sum _{s=0}^{\lfloor {\displaystyle \frac{r-m}{2}}\rfloor }\sum _{L=0}^s{\displaystyle \frac{A^{4L}{B}^{-2L+r}(2L+r-1)!}{\left(4L\right)!(-2L-m+r)!}}{\overline{H}}_{s-L,r-2L-m}{\varphi }_{r-m-2s}\left(x\right)\hfill \\ \hfill & +2r\sum _{s=0}^{\lfloor {\displaystyle \frac{r-m-1}{2}}\rfloor }\sum _{L=0}^s{\displaystyle \frac{A^{2+4L}{B}^{-1-2L+r}(2L+r)!}{(4L+2)!(-2L-m+r-1)!}}{\overline{H}}_{s-L,r-2L-m-1}{\varphi }_{r-m-2s-1}\left(x\right).\hfill \end{array}$$

This completes the proof of Formula (99). The proof of Formula (100) can be proved similarly. □

We apply Theorem 8 in the following corollary to obtain the derivatives of the GJLPs in terms of the ultraspherical polynomials defined in (14).

Corollary 15.

Let m and r be NNIs with $r\ge m$. The following two derivative formulas hold:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d^{m}J{L}_{2r}^{A,B}\left(x\right)}{d{x}^m}}=& {\displaystyle \frac{2^{2+m-r-2\gamma }{B}^r\sqrt{\pi }r!}{\mathrm{\Gamma }\left(\frac{1}{2}+\gamma \right)}}\sum _{s=0}^{\lfloor \frac{1}{2}(r-m)\rfloor }{\displaystyle \frac{(-m+r-2s+\gamma )\mathrm{\Gamma }(-m+r-2s+2\gamma )}{(-m+r-2s)!s!\mathrm{\Gamma }(1-m+r-s+\gamma )}}\hfill \\ \hfill & \times {}_{4}F_3\left(\begin{array}{c}-s,\frac{r}{2},\frac{1}{2}+\frac{r}{2},m-r+s-\gamma \\ \frac{1}{4},\frac{1}{2},\frac{3}{4}\end{array}|\frac{A^4}{16B^2}\right)C_{r-m-2s}^{\left(\gamma \right)}\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{2^{2+m-r-2\gamma }{A}^2{B}^{r-1}\sqrt{\pi }rr!}{\mathrm{\Gamma }\left(\frac{1}{2}+\gamma \right)}}\sum _{s=0}^{\lfloor \frac{1}{2}(r-m-1)\rfloor }{\displaystyle \frac{(-1-m+r-2s+\gamma )\mathrm{\Gamma }(-1-m+r-2s+2\gamma )}{(-1-m+r-2s)!s!\mathrm{\Gamma }(-m+r-s+\gamma )}}\hfill \\ \hfill & \times {}_{4}F_3\left(\begin{array}{c}-s,1+\frac{r}{2},\frac{1}{2}+\frac{r}{2},1+m-r+s-\gamma \\ \frac{3}{4},\frac{5}{4},\frac{3}{2}\end{array}|\frac{A^4}{16B^2}\right)C_{r-m-2s-1}^{\left(\gamma \right)}\left(x\right),\hfill \end{array}\end{array}$$

(103)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d^{m}J{L}_{2r+1}^{A,B}\left(x\right)}{d{x}^m}}=& {\displaystyle \frac{2^{1+m-r-2\gamma }A{B}^r\sqrt{\pi }(2r+1)r!}{\mathrm{\Gamma }\left(\frac{1}{2}+\gamma \right)}}\sum _{s=0}^{\lfloor \frac{1}{2}(r-m)\rfloor }{\displaystyle \frac{(-m+r-2s+\gamma )\mathrm{\Gamma }(-m+r-2s+2\gamma )}{s!(-m+r-2s)!\mathrm{\Gamma }(1-m+r-s+\gamma )}}\hfill \\ \hfill & \times {}_{4}F_3\left(\begin{array}{c}-s,1+\frac{r}{2},\frac{1}{2}+\frac{r}{2},m-r+s-\gamma \\ \frac{1}{2},\frac{3}{4},\frac{5}{4}\end{array}|\frac{A^4}{16B^2}\right)C_{r-m-2s}^{\left(\gamma \right)}\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{2^{1+m-r-2\gamma }{A}^3{B}^{r-1}\sqrt{\pi }(2r+1)(r+1)!}{3\mathrm{\Gamma }\left(\frac{1}{2}+\gamma \right)}}\sum _{s=0}^{\lfloor \frac{1}{2}(r-m-1)\rfloor }{\displaystyle \frac{(-1-m+r-2s+\gamma )\mathrm{\Gamma }(-1-m+r-2s+2\gamma )}{s!(-1-m+r-2s)!\mathrm{\Gamma }(-m+r-s+\gamma )}}\hfill \\ \hfill & \times {}_{4}F_3\left(\begin{array}{c}-s,\frac{3}{2}+\frac{r}{2},1+\frac{r}{2},1+m-r+s-\gamma \\ \frac{5}{4},\frac{3}{2},\frac{7}{4}\end{array}|\frac{A^4}{16B^2}\right)C_{r-m-2s-1}^{\left(\gamma \right)}\left(x\right).\hfill \end{array}\end{array}$$

(104)

Proof. 

This is a direct consequence of Theorem 8. □

Remark 13.

The formulas that express the derivatives of symmetric and non-symmetric polynomials can also be derived as combinations of JLPs. We state the following theorem in this regard.

We now give a closed formula for the derivatives of non-symmetric polynomials in terms of GJLPs.

Theorem 9.

For all positive integers $r,m$ with $r\ge m$, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d^m{\psi }_r\left(x\right)}{d{x}^m}}=& {\displaystyle \frac{1}{2}}\sum _{s=0}^r\sum _{L=0}^s{(-A)}^{-2L+2s-m}{B}^{-r+L+m}\left({\displaystyle \genfrac{}{}{0pt}{}{r-L-m}{r+L-2s}}\right)F_{L,r}{(1+r-L-m)}_{m}J{L}_{2r-2s-m}^{A,B}\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\sum _{s=0}^{r-1}\sum _{L=0}^s{(-A)}^{1-2L+2s-m}{B}^{-r+L+m}\left({\displaystyle \genfrac{}{}{0pt}{}{r-L-m}{-1+r+L-2s}}\right)F_{L,r}{(1+r-L-m)}_{m}J{L}_{2r-2s-m-1}^{A,B}\left(x\right).\hfill \end{array}\end{array}$$

(105)

Proof. 

Based on the series representation in (10), together with the inversion formula of the GJLPs in (29), the formula can be obtained. □

6. Some New Definite Integrals

In this section, based on the formulas developed in this paper, some new definite integral formulas in closed form are deduced.

We comment here that the definite integrals for a given sequence of polynomials are useful in many applications. For example, they are used to evaluate inner products arising in spectral methods, and in particular in tau and Galerkin methods. Thus, they serve to obtain the matrix system resulting from the application of these methods, and thereby obtain the numerical solutions to solve several types of differential equations.

Now, the first corollary gives an important definite integral formula for the GJLPs.

Corollary 16.

The following definite integral formula follows for every non-negative integer m:

$${\int }_0^{1}J{L}_m^{A,B}\left(x\right)dx=Q_m^{A,B}=\left\{\begin{array}{cc}{\displaystyle \frac{-{\displaystyle \frac{A^{m+2}m}{B(m+1)}}+4B^{m/2}{}_{2}F_1\left(\begin{array}{c}-1-\frac{m}{2},\frac{m}{2}\\ \frac{1}{2}\end{array}|-\frac{A^2}{4B}\right)}{m+2},}\hfill & \mathit{if}m\mathit{is}\mathit{even},\hfill \\ {\displaystyle \frac{Am\left(-A^{m+1}+2B^{(m+1)/2}(m+2){}_{2}F_1\left(\begin{array}{c}\frac{-1-m}{2},\frac{m+1}{2}\\ \frac{3}{2}\end{array}|-\frac{A^2}{4B}\right)\right)}{B(m+1)(m+2)},}\hfill & \mathit{if}m\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(106)

Proof. 

Based on the two connection Formulas (97) and (98), integrating over $[0,1]$ and using the well-known integral

$${\int }_0^1{B}_j\left(x\right)dx=\left\{\begin{array}{cc}1,\hfill & j=0,\hfill \\ 0,\hfill & j>0,\hfill \end{array}\right.$$

the following two integral formulas hold:

$$\begin{array}{cc}\hfill {\int }_0^{1}J{L}_{2m}^{A,B}\left(x\right)dx& ={\displaystyle \frac{-{\displaystyle \frac{A^{2+2m}m}{B(1+2m)}}+2B^m{}_{2}F_1\left(\begin{array}{c}-1-m,m\\ \frac{1}{2}\end{array}|-\frac{A^2}{4B}\right)}{m+1}},\hfill \end{array}$$

(107)

$$\begin{array}{cc}\hfill {\int }_0^{1}J{L}_{2m+1}^{A,B}\left(x\right)dx& ={\displaystyle \frac{A(1+2m)}{2(m+1)}}\left(-{\displaystyle \frac{A^{2+2m}}{B(3+2m)}}+2B^m{}_{2}F_1\left(\begin{array}{c}-1-m,m+1\\ \frac{3}{2}\end{array}|-\frac{A^2}{4B}\right)\right).\hfill \end{array}$$

(108)

□

The last two formulas can be unified to give (106).

Remark 14.

Note that, for specific Jacobsthal classes, more simplified closed forms may be derived from (107) and (108) for certain choices of the parameters.

Corollary 17.

For $B=-{\displaystyle \frac{A^2}{4}}$, the below integral formula follows:

$$G_m^A={\int }_0^{1}J{L}_m^A\left(x\right)dx=\left\{\begin{array}{cc}{\displaystyle \frac{2^{2-m}\left(2^m{A}^{m}m+{(-A^2)}^{m/2}cos\left(\frac{m\pi }{2}\right)\right)}{(m+1)(m+2)},}\hfill & \mathit{if}m\mathit{is}\mathit{even},\hfill \\ {\displaystyle \frac{4\left(A^{m+1}m+{(-1)}^{(3+m)/2}{2}^{-m}{A}^{m+1}sin\left(\frac{m\pi }{2}\right)\right)}{A(m+1)(m+2)},}\hfill & \mathit{if}m\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(109)

Proof. 

Setting $B=-{\displaystyle \frac{A^2}{4}}$ in Formula (106), we get

$${\int }_0^{1}J{L}_m^A\left(x\right)dx=G_m^A,$$

where

$$G_m^A=\left\{\begin{array}{cc}{\displaystyle \frac{{\displaystyle \frac{4A^{m}m}{m+1}}+{(-1)}^{\frac{m}{2}}{2}^{2-m}{A}^m{}_{2}F_1\left(\begin{array}{c}-1-\frac{m}{2},\frac{m}{2}\\ \frac{1}{2}\end{array}|1\right)}{m+2},}\hfill & \mathrm{if}m\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \frac{4A^{m+2}m\left(1-{(-1)}^{\frac{m+1}{2}}{2}^{-m}(m+2){}_{2}F_1\left(\begin{array}{c}-\frac{m+1}{2},\frac{m+1}{2}\\ \frac{3}{2}\end{array}|1\right)\right)}{A^2(m+1)(m+2)},}\hfill & \mathrm{if}m\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

(110)

Based on the well-known Gauss summation formula, the two ${}_{2}F_1\left(1\right)$ that appear in (110) can be summed in closed forms to give

$$\begin{array}{cc}\hfill {}_{2}F_1\left(\begin{array}{c}-1-\frac{m}{2},\frac{m}{2}\\ \frac{1}{2}\end{array}|1\right)& ={\displaystyle \frac{\pi }{2\mathrm{\Gamma }\left(\frac{1-m}{2}\right)\mathrm{\Gamma }\left(\frac{3+m}{2}\right)}},\hfill \end{array}$$

(111)

$$\begin{array}{cc}\hfill {}_{2}F_1\left(\begin{array}{c}\frac{1}{2}(-1-m),\frac{m+1}{2}\\ \frac{3}{2}\end{array}|1\right)& ={\displaystyle \frac{\pi }{4\mathrm{\Gamma }\left(1-\frac{m}{2}\right)\mathrm{\Gamma }\left(2+\frac{m}{2}\right)}}.\hfill \end{array}$$

(112)

It is clear that they can also be written as

$$\begin{array}{cc}\hfill {}_{2}F_1\left(\begin{array}{c}-1-\frac{m}{2},\frac{m}{2}\\ \frac{1}{2}\end{array}|1\right)& =\left\{\begin{array}{cc}{\displaystyle \frac{{(-1)}^{m/2}}{m+1}},\hfill & \mathrm{if}m\mathrm{is}\mathrm{even},\hfill \\ 0,\hfill & \mathrm{if}m\mathrm{is}\mathrm{odd},\hfill \end{array}\right.\hfill \end{array}$$

(113)

$$\begin{array}{cc}\hfill {}_{2}F_1\left(\begin{array}{c}\frac{1}{2}(-1-m),\frac{m+1}{2}\\ \frac{3}{2}\end{array}|1\right)& =\left\{\begin{array}{cc}{\displaystyle \frac{\pi }{4}},\hfill & \mathrm{if}m=0,\hfill \\ 0,\hfill & \mathrm{if}m\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \frac{{(-1)}^{(m-1)/2}}{m(m+2)}},\hfill & \mathrm{if}m\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.\hfill \end{array}$$

(114)

Using the above two simple forms, the coefficients $G_m^A$ can be written as

$$G_m^A=\left\{\begin{array}{cc}{\displaystyle \frac{2^{2-m}\left(2^m{A}^{m}m+{(-A^2)}^{m/2}cos\left(\frac{m\pi }{2}\right)\right)}{(m+1)(m+2)},}\hfill & \mathrm{if}m\mathrm{is}\mathrm{even},\hfill \\ {\displaystyle \frac{4\left(A^{m+1}m+{(-1)}^{(3+m)/2}{2}^{-m}{A}^{m+1}sin\left(\frac{m\pi }{2}\right)\right)}{A(m+1)(m+2)},}\hfill & \mathrm{if}m\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

(115)

This proves formula (109). □

Corollary 18.

Let $m,n$ be two NNIs. We have

$${\int }_0^{1}J{L}_m^{A,B}\left(x\right)J{L}_n^{A,B}\left(x\right)dx=S_{m,n}^{A,B},$$

(116)

where

$$S_{m,n}^{A,B}=m!\left\{\begin{array}{cc}{\displaystyle \sum _{s=0}^{⌊m/2⌋}{\displaystyle \frac{A^{2s}{Q}_{m+n-2s}^{A,B}}{c_s(m-2s)!\left(2s\right)!}}-\sum _{s=0}^{⌊m/2⌋-1}{\displaystyle \frac{A^{2s+1}(m-2s)Q_{m+n-2s-1}^{A,B}}{(2s+1)(m-2s)!\left(2s\right)!}},}\hfill & \mathit{if}m\mathit{is}\mathit{even},\hfill \\ {\displaystyle \sum _{s=0}^{⌊(m-1)/2⌋}{\displaystyle \frac{A^{2s+1}{Q}_{m+n-2s-1}^{A,B}}{(m-2s-1)!(2s+1)!}}-\sum _{s=0}^{⌊(m-3)/2⌋}{\displaystyle \frac{A^{2s+2}{Q}_{m+n-2s-2}^{A,B}}{(m-2s-2)!(2s+2)!}},}\hfill & \mathit{if}m\mathit{is}\mathit{odd},\hfill \end{array}\right.$$

(117)

where $Q_m^{A,B}$ is given in (106), and $c_s$ is defined by (42).

Proof. 

Formula (117) can be obtained using the two standard linearization Formulas (40) and (41) and integrating over $[0,1]$. □

Corollary 19.

Let $m,n$ be two NNIs. The below integral formula follows:

$$\begin{array}{cc}\hfill {\int }_0^1{\tilde{R}}_m^{({\displaystyle \frac{-1}{2}},\delta )}\left(x\right)J{L}_n^A\left(x\right)dx=& {\displaystyle \frac{{(-1)}^{m}m!}{\mathrm{\Gamma }\left(\frac{1}{2}+m+\delta \right)}}[\sum _{s=0}^m{\displaystyle \frac{2^{4m-2s}{A}^{2s-2m}\mathrm{\Gamma }\left(\frac{1}{2}+2m-s+\delta \right){(1+2m-2s+\delta )}_s}{(2m-2s)!\left(2s\right)!}}{G}_{n+2m-2s}^A\hfill \\ \hfill & -\sum _{s=0}^{m-1}{\displaystyle \frac{2^{-1+4m-2s}{A}^{2s-2m+1}\mathrm{\Gamma }\left(\frac{1}{2}+2m-s+\delta \right){(2m-2s+\delta )}_s}{(2m-2s-1)!(2s+1)!}}{G}_{n+2m-2s-1}^A],\hfill \end{array}$$

(118)

and $G_m^A$ is given in (115).

Proof. 

This demonstrates a direct use of Formula (57) together with the definite integral in (109). □

Corollary 20.

As two special cases of Formula (118), the following specific integral formulas hold:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^1{T}_m^{*}\left(x\right)J{L}_n^A\left(x\right)dx& ={(-1)}^{m}m\sum _{s=0}^m{\displaystyle \frac{2^{1+4m-4s}{A}^{2s-2m}(4m-2s-1)!}{\left(2s\right)!(4m-4s)!}}{G}_{n+2m-2s}^A\hfill \\ \hfill & -\sum _{s=0}^{m-1}{\displaystyle \frac{2^{-1+4m-4s}{A}^{2s-2m+1}(4m-2s-2)!}{(2s+1)!(4m-4s-2)!}}{G}_{n+2m-2s-1}^A,\hfill \end{array}\end{array}$$

(119)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^1{V}_m^{*}\left(x\right)J{L}_n^A\left(x\right)dx& ={(-1)}^m\sum _{s=0}^m{\displaystyle \frac{4^{2m-2s}{A}^{2s-2m}(4m-2s+1)!}{\left(2s\right)!(4m-4s+1)!}}{G}_{n+2m-2s}^A\hfill \\ \hfill & -\sum _{s=0}^{m-1}{\displaystyle \frac{4^{-1+2m-2s}{A}^{2s-2m+1}(4m-2s)!}{(2s+1)!(4m-4s-1)!}}{G}_{n+2m-2s-1}^A.\hfill \end{array}\end{array}$$

(120)

Proof. 

Formulas (119) and (120) can be deduced by setting $\delta ={\displaystyle \frac{-1}{2}}$ and $\delta ={\displaystyle \frac{1}{2}}$, respectively, in (118). □

Corollary 21.

Let $m,n$ be two NNIs. The below integral formula follows:

$${\int }_0^1{\displaystyle \frac{J{L}_r^A\left(x\right)T_m^{*}\left(x\right)}{\sqrt{x(1-x)}}}dx={\displaystyle \frac{{(-1)}^m{2}^{1-2r}{A}^r\pi \left(2r\right)!}{(-2m+r)!(2m+r)!}}.$$

(121)

Proof. 

Starting from the connection Formula (90), multiplying both sides by ${\displaystyle \frac{T_m^{*}\left(x\right)}{\sqrt{x(1-x)}}}$, and making use of the orthogonality relation of $T_m^{*}\left(x\right)$, we get

$${\int }_0^1{\displaystyle \frac{J{L}_{2r}^A\left(x\right)T_m^{*}\left(x\right)}{\sqrt{x(1-x)}}}dx={(-1)}^r\pi 2^{1-4r}{A}^{2r}\left(4r\right)!\sum _{s=0}^r{\displaystyle \frac{{(-1)}^s{\delta }_{r-s,m}}{(4r-2s)!\left(2s\right)!}},$$

(122)

where ${\delta }_{i,j}$ denotes the Kronecker delta function.

The last formula reduces to the following form:

$${\int }_0^1{\displaystyle \frac{J{L}_{2r}^A\left(x\right)T_m^{*}\left(x\right)}{\sqrt{x(1-x)}}}dx={\displaystyle \frac{{(-1)}^m{2}^{-4r+1}{A}^{2r}\pi \left(4r\right)!}{\left(2\right(m+r\left)\right)!(-2m+2r)!}}.$$

(123)

Following the same procedures for Formula (91), we get

$${\int }_0^1{\displaystyle \frac{J{L}_{2r+1}^A\left(x\right)T_m^{*}\left(x\right)}{\sqrt{x(1-x)}}}dx={(-1)}^r\pi 2^{-4r-1}{A}^{2r+1}(2+4r)!\sum _{s=0}^r{\displaystyle \frac{{(-1)}^s{\delta }_{r-s,m}}{(4r-2s+1)!(2s+1)!}},$$

(124)

Therefore, we have

$${\int }_0^1{\displaystyle \frac{J{L}_{2r+1}^A\left(x\right)T_m^{*}\left(x\right)}{\sqrt{x(1-x)}}}dx={\displaystyle \frac{{(-1)}^m{2}^{-1-4r}{A}^{2r+1}\pi (2+4r)!}{(1-2m+2r)!(1+2m+2r)!}}.$$

(125)

Formulas (123) and (125) give the following unified formula:

$${\int }_0^1{\displaystyle \frac{J{L}_r^A\left(x\right)T_m^{*}\left(x\right)}{\sqrt{x(1-x)}}}dx={\displaystyle \frac{{(-1)}^m{2}^{1-2r}{A}^r\pi \left(2r\right)!}{(r-2m)!(r+2m)!}}.$$

(126)

This completes the proof. □

Corollary 22.

The below integral formula follows:

$${\int }_0^{1}J{L}_r^A\left(x\right)V_m^{*}\left(x\right)\sqrt{{\displaystyle \frac{x}{1-x}}}dx={\displaystyle \frac{{(-1)}^m{2}^{1-2r}{A}^r(1+2m)\pi (2r+1)!}{(r-2m)!(r+2m+2)!}}.$$

(127)

Proof. 

The proof is based on the two formulas in (92) and (93), along with the orthogonality relation of $V_m^{*}\left(x\right)$. □

7. Concluding Remarks

This paper was devoted to theoretically investigating a generalized class of polynomials, namely, the generalized Jacobsthal–Lucas class. Many vital expressions in the scope of the special functions for the investigated sequence of polynomials were found. More definitely, the linearization formulas for the GJLPs were derived using the moment formula we obtained. Novel expressions for the derivatives of the GJLPs were represented as combinations of different symmetric and non-symmetric polynomials. Also, it was shown that for a particular class of Jacobi polynomials, a class of GJLPs of one parameter can be connected to them by a simplified connection formula; therefore, a novel trigonometric representation for this class was given. Most of the findings reported in this work are new. The Mathematica (version 12) and Maple (version 17) programs were used to check all formulas presented in this paper. In addition, some Maple packages were used to simplify some formulas. Future work may include applying the obtained formulas in spectral and collocation methods, as well as exploring their use in fractional differential equations and other applied models. Furthermore, other generalizations and computational studies would be of interest for other generalized sequences of polynomials.