New Results for Certain Jacobsthal-Type Polynomials

Abstract

This paper investigates a class of Jacobsthal-type polynomials (JTPs) that involves one parameter. We present several new formulas for these polynomials, including expressions for their derivatives, moments, and linearization formulas. The key idea behind the derivation of these formulas is based on developing a new connection formula that expresses the shifted Chebyshev polynomials of the third kind in terms of the JTPs. This connection formula is used to deduce a new inversion formula of the JTPs. Therefore, by utilizing the power form representation of these polynomials and their corresponding inversion formula, we can derive additional expressions for them. Additionally, we compute some definite integrals based on some formulas of these polynomials.

1. Introduction

Special functions are fundamental in many fields of applied sciences. They help in addressing many problems in the applied sciences. Applications for some special functions can be found in [1,2]. The authors of [3] discussed the role of special functions in mathematical physics. Regarding some theoretical studies of special polynomials, the authors of [4] discussed probabilistic degenerate Fubini polynomials. In [5], the authors investigated probabilistic degenerate Stirling polynomials accompanied by some applications. The authors in [6] developed some results of Appell polynomials. Some generalized polynomials were discussed in [7], accompanied by some applications. Some new formulas concerned with the convolved Fibonacci polynomials were developed in [8]. A generalized class of polynomials associated with Hermite and Euler polynomials was introduced in [9]. A class of Poly-Genocchi polynomials was introduced in [10]. Some formulas and recurrence relations for general polynomial sequences were introduced in [11]. Some generalized Fibonacci and Lucas polynomials were investigated in [12]. Unified Chebyshev polynomials (CPs) were introduced and explored from a theoretical point of view in [13].

From a practical point of view, the different sequences of polynomials are crucial. For example, some applications of Hermite polynomials can be found in [14,15]. Many sequences of polynomials were utilized to solve various types of differential equations (DEs). For example, the Bernstein polynomial basis was employed in [16] to treat the Korteweg–de Vries equation. The authors of [17] solved the Sobolev equation using mixed Fibonacci and Lucas polynomials. A certain Lucas polynomial sequence was used in [18] to solve the time-fractional generalized Kawahara equation. The shifted Vieta–Lucas polynomials were employed in [19] to treat the fractional advection-dispersion equation. In [20], Genocchi polynomials were used to treat a specific fractional model. The authors of [21] used Schröder polynomials to treat a particular fractional model. The authors of [22,23] used various types of CPs to solve some DEs. In [24], the authors followed a spectral method based on Genocchi polynomials to treat some singular fractional DEs. The authors of [25] used Changhee polynomials to treat high-dimensional chaotic Lorenz systems.

Generalized hypergeometric functions are a crucial family of special functions with extensive applications in engineering, mathematics, and physics. They serve to generalize a large class of special and elementary functions, enabling many essential functions to be expressed in terms of them. These functions significantly address important problems related to special functions, including connection and linearization formulas. Various approaches have been used to compute the connection coefficients between different polynomials. For more details, see, for example [26,27,28,29,30,31]. In addition, the expressions for the derivatives of different polynomials in terms of their original ones are crucial in numerical analysis. For example, the author in [32] has developed new derivative expressions for the sixth-kind CPs. Some terminating hypergeometric functions of the type ${}_{4}F_3(1)$ were involved in these equations.

Numerous studies have been conducted on some classes of the Lucas polynomial sequence. For example, the authors of [33] derived sums of Lucas and Pell polynomials. The authors of [34] studied some generalized polynomials. In [35], a study on generalized Fibonacci polynomials was presented. Some other contributions regarding these sequences can be found in [36,37,38,39,40]. However, there are fewer studies on Jacobsthal and Jacobsthal–Lucas polynomials compared to those on Fibonacci and Lucas polynomials. For example, Koshy in [41] has developed infinite sums involving Jacobsthal polynomials. The same author in [42,43] has derived some formulas related to Jacobsthal polynomials. Djordjevic in [44] has developed some results regarding the derivative sequences of generalized Jacobsthal and Jacobsthal–Lucas polynomials. Some results on third-order Jacobsthal polynomials were derived in [45]. In [46], some infinite sums involving Jacobsthal polynomials were developed. Other sums involving a class of Jacobsthal polynomial squares were developed in [47]. For some other studies regarding Jacobsthal polynomials and their related polynomials, one can refer to [48,49,50].

This paper introduces a class of JTPs. New formulas for these polynomials will be developed. We can summarize the main objectives of the current paper in the following items:

• Developing a new connection formula between the shifted CPs of the third kind and the JTPs.
• Deriving a new inversion formula of JTPs based on the above connection formula.
• Derivation of the moment and linearization formulas of the JTPs.
• Developing new derivative formulas for the JTPs using other polynomials. Additionally, we will also derive the inverse derivative formulas.
• Introducing some definite integrals using some of the introduced formulas.

The paper is structured as follows: Section 2 presents some properties of a certain generalized Lucas polynomial sequence. Some other polynomials are also accounted for. Two basic formulas of the JTPs are developed in Section 3. Moment and linearization formulas of the JTPs are presented in Section 4. New derivative expressions of the JTPs in terms of different polynomials are given in Section 5. Section 6 gives the inverse formulas to those provided in Section 5. Section 7 developed some definite integral formulas that are deduced as consequences of some of the developed formulas. Finally, Section 8 reports some findings.

2. Some Fundamentals and Basic Formulas

This section is devoted to presenting some characteristics of the Lucas polynomial sequence. In addition, some particular classes of this sequence are accounted for. This section also provides an account of certain other polynomials.

Horadam in [51] introduced two sequences of polynomials that can be generated, respectively, by the following two recursive formulas:

$$\begin{array}{cc}\hfill W_n(x)& =p(x)W_{n-1}(x)+q(x)W_{n-2}(x),W_0(x)=0,W_1(x)=1,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill w_n(x)& =\overline{p}(x)w_{n-1}(x)+\overline{q}(x)w_{n-2}(x),w_0(x)=2,w_1(x)=\overline{p}(x).\hfill \end{array}$$

(2)

They can be written explicitly in the following forms:

$$\begin{array}{cc}\hfill W_n(x)& ={\displaystyle \frac{{\left[p(x)+\sqrt{p^2(x)+4q(x)}\right]}^n-{\left[p(x)-\sqrt{p^2(x)+4q(x)}\right]}^n}{2^n\sqrt{p^2(x)+4q(x)}}},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill w_n(x)& ={\displaystyle \frac{{\left[\overline{p}(x)+\sqrt{{\overline{p}}^2(x)+4\overline{q}(x)}\right]}^n+{\left[\overline{p}(x)-\sqrt{{\overline{p}}^2(x)+4\overline{q}(x)}\right]}^n}{2^n}}.\hfill \end{array}$$

(4)

Remark 1. 

Many celebrated sequences are particular ones of the two polynomial sequences $W_n(x)$ and $w_n(x)$.

Table 1. Some special classes of the two sequences.

$\begin{array}{c}\mathit{p}(\mathit{x})\\ \overline{\mathit{p}}(\mathit{x})\end{array}$	$\begin{array}{c}\mathit{q}(\mathit{x})\\ \overline{\mathit{q}}(\mathit{x})\end{array}$	${\mathit{W}}_{\mathit{n}}(\mathit{x})$	${\mathit{w}}_{\mathit{n}}(\mathit{x})$
x	1	Fibonacci polynomial $F_n(x)$	Lucas polynomial $L_n(x)$
$2x$	1	Pell polynomial $P_n(x)$	Pell–Lucas polynomial $Q_n(x)$
1	$2x$	Jacobsthal polynomial $J_n(x)$	Jacobsthal–Lucas polynomial $j_n(x)$
$3x$	−2	Fermat polynomial ${\mathcal{F}}_n(x)$	Fermat–Lucas polynomial $f_n(x)$
$2x$	−1	CPs of the 2nd-kind $U_{n-1}(x)$	CPs of 1st-kind $2T_n(x)$

exhibits some of these polynomials.

Many sequences generalize the two standard sequences of Fibonacci and Lucas sequences, which can also be extracted from the two sequences $W_n(x)$ and $w_n(x)$ as special cases. The two sequences $F_k^{a,b}(x)$ and $L_k^{c,d}(x)$, which are considered generalizations of Fibonacci and Lucas polynomials, can be generated from the following two recursive formulas:

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

(5)

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

(6)

Remark 2. 

It is evident from Table 1 that the Jacobsthal polynomials can be constructed using the following recurrence relation:

$$J_n(x)=J_{n-1}(x)+2x{J}_{n-2}(x),J_0(x)=0,J_1(x)=1.$$

(7)

These polynomials can be constructed using the following generating function [52]:

$${\displaystyle \sum _{n=0}^{\infty }{t}^n{J}_n(x)=\frac{t}{1-t-2x{t}^2}.}$$

(8)

The series representation of the Jacobsthal polynomials is given by [53]

$$J_n(x)=\sum _{i=0}^{⌊{\displaystyle \frac{n-1}{2}}⌋}\left({\displaystyle \genfrac{}{}{0pt}{}{n-i-1}{i}}\right)x^i,$$

(9)

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

Remark 3. 

Many sequences generalize the sequence of Jacobsthal polynomials. Within these sequences is the sequence ${\left\{J_n^{a,b}(x)\right\}}_{n\ge 0}$ that can be constructed with the aid of the following recurrence relation:

$$J_n^{a,b}(x)=a{J}_{n-1}^{a,b}(x)+bx{J}_{n-2}^{a,b}(x),J_0^{a,b}(x)=0,J_1^{a,b}(x)=1.$$

(10)

Based on the recurrence relation (10), it can be shown that the polynomials ${\left\{J_n^{a,b}(x)\right\}}_{n\ge 0}$ can be generated using the following generating function:

$${\displaystyle \sum _{n=0}^{\infty }{t}^n{J}_n^{a,b}(x)=\frac{t}{1-at-bx{t}^2}.}$$

(11)

Remark 4. 

From the representation in (9), we can write the following representation for $J_{n+1}(x)$:

$${\widehat{J}}_n(x)=J_{n+1}(x)=\sum _{i=0}^{⌊{\displaystyle \frac{n}{2}}⌋}\left({\displaystyle \genfrac{}{}{0pt}{}{i+n-⌊\frac{n}{2}⌋}{⌊\frac{n}{2}⌋-i}}\right)x^{⌊{\displaystyle \frac{n}{2}}⌋-i}.$$

(12)

In addition, the above representation can be split into the following two representations:

$$\begin{array}{cc}\hfill {\widehat{J}}_{2n}(x)=& \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n+i}{n-i}}\right)x^{n-i},\hfill \\ \hfill {\widehat{J}}_{2n+1}(x)=& \sum _{i=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n+i+1}{n-i}}\right)x^{n-i}.\hfill \end{array}$$

Remark 5. 

To the best of our knowledge, many formulas related to the Jacobsthal polynomials and their related polynomials are traceless in the literature, such as their derivative, moment, and linearization formulas. We also found no correlations between these formulas and orthogonal polynomials. This motivates us to investigate the JTPs further.

This paper considers a kind of JTPs generated by the following recursive formula:

$$J_n^a(x)=a{J}_{n-1}^a(x)-{\displaystyle \frac{a^2}{4}}x{J}_{n-2}^a(x),J_0(x)=1,J_1(x)=a.$$

(13)

Remark 6. 

In this paper, we restrict our study of the generalized Jacobsthal polynomials generated by (10) to the case corresponding to: $b=-{\displaystyle \frac{a^2}{4}}$, since in such a case, we will prove that the Chebyshev polynomials of the third kind can be written as a simple combination of them that does not involve any hypergeometric term. This pivotal formula in our study helped us develop further formulas related to these polynomials.

An Overview of Some Polynomials

We will present an overview of a selection of orthogonal and non-orthogonal polynomials. Two families of polynomials, one symmetric and the other non-symmetric, are considered. We will refer to them as ${\varphi }_m(x)$ and ${\psi }_m(x)$, and they have the following analytic representations:

$$\begin{array}{cc}\hfill {\varphi }_m(x)=& \sum _{\ell =0}^{⌊{\displaystyle \frac{m}{2}}⌋}{H}_{\ell ,m}{x}^{m-2\ell },\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {\psi }_m(x)=& \sum _{\ell =0}^m{F}_{\ell ,m}{x}^{m-\ell },\hfill \end{array}$$

(15)

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

In addition, let us assume that the inversion Formulas for (14) and (15) are as follows:

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

(16)

$$\begin{array}{cc}\hfill x^m=& \sum _{\ell =0}^m{\overline{F}}_{\ell ,m}{\psi }_{m-\ell }(x),\hfill \end{array}$$

(17)

where ${\overline{H}}_{\ell ,m}$ and ${\overline{F}}_{\ell ,m}$ are known coefficients.

We provide some famous polynomials that may be defined as in (14) and (15). The normalized shifted Jacobi polynomials are expressed as [8]:

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

(18)

${\tilde{R}}_{\ell }^{(\rho ,\gamma )}(x)$ are orthogonal on $[0,1]$ in the sense that

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

(19)

where

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

(20)

Specifically, the ultraspherical polynomials are symmetric Jacobi polynomials. We have

$$C_m^{(\gamma )}(x)={\tilde{R}}_m^{(\gamma -{\displaystyle \frac{1}{2}},\gamma -{\displaystyle \frac{1}{2}})}(x).$$

(21)

Moreover, we note that Jacobi polynomials involve four kinds of CPs. The following recurrence relation can generate all these kinds:

$$C_m(x)=2x{C}_{m-1}(x)-C_{m-2}(x),$$

(22)

but with different initials.

In addition, the moment formula for $C_m(x)$ is given by

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

(23)

Remark 7. 

Examples of important non-symmetric polynomials are the shifted Jacobi polynomials ${\tilde{R}}_m^{(\rho ,\gamma )}(x)$ that are expressed in (18), Schröder polynomials $S_{\ell }(x)$, and Bernoulli polynomials $B_{\ell }(x)$.

Table 2. Power form and inversion coefficients for some non-symmetric polynomials.

Polynomial	${\mathit{F}}_{\ell ,\mathit{m}}$ in (15)	${\overline{\mathit{F}}}_{\ell ,\mathit{m}}$ in (17)
${\tilde{R}}_{\ell }^{(\rho ,\gamma )}(x)$	${\displaystyle \frac{{(-1)}^{\ell }m!\Gamma (1+\rho ){(1+\gamma )}_m{(1+\rho +\gamma )}_{2m-\ell }}{(m-\ell )!\ell !\Gamma (1+m+\rho ){(1+\gamma )}_{m-\ell }{(1+\rho +\gamma )}_m}}$	${\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m}{\ell }\right){(1+\rho )}_{m-\ell }{(1+m-\ell +\gamma )}_{\ell }}{{(2+2m-2\ell +\rho +\gamma )}_L{(1+m-\ell +\rho +\gamma )}_{m-\ell }}}$
$S_{\ell }(x)$	${\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m}{m-\ell }\right)\left(\genfrac{}{}{0pt}{}{2m-\ell }{m-\ell }\right)}{1+m-\ell }}$	${\displaystyle \frac{i!(i+1)!\left({(-1)}^m(1+2i-2m)\right)}{(2i-m+1)!m!}}$
$B_{\ell }(x)$	${\displaystyle B_{\ell }\left(\genfrac{}{}{0pt}{}{m}{m-\ell }\right)}$	${\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{m+1}{m-\ell }\right)}{m+1}}$

presents the power form and inversion coefficients for these non-symmetric polynomials.

Remark 8. 

Some examples of important symmetric polynomials are the ultraspherical polynomials that are given by (21), Hermite polynomials $H_{\ell }(x)$, and the two generalized classes of Fibonacci and Lucas polynomials that are generated, respectively, by (5) and (6).

Table 3. Power form and inversion coefficients for some symmetric polynomials.

Polynomial	${\mathit{H}}_{\ell ,\mathit{m}}$ in (14)	${\overline{\mathit{H}}}_{\ell ,\mathit{m}}$ in (16)
$C_{\ell }^{(\lambda )}(x)$	${\displaystyle \frac{{(-1)}^{\ell }{2}^{-1+m-2\ell }m!\Gamma (m-\ell +\lambda )\Gamma (1+2\lambda )}{(m-2\ell )!\ell !\Gamma (1+\lambda )\Gamma (m+2\lambda )}}$	${\displaystyle \frac{2^{-m+1}(m-2\ell +\lambda )m!\Gamma (\lambda +1)\Gamma (m-2\ell +2\lambda )}{(m-2\ell )!\ell !\Gamma (2\lambda +1)\Gamma (1+m-\ell +\lambda )}}$
$H_{\ell }(x)$	${\displaystyle \frac{{(-1)}^{\ell }{2}^{-2\ell +m}m!}{\ell !(-2\ell +m)!}}$	${\displaystyle \frac{2^{-m}m!}{\ell !(m-2\ell )!}}$
$F_{\ell }^{A,B}(x)$	${\displaystyle \frac{A^{m-2\ell }{B}^{\ell }{(m-2\ell +1)}_{\ell }}{\ell !}}$	${\displaystyle \frac{{(-1)}^{\ell }(m-2\ell +1){(m-\ell +2)}_{\ell -1}{B}^{\ell }}{\ell !A^m}}$
$L_{\ell }^{\overline{A},\overline{B}}(x)$	${\displaystyle \frac{{\overline{A}}^{m-2\ell }{\overline{B}}^{\ell }m{(1-2\ell +m)}_{\ell -1}}{\ell !}}$	${\displaystyle \frac{c_{m-2\ell }{(-1)}^{\ell }{\overline{A}}^{-m}{\overline{B}}^{\ell }{(1-\ell +m)}_{\ell }}{\ell !}}$

represents the power form and inversion coefficients for these polynomials.

3. Two Basic Formulas of the Generalized Jacobsthal Polynomials

This section is devoted to developing two basic formulas of the JTPs: $J_i^a(x)$ generated by (7). The first is the series representation of $J_i^a(x)$, while the second formula exhibits the expression of $x^m$ in terms of $J_i^a(x)$. These two formulas will be the backbone for many formulas in this paper.

Lemma 1. 

For any non-negative integer m, one has

$$J_m^a(x)=a^m\sum _{r=0}^{⌊{\displaystyle \frac{m}{2}}⌋}{\left(-{\displaystyle \frac{1}{4}}\right)}^r\left({\displaystyle \genfrac{}{}{0pt}{}{m-r}{r}}\right)x^r.$$

(24)

Proof. 

First, let

$${\xi }_m^a(x)=a^m\sum _{r=0}^{⌊{\displaystyle \frac{m}{2}}⌋}{\left(-{\displaystyle \frac{1}{4}}\right)}^r\left({\displaystyle \genfrac{}{}{0pt}{}{m-r}{r}}\right)x^r.$$

It is easy to see that

$${\xi }_0^a(x)=J_0^a(x)=1,{\xi }_1^a(x)=J_1^a(x)=a.$$

In addition, by performing some computations, we can show that the following relation holds:

$${\xi }_m^a(x)-a{\xi }_{m-1}^a(x)+{\displaystyle \frac{a^2}{4}}x{\xi }_{m-2}^a(x)=0.$$

(25)

This proves that

$${\xi }_m^a(x)=J_m^a(x),m\ge 0.$$

Lemma 1 is now proved. □

Remark 9. 

Two representations may be obtained from the power form given in (24):

$$\begin{array}{cc}\hfill J_{2m}^a(x)=& a^{2m}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{m-r}(m+r)!}{(2r)!(m-r)!}}{x}^{m-r},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill J_{2m+1}^a(x)=& a^{2m+1}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{m-r}(m+r+1)!}{(2r+1)!(m-r)!}}{x}^{m-r}.\hfill \end{array}$$

(27)

Now, we are going to show that the shifted CPs of the third-kind $V_r^{*}(x),r\ge 0$, can be written as a combination of the polynomials $J_i^a(x)$.

Theorem 1. 

The formula that connects the third-kind CPs and the polynomials $J_i^a(x)$ is given by

$$\begin{array}{cc}\hfill V_r^{*}(x)=& {(-1)}^r\sum _{s=0}^r{\left({\displaystyle \frac{4}{a}}\right)}^{2r-2s}\left({\displaystyle \genfrac{}{}{0pt}{}{4r-2s+1}{2s}}\right)J_{2r-2s}^a(x)\hfill \\ \hfill & +{(-1)}^{r+1}\sum _{s=0}^{r-1}{\left({\displaystyle \frac{4}{a}}\right)}^{2r-2s-1}\left({\displaystyle \genfrac{}{}{0pt}{}{4r-2s}{2s+1}}\right)J_{2r-2s-1}^a(x),r\ge 0.\hfill \end{array}$$

(28)

Proof. 

We will prove Formula (28) by induction on r. It is obvious that (28) holds for $r=0$. Now, we are going to prove the following connection formula:

$$V_r^{*}(x)=\sum _{s=0}^r{G}_{s,r}{J}_{2r-2s}^a(x)+\sum _{s=0}^{r-1}{\overline{G}}_{s,r}{J}_{2r-2s-1}^a(x),r\ge 1,$$

(29)

where

$$\begin{array}{c}\hfill G_{s,r}={(-1)}^r{\left({\displaystyle \frac{4}{a}}\right)}^{2r-2s}\left({\displaystyle \genfrac{}{}{0pt}{}{4r-2s+1}{2s}}\right),\\ \hfill {\overline{G}}_{s,r}={(-1)}^{r+1}{\left({\displaystyle \frac{4}{a}}\right)}^{2r-2s-1}\left({\displaystyle \genfrac{}{}{0pt}{}{4r-2s}{2s+1}}\right).\end{array}$$

We proceed with the proof by induction. Assume that Formula (29) is valid for all $n<r$, i.e.,

$$V_n^{*}(x)=\sum _{s=0}^n{G}_{s,n}{J}_{2n-2s}^a(x)+\sum _{s=0}^{n-1}{\overline{G}}_{s,n}{J}_{2n-2s-1}^a(x),$$

(30)

and we must prove that (29) is itself valid.

Now, we start with the recurrence relation satisfied by $V_r^{*}(x)$:

$$V_r^{*}(x)=2(2x-1)V_{r-1}^{*}(x)-V_{r-2}^{*}(x),$$

(31)

and apply the inductive assumption to write $V_{r-1}^{*}(x)$ and $V_{r-2}^{*}(x)$ to obtain

$$\begin{array}{c}\hfill V_{r-1}^{*}(x)=\sum _{s=0}^{r-1}{G}_{s,r-1}{J}_{2r-2s-2}^a(x)+\sum _{s=0}^{r-2}{\overline{G}}_{s,r-1}{J}_{2r-2s-3}^a(x),\\ \hfill V_{r-2}^{*}(x)=\sum _{s=0}^{r-2}{G}_{s,r-2}{J}_{2r-2s-4}^a(x)+\sum _{s=0}^{r-3}{\overline{G}}_{s,r-2}{J}_{2r-2s-5}^a(x).\end{array}$$

The last two expressions, along with (31), enable one to write $V_r^{*}(x)$ in the following form:

$$\begin{array}{cc}\hfill V_{r-1}^{*}(x)=& 2(2x-1)\left(\sum _{s=0}^{r-1}{G}_{s,r-1}{J}_{2j-2s-2}^a+\sum _{s=0}^{r-2}{\overline{G}}_{s,r-1}{J}_{2j-2s-3}^a\right)\hfill \\ \hfill & -\left(\sum _{s=0}^{r-2}{G}_{s,r-2}{J}_{2j-2s-4}^a+\sum _{s=0}^{r-2}{\overline{G}}_{s,r-2}{J}_{2j-2s-5}^a\right).\hfill \end{array}$$

(32)

Now, inserting the recurrence relation (13) in the form

$$x{J}_k^a(x)=-{\displaystyle \frac{4}{a^2}}\left(J_{k+2}^a(x)-a{J}_{k+1}^a(x)\right),$$

(33)

into (32) yields the following formula:

$$\begin{array}{cc}\hfill V_r^{*}(x)=& -{\displaystyle \frac{16}{a^2}}\left(\sum _{s=0}^{r-2}{\overline{G}}_{s,r-1}(-a{J}_{-2+2j-2s}^a(x)+J_{-1+2j-2s}^a(x))\right.\hfill \\ & \left.+\sum _{s=0}^{r-1}{G}_{s,r-1}(-a{J}_{-1+2j-2s}^a(x)+J_{2j-2s}^a(x))\right)\hfill \\ & -2\left(\sum _{s=0}^{r-1}{G}_{s,r-1}{J}_{2j-2s-2}^a(x)+\sum _{s=0}^{r-2}{\overline{G}}_{s,r-1}{J}_{2j-2s-3}^a(x)\right)\hfill \\ & -\left(\sum _{s=0}^{r-2}{G}_{s,r-2}{J}_{2j-2s-4}^a(x)+\sum _{s=0}^{r-2}{\overline{G}}_{s,r-2}{J}_{2j-2s-5}^a(x)\right).\hfill \end{array}$$

Now, we can write

$$\begin{array}{c}\hfill V_r^{*}(x)=\sum _1+\sum _2,\end{array}$$

(34)

where

$$\begin{array}{cc}\hfill \sum _1=& {\displaystyle \frac{16}{a}}\sum _{s=0}^{r-2}{\overline{G}}_{s,r-1}{J}_{-2+2j-2s}^a(x)-{\displaystyle \frac{16}{a^2}}\sum _{s=0}^{r-1}{G}_{s,r-1}{J}_{2j-2s}^a(x)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & -2\sum _{s=0}^{r-1}{G}_{s,r-1}{J}_{2j-2s-2}^a(x)-\sum _{s=0}^{r-2}{G}_{s,r-2}{J}_{2j-2s-4}^a(x),\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill \sum _2=& -{\displaystyle \frac{16}{a^2}}\sum _{s=0}^{r-2}{\overline{G}}_{s,r-1}{J}_{2j-2s-1}^a(x)+{\displaystyle \frac{16}{a}}\sum _{s=0}^{r-1}{G}_{s,r-1}{J}_{2j-2s-1}^a(x)\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & -2\sum _{s=0}^{r-2}{\overline{G}}_{s,r-1}{J}_{2j-2s-3}^a(x)-\sum _{s=0}^{r-2}{\overline{G}}_{s,r-2}{J}_{2j-2s-5}^a(x).\hfill \end{array}$$

(38)

Some algebraic manipulations transform ${\displaystyle \sum _1}$ and ${\displaystyle \sum _2}$ into the following forms:

$$\begin{array}{cc}\hfill {\displaystyle \sum _1=}& \sum _{s=0}^r\left({\displaystyle \frac{16}{a}}{\overline{G}}_{s-1,r-1}-{\displaystyle \frac{16}{a^2}}{G}_{s,r-1}-2G_{s-1,r-1}-G_{s-2,r-2}\right)J_{2j-2s}^a(x),\hfill \\ \hfill {\displaystyle \sum _2=}& \sum _{s=0}^{r-1}\left(-{\displaystyle \frac{16}{a^2}}{\overline{G}}_{s,r-1}+{\displaystyle \frac{16}{a}}{G}_{s,r-1}-2{\overline{G}}_{s-1,r-1}-{\overline{G}}_{s-2,r-2}\right)J_{2j-2s-1}^a(x).\hfill \end{array}$$

It is not difficult to show that the following two identities hold:

$$\begin{array}{cc}\hfill {\displaystyle \frac{16}{a}}{\overline{G}}_{s-1,r-1}-{\displaystyle \frac{16}{a^2}}{G}_{s,r-1}-2G_{s-1,r-1}-G_{s-2,r-2}& =G_{s,r},\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill -{\displaystyle \frac{16}{a^2}}{\overline{G}}_{s,r-1}+{\displaystyle \frac{16}{a}}{G}_{s,r-1}-2{\overline{G}}_{s-1,r-1}-{\overline{G}}_{s-2,r-2}& ={\overline{G}}_{s,r},\hfill \end{array}$$

(40)

and this proves that

$$V_r^{*}(x)=\sum _{s=0}^r{G}_{s,r}{J}_{2r-2s}^a(x)+\sum _{s=0}^{r-1}{\overline{G}}_{s,r}{J}_{2r-2s-1}^a(x).$$

This finalizes the proof of Theorem 1. □

Theorem 2. 

Let r be a non-negative integer. The following inversion formula for the JTPs holds:

$$\begin{array}{cc}\hfill x^r=& {(-1)}^r{2}^{2r}\sum _{s=0}^r{\displaystyle \frac{a^{2s-2r}{(1-2s+r)}_{2s}}{(2s)!}}{J}_{2r-2s}^a(x)\hfill \\ \hfill & +{(-1)}^{r+1}{2}^{2r}\sum _{s=0}^{r-1}{\displaystyle \frac{a^{2s-2r+1}{(-2s+r)}_{1+2s}}{(2s+1)!}}{J}_{2r-2s-1}^a(x).\hfill \end{array}$$

(41)

Proof. 

The inversion formula of $V_j^{*}(x)$ is given by [54]

$$x^r=4^{-r}(2r+1)!\sum _{\ell =0}^r{\displaystyle \frac{1}{\ell !(1-\ell +2r)!}}{V}_{r-\ell }^{*}(x).$$

(42)

By virtue of the connection Formula (28), one can write

$$\begin{array}{cc}\hfill x^r=& \sum _{i=0}^r{\displaystyle \frac{{(-1)}^{i-r}{4}^{-r}(2r+1)!a^{2i-2r}}{i!(1-i+2r)!}}\sum _{s=0}^{r-i}{4}^{-2i-2s+2r}{a}^{2s}\left({\displaystyle \genfrac{}{}{0pt}{}{1-4i-2s+4r}{2s}}\right)J_{2r-2i-2s}^a(x)\hfill \\ \hfill +& \sum _{i=0}^r{\displaystyle \frac{{(-1)}^{i-r+1}{4}^{-r}(2r+1)!a^{2i-2r}}{i!(1-i+2r)!}}\sum _{s=0}^{r-i-1}{2}^{-2-4i-4s+4r}{a}^{2s+1}\left({\displaystyle \genfrac{}{}{0pt}{}{-4i-2s+4r}{1+2s}}\right)J_{2r-2i-2s-1}^a(x).\hfill \end{array}$$

(43)

Some calculations lead to the following formula:

$$\begin{array}{cc}\hfill x^r& =\sum _{s=0}^r{4}^{-2s+r}(2r+1)!a^{2s-2r}\sum _{\ell =0}^s{\displaystyle \frac{{(-1)}^{\ell +r}\left(\genfrac{}{}{0pt}{}{1-2\ell -2s+4r}{-2\ell +2s}\right)}{\ell !(-\ell +2r+1)!}}{J}_{2r-2s}^a(x)\hfill \\ \hfill & +\sum _{s=0}^{r-1}{4}^{-1-2s+r}(2r+1)!a^{2s-2r+1}\sum _{\ell =0}^s{\displaystyle \frac{{(-1)}^{\ell +r+1}\left(\genfrac{}{}{0pt}{}{-2(\ell +s-2r)}{1-2\ell +2s}\right)}{\ell !(-\ell +2r+1)!}}{J}_{2r-2s-1}^a(x).\hfill \end{array}$$

(44)

To obtain a simplified formula for (44), let

$$\begin{array}{cc}\hfill W_{s,r}=& \sum _{\ell =0}^s{\displaystyle \frac{{(-1)}^{\ell +r}\left(\genfrac{}{}{0pt}{}{1-2\ell -2s+4r}{-2\ell +2s}\right)}{\ell !(-\ell +2r+1)!}},\hfill \end{array}$$

(45)

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

(46)

We use Zeilberger’s algorithm [55] to find a closed formula for the above two sums. They satisfy the following two recursive formulas:

$$\begin{array}{cc}\hfill & (s+1)(2s+1)W_{s+1,r}-8(r-2s-1)(r-2s)W_{s,r}=0,W_{0,r}={\displaystyle \frac{{(-1)}^r}{(2r+1)!}},\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill & (s+1)(2s+3){\overline{W}}_{s+1,r}-8(r-2s-2)(r-2s-1){\overline{W}}_{s,r}=0,{\overline{W}}_{0,r}={\displaystyle \frac{4r{(-1)}^{r+1}}{(2r+1)!}},\hfill \end{array}$$

(48)

whose solutions are given as follows:

$$\begin{array}{cc}\hfill W_{s,r}=& {\displaystyle \frac{{(-1)}^r{4}^{2s}{(1-2s+r)}_{2s}}{(2r+1)!(2s)!}},\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill {\overline{W}}_{s,r}=& {\displaystyle \frac{4^{2s+1}{(-1)}^{r+1}{(-2s+r)}_{2s+1}}{(2r+1)!(2s+1)!}},\hfill \end{array}$$

(50)

and accordingly, Formula (44) reduces to Formula (41). This ends the proof. □

4. Moment and Linearization Formulas of the JTPs

This section is confined to presenting a new moment formula for $J_i^a(x)$. This formula will be the key to developing new linearization formulas for the polynomials $J_i^a(x)$.

4.1. Moment Formula of the JTPs

Now, we are going to state and prove a new moment formula for the polynomials $J_i^a(x)$.

Theorem 3. 

Assume that s and i are two non-negative integers. We have

$$x^s{J}_i^a(x)=2^{2s}\sum _{r=0}^s{(-1)}^r\left({\displaystyle \genfrac{}{}{0pt}{}{s}{r}}\right)a^{-s-r}{J}_{i+s+r}^a(x).$$

(51)

Proof. 

We proceed by induction. For $s=0$, it is clear that the formula holds since each side in such a case is equal to $J_i^a(x)$. Now, assume that Formula (51) is valid, so to prove the inductive step, we have to prove that the following identity holds:

$$x^{s+1}{J}_i^a(x)=2^{2s+2}\sum _{r=0}^{s+1}{(-1)}^r\left({\displaystyle \genfrac{}{}{0pt}{}{s+1}{r}}\right)a^{-s-r-1}{J}_{i+s+r+1}^a(x).$$

(52)

Now, using the recurrence relation (33) and multiplying both sides of (51) by x, we obtain

$$x^{s+1}{J}_i^a(x)=2^{2s+2}\sum _{r=0}^s{(-1)}^{r+1}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{r}}\right)a^{-s-r-2}\left(J_{i+s+r+2}^a(x)-a{J}_{i+s+r+1}^a(x)\right),$$

(53)

that can be written in the following form:

$$\begin{array}{cc}\hfill x^{s+1}{J}_i^a(x)=& -2^{2s+2}\sum _{r=0}^s{(-1)}^{r+1}\left({\displaystyle \genfrac{}{}{0pt}{}{s}{r}}\right)a^{-s-r-1}{J}_{i+s+r+1}^a(x)\hfill \\ & +2^{2s+2}\sum _{r=0}^{s+1}{(-1)}^r\left({\displaystyle \genfrac{}{}{0pt}{}{s}{r-1}}\right)a^{-s-r-1}{J}_{i+s+r+1}^a(x).\hfill \end{array}$$

If we note the following identity:

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

then, the following formula can be obtained after performing some simplifications:

$$x^{s+1}{J}_i^a(x)=2^{2s+2}\sum _{r=0}^{s+1}{(-1)}^r\left({\displaystyle \genfrac{}{}{0pt}{}{s+1}{r}}\right)a^{-s-r-1}{J}_{i+s+r+1}^a(x).$$

(54)

This proves Formula (52). □

4.2. Linearization Formulas Involving the JTPs

This section is interested in developing new linearization formulas involving the JTPs. In this concern, the following linearization formulas will be introduced:

• The linearization formulas of $J_i^a(x)$.
• The product formulas of $J_i^a(x)$ and some celebrated polynomials.

Theorem 4. 

For all non-negative integers $r,s$, one has the following linearization formulas:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_{2r}^a(x)J_s^a(x)=& \sqrt{\pi }r!\left(\sum _{m=0}^r{\displaystyle \frac{a^{2m}{}_3\tilde{F}_2\left(\begin{array}{c}-m,\frac{1}{2}-m,r+1\\ \frac{1}{2},r+1-2m\end{array};1\right)}{(2m)!}}{J}_{s+2r-2m}^a(x)\right.\hfill \\ \hfill & \left.-\sum _{m=0}^{r-1}{\displaystyle \frac{a^{2m+1}{}_3\tilde{F}_2\left(\begin{array}{c}-m,-m-\frac{1}{2},r+1\\ \frac{1}{2},r-2m\end{array};1\right)}{(2m+1)!}}{J}_{s+2r-2m-1}^a(x)\right),\hfill \end{array}\end{array}$$

(55)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_{2r+1}^a(x)J_s^a(x)=& {\displaystyle \frac{1}{2}}\sqrt{\pi }(r+1)!\left(\sum _{m=0}^r{\displaystyle \frac{a^{2m+1}{}_3\tilde{F}_2\left(\begin{array}{c}-m,\frac{1}{2}-m,r+2\\ \frac{3}{2},r+1-2m\end{array};1\right)}{(2m)!}}{J}_{s+2r-2m}^a(x)\right.\hfill \\ \hfill & \left.-\sum _{m=0}^{r-1}{\displaystyle \frac{a^{2m+2}{}_3\tilde{F}_2\left(\begin{array}{c}-m,-m-\frac{1}{2},r+2\\ \frac{3}{2},r-2m\end{array};1\right)}{(2m+1)!}}{J}_{s+2r-2m-1}^a(x)\right),\hfill \end{array}\end{array}$$

(56)

where ${}_3\tilde{F}_2(z)$ is the regularized hypergeometric function [56].

Proof. 

We will prove (55). The proof of (56) is similar. The analytic formula of $J_{2r}^a(x)$ enables one to write

$$J_{2r}^a(x)J_s^a(x)=a^{2r}\sum _{\ell =0}^r{(-4)}^{\ell -r}\left({\displaystyle \genfrac{}{}{0pt}{}{r+\ell }{r-\ell }}\right)x^{r-\ell }{J}_s^a(x).$$

(57)

The application of the moment Formula (51) yields

$$J_{2r}^a(x)J_s^a(x)=a^{2r}\sum _{\ell =0}^r{(-1)}^{r-\ell }\left({\displaystyle \genfrac{}{}{0pt}{}{r+\ell }{r-\ell }}\right)\sum _{m=0}^{r-\ell }{(-1)}^m{a}^{\ell -m-r}\left({\displaystyle \genfrac{}{}{0pt}{}{r-\ell }{m}}\right)J_{s+m+r-\ell }^a(x),$$

(58)

which may be written alternatively after some manipulations in the form

$$\begin{array}{cc}\hfill J_{2r}^a(x)J_s^a(x)=& \sum _{m=0}^r{a}^{2m}\sum _{\ell =0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{r-\ell }{\ell -2m+r}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\ell +r}{r-\ell }}\right)J^{s+2r-2m}(x)\hfill \\ \hfill & -\sum _{m=0}^{r-1}{a}^{2m+1}\sum _{\ell =0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{r-\ell }{-1+\ell -2m+r}}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\ell +r}{r-\ell }}\right)J_{s+2r-2m-1}^a(x),\hfill \end{array}$$

(59)

which leads to (55). □

Theorem 5. 

Let r and s be two non-negative integers. Let also $C_s(x)$ be any of the four kinds of CPs. One has the following linearization formulas:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_{2r+1}^a(x)C_s(x)=& {\left(-{\displaystyle \frac{1}{8}}\right)}^r{a}^{2r+1}(r+1)\times \hfill \\ \hfill & \sum _{m=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{r}{m}}\right){}_{4}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{r}{2}}+{\displaystyle \frac{3}{2}},m-r,{\displaystyle \frac{r}{2}}+1\\ {\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}}\end{array}|1\right)C_{r+s-2m}(x)\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}{(-1)}^{r+1}{2}^{2-3r}{a}^{2r+1}(r+2)!\sum _{m=0}^{r-1}{\displaystyle \frac{1}{m!(r-m-1)!}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{r}{2}}+2,m-r+1,{\displaystyle \frac{r}{2}}+{\displaystyle \frac{3}{2}}\\ {\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}},{\displaystyle \frac{7}{4}}\end{array}|1\right)C_{r+s-2m-1}(x),\hfill \end{array}\end{array}$$

(60)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_{2r}^a(x)C_s(x)=& {\left(-{\displaystyle \frac{1}{8}}\right)}^r{a}^{2r}\sum _{m=0}^r\left({\displaystyle \genfrac{}{}{0pt}{}{r}{m}}\right){}_{4}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{r}{2}}+1,m-r,{\displaystyle \frac{r}{2}}+{\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\end{array}|1\right)C_{r+s-2m}(x)\hfill \\ \hfill +& {(-1)}^{r+1}{2}^{2-3r}{a}^{2r}(r+1)!\sum _{m=0}^{r-1}{\displaystyle \frac{1}{m!(r-m-1)!}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{r}{2}}+{\displaystyle \frac{3}{2}},m-r+1,{\displaystyle \frac{r}{2}}+1\\ {\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}}\end{array}|1\right)C_{r+s-2m-1}(x).\hfill \end{array}\end{array}$$

(61)

Proof. 

Based on the analytic Formula (27), one can write

$$J_{2r+1}^a(x)C_s(x)=a^{2r+1}\sum _{\ell =0}^r{\displaystyle \frac{{(-4)}^{\ell -r}(1+\ell +r)!}{(2\ell +1)!(r-\ell )!}}{x}^{r-\ell }{C}_s(x).$$

(62)

Based on the moment formula of $C_s(x)$ in (23), Formula (62) can be written as

$$J_{2r+1}^a(x)C_s(x)=a^{2r+1}\sum _{\ell =0}^r{\displaystyle \frac{{(-8)}^{\ell -r}(1+\ell +r)!}{(2\ell +1)!(r-\ell )!}}\sum _{m=0}^{r-\ell }\left({\displaystyle \genfrac{}{}{0pt}{}{r-\ell }{m}}\right)C_{r+s-\ell -2m},$$

(63)

which may be written again as

$$\begin{array}{cc}\hfill J_{2r+1}^a(x)C_s(x)& =a^{2r+1}\sum _{m=0}^{⌊{\displaystyle \frac{r+s}{2}}⌋}\sum _{\ell =0}^m{\displaystyle \frac{{(-8)}^{2\ell -r}\left(\genfrac{}{}{0pt}{}{-2\ell +r}{-\ell +m}\right)(2\ell +r+1)!}{(-2\ell +r)!(4\ell +1)!}}{C}_{r+s-2m}(x)\hfill \\ \hfill & +a^{2r+1}\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(r+s-1)⌋}\sum _{\ell =0}^m{\displaystyle \frac{{(-8)}^{2\ell -r+1}\left(\genfrac{}{}{0pt}{}{-1-2\ell +r}{-\ell +m}\right)(2\ell +r+2)!}{(4\ell +3)!(-1-2\ell +r)!}}{C}_{r+s-2m-1}(x).\hfill \end{array}$$

(64)

The last formula can be written as in (60). Formula (61) can be similarly proved. □

Theorem 6. 

For non-negative integers r and s. The linearization formulas for $J_r^a(x)$ and generalized Fibonacci polynomials are given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_{2r}^a(x)F_s^{A,B}(x)=& a^{2r}{A}^{-r}\sum _{m=0}^r{\left(-{\displaystyle \frac{1}{4}}\right)}^r{(-B)}^m\left({\displaystyle \genfrac{}{}{0pt}{}{r}{m}}\right)\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,1+{\displaystyle \frac{r}{2}},{\displaystyle \frac{1}{2}}+{\displaystyle \frac{r}{2}},-r+m\\ {\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)F_{r+s-2m}^{A,B}(x)\hfill \\ \hfill & +{(-1)}^{r+1}{2}^{1-2r}{a}^{2r}{A}^{1-r}(r+1)!\sum _{m=0}^{r-1}{\displaystyle \frac{1}{m!(r-m-1)!}}{(-B)}^m\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{3}{2}}+{\displaystyle \frac{r}{2}},1+{\displaystyle \frac{r}{2}},1-r+m\\ {\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)F_{r+s-2m-1}^{A,B}(x),\hfill \end{array}\end{array}$$

(65)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill J_{2r+1}^a(x)F_s^{A,B}(x)=& {\left(-{\displaystyle \frac{1}{4}}\right)}^r{a}^{1+2r}{A}^{-r}(r+1)\sum _{m=0}^r{(-B)}^m\left({\displaystyle \genfrac{}{}{0pt}{}{r}{m}}\right)\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{3}{2}}+{\displaystyle \frac{r}{2}},1+{\displaystyle \frac{r}{2}},-r+m\\ {\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)F_{r+s-2m}^{A,B}(x)\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}{(-1)}^{r+1}{2}^{1-2r}{a}^{1+2r}{A}^{1-r}(r+2)!\sum _{m=0}^{r-1}{\displaystyle \frac{1}{m!(r-m-1)!}}{(-B)}^m\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,2+{\displaystyle \frac{r}{2}},{\displaystyle \frac{3}{2}}+{\displaystyle \frac{r}{2}},1-r+m\\ {\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}},{\displaystyle \frac{7}{4}}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)F_{r+s-2m-1}^{A,B}(x).\hfill \end{array}\end{array}$$

(66)

Proof. 

The proof is similar to the proof of Theorem 5. □

5. New Derivative Expressions of the JTPs in Terms of Different Polynomials

This section focuses on developing new expressions for the derivatives of $J_k^a(x)$. We begin by providing derivative expressions for these polynomials as combinations of their original forms. Additionally, we present alternative expressions for the derivatives in terms of both symmetric and non-symmetric polynomials.

5.1. Derivatives in Terms of Their Original Ones

Theorem 7. 

Consider the two non-negative integers k and p with $k\ge p$. The derivatives of $J_k^a(x)$ can be expressed as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k}^a(x)& =4^{-p}{a}^{2p}\sqrt{\pi }{(-1)}^{p}k!\left(\sum _{\ell =0}^{k-p}{\displaystyle \frac{a^{2\ell }{}_3\tilde{F}_2\left(\begin{array}{c}-\ell ,\frac{1}{2}-\ell ,1+k\\ \frac{1}{2},1+k-2\ell -p\end{array}|1\right)}{(2\ell )!}}{J}_{2k-2p-2\ell }^a(x)\right.\hfill \\ \hfill & -\left.\sum _{\ell =0}^{k-p-1}{\displaystyle \frac{a^{2\ell +1}{}_3\tilde{F}_2\left(\begin{array}{c}-\ell ,-\frac{1}{2}-\ell ,1+k\\ \frac{1}{2},k-2\ell -p\end{array}|1\right)}{(2\ell +1)!}}{J}_{2k-2p-2\ell -1}^a(x)\right),\hfill \end{array}\end{array}$$

(67)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)& ={(-1)}^p{2}^{-1-2p}\sqrt{\pi }(k+1)!\left({\displaystyle \sum _{\ell =0}^{k-p}\frac{a^{1+2\ell +2p}{}_3\tilde{F}_2\left(\begin{array}{c}-\ell ,\frac{1}{2}-\ell ,2+k\\ \frac{3}{2},1+k-2\ell -p\end{array}|1\right)}{(2\ell )!}{J}_{2k-2p-2\ell }^a(x)}\right.\hfill \\ \hfill & \left.-\sum _{\ell =0}^{k-p-1}{\displaystyle \frac{a^{2(1+\ell +p)}{}_3\tilde{F}_2\left(\begin{array}{c}-\ell ,-\frac{1}{2}-\ell ,2+k\\ \frac{3}{2},k-2\ell -p\end{array}|1\right)}{(2\ell +1)!}}{J}_{2k-2p-2\ell -1}^a(x)\right).\hfill \end{array}\end{array}$$

(68)

Proof. 

We prove (67). The analytic form (26) enables one to write

$$D^p{J}_{2k}^a(x)=a^{2k}\sum _{r=0}^{k-p}{\displaystyle \frac{{(-4)}^{r-k}(k+r)!}{(k-p-r)!(2r)!}}{x}^{k-r-p},$$

(69)

which may be written in terms of $J_i^a(x)$ using the inversion Formula (41) as

$$\begin{array}{cc}\hfill D^p{J}_{2k}^a(x)& =a^{2k}\sum _{r=0}^{k-p}{\displaystyle \frac{{(-1)}^{k-p-r}{(-4)}^{r-k}(k+r)!{\left(\frac{2}{a}\right)}^{2(k-p-r)}}{(k-p-r)!(2r)!}}\hfill \\ \hfill & \times \left(\sum _{\ell =0}^{k-r-p}{\displaystyle \frac{a^{2\ell }{(1+k-2\ell -p-r)}_{2\ell }}{(2\ell )!}}{J}_{2k-2r-2p-2\ell }^a(x)\right.\hfill \\ \hfill & \left.-\sum _{\ell =0}^{k-r-p-1}{\displaystyle \frac{a^{2\ell +1}{(k-2\ell -p-r)}_{2\ell +1}}{(2\ell +1)!}}{J}_{2k-2r-2p-2\ell -1}^a(x)\right).\hfill \end{array}$$

(70)

The last formula can be rearranged to be written as

$$\begin{array}{cc}\hfill D^p{J}_{2k}^a(x)=& 4^{-p}{a}^{2p}{(-1)}^p\left(\sum _{\ell =0}^{k-p}{a}^{2\ell }\sum _{m=0}^{\ell }{\displaystyle \frac{(k+m)!}{(2m)!(2\ell -2m)!(k-2\ell +m-p)!}}{J}_{2k-2p-2\ell }^a(x)\right.\hfill \\ \hfill & \left.-\sum _{\ell =0}^{k-p-1}{a}^{2\ell +1}\sum _{m=0}^{\ell }{\displaystyle \frac{(k+m)!}{(2m)!(2\ell -2m+1)!(k-2\ell +m-p-1)!}}{J}_{2k-2p-2\ell -1}^a(x)\right),\hfill \end{array}$$

(71)

which may be written alternatively as in (67). Formula (68) can be similarly proved. □

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

In this part, we will find a general expression for the derivatives of $J_k^a(x)$ in terms of any non-symmetric polynomial. In addition, some specific derivative expressions are given.

Theorem 8. 

If we consider any non-symmetric polynomial ${\psi }_i(x)$ expressed as in (15), then for all non-negative integers p and k with $k\ge p$, we have the following two derivative expressions:

$$\begin{array}{cc}\hfill D^p{J}_{2k}^a(x)=& \sum _{m=0}^{k-p}{R}_{m,k}{\psi }_{k-p-m}(x),\hfill \end{array}$$

(72)

$$\begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)=& \sum _{m=0}^{k-p}{\overline{R}}_{m,k}{\psi }_{k-p-m}(x),\hfill \end{array}$$

(73)

where

$$\begin{array}{cc}\hfill R_{m,k}=& a^{2k}\sum _{r=0}^m{(-4)}^{r-k}\left({\displaystyle \genfrac{}{}{0pt}{}{k+r}{k-r}}\right){(1+k-p-r)}_p{\overline{F}}_{m-r,k-r-p},\hfill \end{array}$$

(74)

$$\begin{array}{cc}\hfill {\overline{R}}_{m,k}=& a^{2k+1}\sum _{r=0}^m{(-4)}^{r-k}\left({\displaystyle \genfrac{}{}{0pt}{}{k+r+1}{k-r}}\right){(1+k-p-r)}_p{\overline{F}}_{m-r,k-r-p},\hfill \end{array}$$

(75)

and ${\overline{F}}_{\ell ,m}$ are the inversion coefficients appear in (17).

Proof. 

We start from Formula (69), and after that, apply the inversion formula in (17) to obtain the following formula:

$$D^p{J}_{2k}^a(x)=a^{2k}\sum _{r=0}^{k-p}{(-4)}^{r-k}\left({\displaystyle \genfrac{}{}{0pt}{}{k+r}{k-r}}\right){(1+k-p-r)}_p\sum _{t=0}^{k-r-p}{\overline{F}}_{t,k-r-p}{\psi }_{k-r-p-t}(x),$$

(76)

which can be arranged to give the following formula:

$$D^p{J}_{2k}^a(x)=a^{2k}\sum _{m=0}^{k-p}\sum _{r=0}^m{(-4)}^{r-k}\left({\displaystyle \genfrac{}{}{0pt}{}{k+r}{k-r}}\right){(1+k-p-r)}_p{\overline{F}}_{m-r,k-r-p}{\psi }_{k-p-m}(x).$$

(77)

This proves (72). Formula (73) can be similarly proved. □

As a result of Theorem 8, we will present some derivative formulas for the JTPs as combinations of specific non-symmetric polynomials. The following corollaries display these results.

Corollary 1. 

Consider the two non-negative integers k and p with $k\ge p$. In terms of the shifted Jacobi polynomials ${\tilde{R}}_{\ell }^{(\rho ,\gamma )}(x)$, the derivatives $D^p{J}_k^a(x)$ have the following expressions:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k}^a(x)=& {\displaystyle \frac{{\left(-\frac{1}{4}\right)}^k{a}^{2k}\Gamma (1+k-p+\gamma )k!}{\Gamma (1+\rho )}}\times \hfill \\ \hfill & \sum _{m=0}^{k-p}{\displaystyle \frac{(1+2k-2m-2p+\rho +\gamma )\Gamma (1+k-m-p+\rho )\Gamma (1+k-m-p+\rho +\gamma )}{m!(k-m-p)!\Gamma (1+k-m-p+\gamma )\Gamma (2+2k-m-2p+\rho +\gamma )}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}-m,1+k,-1-2k+m+2p-\rho -\gamma \\ {\displaystyle \frac{1}{2}},-k+p-\gamma \end{array}|1\right){\tilde{R}}_{k-p-m}^{(\rho ,\gamma )}(x),\hfill \end{array}\end{array}$$

(78)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)=& {\displaystyle \frac{{\left(-\frac{1}{4}\right)}^k{a}^{2k+1}\Gamma (1+k-p+\gamma )(k+1)!}{\Gamma (1+\rho )}}\times \hfill \\ \hfill & \sum _{m=0}^{k-p}{\displaystyle \frac{(1+2k-2m-2p+\rho +\gamma )\Gamma (1+k-m-p+\rho )\Gamma (1+k-m-p+\rho +\gamma )}{m!(k-m-p)!\Gamma (1+k-m-p+\gamma )\Gamma (2+2k-m-2p+\rho +\gamma )}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}-m,2+k,-1-2k+m+2p-\rho -\gamma \\ {\displaystyle \frac{3}{2}},-k+p-\gamma \end{array}|1\right){\tilde{R}}_{k-p-m}^{(\rho ,\gamma )}(x).\hfill \end{array}\end{array}$$

(79)

Proof. 

Based on Theorem 8 along with the inversion coefficients of ${\tilde{R}}_{\ell }^{(\rho ,\gamma )}(x)$ in the first row and last column of Table 2. □

Corollary 2. 

Consider the two non-negative integers k and p with $k\ge p$. In terms of Schröder polynomials, $D^p{J}_{2k}^a(x)$ have the following expressions:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k}^a(x)& =4^{-k}{a}^{2k}(1+k-p)!k!\sum _{m=0}^{k-p}{\displaystyle \frac{{(-1)}^{k+m}(1+2k-2m-2p)}{m!(1+2k-m-2p)!}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}-m,1+k,-1-2k+m+2p\\ {\displaystyle \frac{1}{2}},-1-k+p\end{array}|-1\right)S_{k-p-m}(x),\hfill \end{array}\end{array}$$

(80)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)& =4^{-k}{a}^{1+2k}(k+1)!(1+k-p)!\sum _{m=0}^{k-p}{\displaystyle \frac{{(-1)}^{k+m}(1+2k-2m-2p)}{m!(1+2k-m-2p)!}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}-m,2+k,-1-2k+m+2p\\ {\displaystyle \frac{3}{2}},-1-k+p\end{array}|-1\right)S_{k-p-m}(x).\hfill \end{array}\end{array}$$

(81)

Proof. 

Based on Theorem 8 along with the inversion coefficients of $S_{\ell }(x)$ in the second row and last column of Table 2. □

Corollary 3. 

Consider the two non-negative integers k and p with $k\ge p$. In terms of Bernoulli polynomials, $D^p{J}_{2k}^a(x)$ have the following expressions:

$$\begin{array}{cc}\hfill D^p{J}_{2k}^a(x)=& {\displaystyle \frac{{\left(-\frac{1}{4}\right)}^k{a}^{2k}\sqrt{\pi }}{\Gamma \left(-\frac{1}{2}-k\right)}}\sum _{m=0}^{k-p}{\displaystyle \frac{k!\Gamma \left(\frac{1}{2}-k+m\right)+{(-1)}^m\Gamma \left(-\frac{1}{2}-k\right)(1+k+m)!}{\Gamma \left(\frac{3}{2}+m\right)(m+1)!(k-m-p)!}}{B}_{k-p-m}(x),\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)=& {\displaystyle \frac{a^{2k+1}}{\sqrt{\pi }}\sum _{m=0}^{k-p}\frac{4^{-2k+m}\left(-(2k+2)!\Gamma \left(\frac{1}{2}-k+m\right)+{(-1)}^{m+k}{4}^{1+k}\sqrt{\pi }(2+k+m)!\right)}{(2m+3)!(k-m-p)!}{B}_{k-p-m}(x).}\hfill \end{array}$$

(83)

Proof. 

By applying Theorem 8 using the inversion coefficients of $B_m(x)$ in the third row and last column of Table 2. □

5.3. Derivatives of Jacobsthal Polynomials in Terms of Symmetric Polynomials

In this part, we will find a general expression for the derivatives of $J_k^a(x)$ in terms of any symmetric polynomial. In addition, some specific derivatives of the JTPs in terms of symmetric polynomials are given.

Theorem 9. 

If we consider any symmetric polynomial ${\varphi }_i(x)$ expressed as in (14), then for any non-negative integers p and k with $k\ge p$, then we have the following two derivative expressions:

$$\begin{array}{cc}\hfill D^p{J}_{2k}^a(x)=& \sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{M}_{m,k,p}{\varphi }_{k-p-2m}(x)+\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}\sum _{r=0}^m{\overline{M}}_{m,k,p}{\varphi }_{k-p-2m-1}(x),\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)=& \sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{W}_{m,k,p}{\varphi }_{k-p-2m}(x)+\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}\sum _{r=0}^m{\overline{W}}_{m,k,p}{\varphi }_{k-p-2m-1}(x),\hfill \end{array}$$

(85)

where

$$\begin{array}{cc}\hfill M_{m,k,p}=& a^{2k}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-2r}(k+2r)!}{(k-p-2r)!(4r)!}}{\overline{H}}_{m-r,k-2r-p},\hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill {\overline{M}}_{m,k,p}=& a^{2k}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-2r-1}(k+2r+1)!}{(k-p-2r-1)!(4r+2)!}}{\overline{H}}_{m-r,k-2r-p-1},\hfill \end{array}$$

(87)

$$\begin{array}{ccc}\hfill W_{m,k,p}=& a^{2k+1}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-2r}(1+k+2r)!{(1+k-p-2r)}_p}{(k-2r)!(1+4r)!}}{\overline{H}}_{m-r,k-2r-p},\hfill & \end{array}$$

(88)

$$\begin{array}{cc}\hfill {\overline{W}}_{m,k,p}=& a^{2k+1}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-2r-1}(2+k+2r)!{(k-p-2r)}_p}{(k-2r-1)!(3+4r)!}}{\overline{H}}_{m-r,k-2r-p-1}.\hfill \end{array}$$

(89)

and ${\overline{H}}_{\ell ,m}$ are the inversion coefficients of ${\varphi }_i(x)$ appear in (16).

Proof. 

The proofs of (84) and (85) are similar. We prove (85). Starting from Formula (69), and making use of the inversion formula of ${\varphi }_k(x)$ in (16), we get

$$D^p{J}_{2k+1}^a(x)=a^{2k+1}\sum _{r=0}^{k-p}{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-r}(1+k+r)!{(1+k-p-r)}_p}{(k-r)!(2r+1)!}}\sum _{t=0}^{⌊{\displaystyle \frac{1}{2}}(k-r-p)⌋}{\overline{H}}_{t,k-r-p}{\varphi }_{k-r-p-2t}(x).$$

(90)

The following formula is the result of some manipulations:

$$\begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)=& a^{2k+1}\sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-2r}(k+1+2r)!{(k+1-p-2r)}_p}{(k-2r)!(4r+1)!}}{\overline{H}}_{m-r,k-2r-p}{\varphi }_{k-p-2m}(x)\hfill \\ \hfill & +a^{2k+1}\sum _{m=0}^{⌊{\displaystyle \frac{k-p-1}{2}}⌋}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-2r-1}(k+2+2r)!{(k-p-2r)}_p}{(k-2r-1)!(4r+3)!}}{\overline{H}}_{m-r,k-2r-p-1}{\varphi }_{k-p-2m-1}(x).\hfill \end{array}$$

(91)

Similarly, we can show that

$$\begin{array}{cc}\hfill D^p{J}_{2k}^a(x)=& a^{2k}\sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-2r}(k+2r)!}{(k-p-2r)!(4r)!}}{\overline{H}}_{m-r,k-2r-p}{\varphi }_{k-p-2m}(x)\hfill \\ \hfill & +a^{2k}\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}\sum _{r=0}^m{\displaystyle \frac{{\left(-\frac{1}{4}\right)}^{k-2r-1}(k+2r+1)!}{(k-p-2r-1)!(4r+2)!}}{\overline{H}}_{m-r,k-2r-p-1}{\varphi }_{k-p-2m-1}(x).\hfill \end{array}$$

(92)

The two Formulas (91), and (92) can be written as:

$$D^p{J}_{2k}^a(x)=\sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{M}_{m,k,p}{\varphi }_{k-p-2m}(x)+\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}{\overline{M}}_{m,k,p}{\varphi }_{k-p-2m-1}(x),$$

(93)

$$D^p{J}_{2k+1}^a(x)=\sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{W}_{m,k,p}{\varphi }_{k-p-2m}(x)+\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}{\overline{W}}_{m,k,p}{\varphi }_{k-p-2m-1}(x),$$

(94)

where the coefficients $M_{m,k,p},{\overline{M}}_{m,k,p},W_{m,k,p},{\overline{W}}_{m,k,p}$ are given, respectively, as in (86)–(89). This proves Theorem 9. □

In the following, we give two expressions for the derivatives of the JTPs in terms of the ultraspherical and Hermite polynomials as applications of Theorem 9.

Corollary 4. 

Consider the two non-negative integers k and p with $k\ge p$. In terms of the ultraspherical polynomials $C_i^{(\gamma )}(x)$, the derivatives $D^p{J}_k^a(x)$ have the following expressions:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k}^a(x)& ={\displaystyle \frac{{(-1)}^k{2}^{1-3k+p-2\gamma }{a}^{2k}\sqrt{\pi }k!}{\Gamma \left(\frac{1}{2}+\gamma \right)}}\sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{\displaystyle \frac{(k-2m-p+\gamma )\Gamma (k-2m-p+2\gamma )}{m!(k-2m-p)!\Gamma (1+k-m-p+\gamma )}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{1}{2}}+{\displaystyle \frac{k}{2}},1+{\displaystyle \frac{k}{2}},-k+m+p-\gamma \\ {\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\end{array}|1\right)C_{k-p-2m}^{(\gamma )}(x)\hfill \\ \hfill & +{\displaystyle \frac{{(-1)}^{1+k}{2}^{3-3k+p-2\gamma }{a}^{2k}\sqrt{\pi }(1+k)!}{\Gamma \left(\frac{1}{2}+\gamma \right)}}\times \hfill \\ \hfill & \sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}{\displaystyle \frac{(-1+k-2m-p+\gamma )\Gamma (-1+k-2m-p+2\gamma )}{m!(-1+k-2m-p)!\Gamma (k-m-p+\gamma )}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,1+{\displaystyle \frac{k}{2}},{\displaystyle \frac{3}{2}}+{\displaystyle \frac{k}{2}},1-k+m+p-\gamma \\ {\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}}\end{array}|1\right)C_{k-p-2m-1}^{(\gamma )}(x),\hfill \end{array}\end{array}$$

(95)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)& ={\displaystyle \frac{{(-1)}^k{2}^{1-3k+p-2\gamma }{a}^{1+2k}\sqrt{\pi }(1+k)!}{\Gamma \left(\frac{1}{2}+\gamma \right)}}\sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{\displaystyle \frac{(k-2m-p+\gamma )\Gamma (k-2m-p+2\gamma )}{m!(k-2m-p)!\Gamma (1+k-m-p+\gamma )}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,1+{\displaystyle \frac{k}{2}},{\displaystyle \frac{3}{2}}+{\displaystyle \frac{k}{2}},-k+m+p-\gamma \\ {\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}}\end{array}|1\right)C_{k-p-2m}^{(\gamma )}(x)\hfill \\ \hfill & +{\displaystyle \frac{{(-1)}^{k+1}{2}^{3-3k+p-2\gamma }{a}^{1+2k}\sqrt{\pi }(2+k)!}{3\Gamma \left(\frac{1}{2}+\gamma \right)}}\times \hfill \\ \hfill & \sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}{\displaystyle \frac{(-1+k-2m-p+\gamma )\Gamma (-1+k-2m-p+2\gamma )}{m!(-1+k-2m-p)!\Gamma (k-m-p+\gamma )}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{3}{2}}+{\displaystyle \frac{k}{2}},2+{\displaystyle \frac{k}{2}},1-k+m+p-\gamma \\ {\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}},{\displaystyle \frac{7}{4}}\end{array}|1\right)C_{k-p-2m-1}^{(\gamma )}(x).\hfill \end{array}\end{array}$$

(96)

Proof. 

By applying Theorem 9 using the inversion coefficients of $C_{\ell }^{(\gamma )}(x)$ in the first row, last column of Table 3. □

Corollary 5. 

Consider the two non-negative integers k and p with $k\ge p$. In terms of Hermite polynomials $H_{\ell }(x)$, the derivatives $D^p{J}_k^a(x)$ have the following expressions:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k}^a(x)& ={(-1)}^k{2}^{-3k+p}{a}^{2k}k!\sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{\displaystyle \frac{1}{m!(k-2m-p)!}}\times \hfill \\ \hfill & {}_{3}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{1}{2}}+{\displaystyle \frac{k}{2}},1+{\displaystyle \frac{k}{2}}\\ {\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\end{array}|-1\right)H_{k-p-2m}(x)\hfill \\ \hfill & +{(-1)}^{1+k}{2}^{2-3k+p}{a}^{2k}(k+1)!\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}{\displaystyle \frac{1}{m!(k-2m-p-1)!}}\times \hfill \\ \hfill & {}_{3}F_3\left(\begin{array}{c}-m,1+{\displaystyle \frac{k}{2}},{\displaystyle \frac{3}{2}}+{\displaystyle \frac{k}{2}}\\ {\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}}\end{array}|-1\right)H_{k-p-2m-1}(x),\hfill \end{array}\end{array}$$

(97)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^p{J}_{2k+1}^a(x)& ={(-1)}^k{2}^{-3k+p}{a}^{1+2k}(k+1)!\sum _{m=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{\displaystyle \frac{1}{m!(k-2m-p)!}}\times \hfill \\ \hfill & {}_{3}F_3\left(\begin{array}{c}-m,1+{\displaystyle \frac{k}{2}},{\displaystyle \frac{3}{2}}+{\displaystyle \frac{k}{2}}\\ {\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}}\end{array}|-1\right)H_{k-p-2m}(x)\hfill \\ \hfill & +{\displaystyle \frac{1}{3}}{(-1)}^{k+1}{2}^{2-3k+p}{a}^{1+2k}(k+2)!\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(k-p-1)⌋}{\displaystyle \frac{1}{m!(k-2m-p-1)!}}\times \hfill \\ \hfill & {}_{3}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{3}{2}}+{\displaystyle \frac{k}{2}},2+{\displaystyle \frac{k}{2}}\\ {\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}},{\displaystyle \frac{7}{4}}\end{array}|-1\right)H_{k-p-2m-1}(x).\hfill \end{array}\end{array}$$

(98)

Proof. 

By applying Theorem 9 using the inversion coefficients of $H_{\ell }(x)$ in the second row, last column of Table 3. □

6. Derivatives of Different Polynomials in Terms of the JTPs

This section is confined to presenting new derivative expressions for various polynomials as combinations of the JTPs.

6.1. Derivatives of Non-Symmetric Polynomials in Terms of Jacobsthal Polynomials

We give a general theorem in which we express the derivatives of any non-symmetric polynomial represented as (15) in terms of the JTPs.

Theorem 10. 

Consider any symmetric polynomial ${\psi }_k(x)$. In addition, consider the two non-negative integers k and p with $k\ge p$. We have

$$\begin{array}{cc}\hfill & D^p{\psi }_k(x)=\sum _{t=0}^{k-p}{a}^{2(-k+t+p)}\times \hfill \\ \hfill & \left(\sum _{m=0}^t{\displaystyle \frac{F_{m,k}{(1+k-m-p)}_p{(-1)}^{k-m-p}{2}^{2(k-m-p)}{(1+k+m-2t-p)}_{-2m+2t}}{(2t-2m)!}}\right)J_{2k-2p-2t}^a(x)\hfill \\ \hfill & +\sum _{t=0}^{k-p-1}{a}^{1-2k+2t+2p}\left(\sum _{m=0}^t{\displaystyle \frac{F_{m,k}{(1+k-m-p)}_p{(-1)}^{1+k-m-p}{2}^{2(k-m-p)}{(k+m-2t-p)}_{1-2m+2t}}{(2t-2m+1)!}}\right)J_{2k-2p-2t-1}^a(x),\hfill \end{array}$$

(99)

where $F_{m,k}$ are the power form coefficients given in the second column of Table 2.

Proof. 

If we consider the power form representation in (15), then we can write

$$D^p{\psi }_k(x)=\sum _{r=0}^{k-p}{F}_{r,k}{(k-r-p+1)}_p{x}^{k-r-p}.$$

If the inversion Formula (41) is applied, then the last formula transforms into the following form:

$$\begin{array}{cc}\hfill D^p{\psi }_k(x)=& \sum _{r=0}^{k-p}{F}_{r,k}{(1+k-p-r)}_p\times \hfill \\ \hfill & \left(\sum _{t=0}^{k-r-p}{\displaystyle \frac{{(-1)}^{k-p-r}{2}^{2(k-p-r)}{a}^{2(-k+t+p+r)}{(1+k-2t-p-r)}_{2t}}{(2t)!}}{J}_{2k-2r-2p-2t}^a(x)\right.\hfill \\ \hfill & +\left.\sum _{t=0}^{k-r-p-1}{\displaystyle \frac{{(-1)}^{1+k-p-r}{2}^{2(k-p-r)}{a}^{1-2k+2t+2p+2r}{(k-2t-p-r)}_{1+2t}}{(2t+1)!}}{J}_{2k-2r-2p-2t-1}^a(x)\right).\hfill \end{array}$$

(100)

Some tedious manipulations on the right-hand side of the last formula enable one to write (100) in the following form

$$\begin{array}{cc}\hfill & D^p{\psi }_k(x)=\sum _{t=0}^{k-p}{a}^{2(-k+t+p)}\times \hfill \\ \hfill & \left(\sum _{m=0}^t{\displaystyle \frac{F_{m,k}{(1+k-m-p)}_p{(-1)}^{k-m-p}{2}^{2(k-m-p)}{(1+k+m-2t-p)}_{-2m+2t}}{(2t-2m)!}}\right)J_{2k-2p-2t}^a(x)\hfill \\ \hfill & +\sum _{t=0}^{k-p-1}{a}^{1-2k+2t+2p}\left(\sum _{m=0}^t{\displaystyle \frac{F_{m,k}{(1+k-m-p)}_p{(-1)}^{1+k-m-p}{2}^{2(k-m-p)}{(k+m-2t-p)}_{1-2m+2t}}{(2t-2m+1)!}}\right)J_{2k-2p-2t-1}^a(x).\hfill \end{array}$$

(101)

This completes the proof. □

The following corollary gives an application to Theorem 10. The formula expressing the Jacobi polynomials’ derivatives in terms of the JTPs is given.

Corollary 6. 

Consider the two non-negative integers k and p with $k\ge p$. The following expression holds:

$$\begin{array}{cc}\hfill & D^p{\tilde{R}}_k^{(\gamma ,\rho )}(x)=\sum _{t=0}^{k-p}{V}_{t,k,p}{J}_{2k-2p-2t}^a(x)+\sum _{t=0}^{k-p-1}{\overline{V}}_{t,k,b}{J}_{2k-2p-2t-1}^a(x),\hfill \end{array}$$

(102)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill V_{t,k,p}=& {\displaystyle \frac{{(-1)}^{k-p}{2}^{2(k-p)}{a}^{2(-k+p+t)}k!\Gamma (1+\gamma )\Gamma (1+2k+\gamma +\rho )}{(k-p-1-2t)!(2t)!k!\Gamma (1+k+\gamma +\rho )}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}-t,{\displaystyle \frac{1}{2}}-t,-k-\rho \\ 1+k-p-2t,-2k-\gamma -\rho \end{array}|1\right),\hfill \end{array}\end{array}$$

(103)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\overline{V}}_{t,k,p}=& {\displaystyle \frac{{(-1)}^{1+k-p}{2}^{2(k-p)}{a}^{1-2k+2p+2t}\Gamma (1+k)\Gamma (1+\gamma )\Gamma (1+2k+\gamma +\rho )}{\Gamma (k-p-2t)\Gamma (2(1+t))\Gamma (1+k+\gamma )\Gamma (1+k+\gamma +\rho )}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}-t,-{\displaystyle \frac{1}{2}}-t,-k-\rho \\ k-p-2t,-2k-\gamma -\rho \end{array}|1\right).\hfill \end{array}\end{array}$$

(104)

Proof. 

Direct application of Theorem 10 considering the power form coefficients of ${\tilde{R}}_k^{(\gamma ,\rho )}(x)$ in the first row, second column of Table 2. □

6.2. Derivatives of Symmetric Polynomials in Terms of the JTPs

Theorem 11. 

Consider any symmetric polynomial ${\varphi }_k(x)$. For all non-negative integers $k,p$ with $k\ge p$, the following derivative expression for ${\varphi }_k(x)$ is valid:

$$\begin{array}{cc}\hfill D^p{\varphi }_k(x)=& {(-1)}^{k-p}\sum _{t=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{a}^{-2k+2p+4t}\left(\sum _{m=0}^t{\displaystyle \frac{\left(2^{2(k-2m-p)}(k-2m)!\right)H_{m,k}}{(4t-4m)!(k+2m-p-4t)!}}\right)J_{2k-2p-4t}^a(x)\hfill \\ \hfill & +{(-1)}^{1+k-p}\sum _{t=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{a}^{1-2k+2p+4t}\left(\sum _{m=0}^t{\displaystyle \frac{\left(2^{2(k-2m-p)}(k-2m)!\right)H_{m,k}}{(4t-4m+1)!(k+2m-p-4t-1)!}}\right)J_{2k-2p-4t-1}^a(x)\hfill \\ \hfill & +{(-1)}^{k-p}\sum _{t=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋-1}{a}^{2-2k+2p+4t}\left(\sum _{m=0}^t{\displaystyle \frac{\left(2^{2(k-2m-p)}(k-2m)!\right)H_{m,k}}{(4t-4m+2)!(k+2m-p-4t-2)!}}\right)J_{2k-2p-4t-2}^a(x)\hfill \\ \hfill & +{(-1)}^{1+k-p}\sum _{t=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋-1}{a}^{3-2k+2p+4t}\left(\sum _{m=0}^t{\displaystyle \frac{\left(2^{2(k-2m-p)}(k-2m)!\right)H_{m,k}}{(4t-4m+3)!(k+2m-p-4t-3)!}}\right)J_{2k-2p-4t-3}^a(x),\hfill \end{array}$$

(105)

where $H_{m,k}$ are the power form coefficients given in the second column of Table 3.

Proof. 

The analytic form of a symmetric polynomial given in (14) allows one to write

$$\begin{array}{c}\hfill D^p{\varphi }_k(x)=\sum _{r=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{H}_{r,k}{(1+k-p-2r)}_p{x}^{k-2r-p}.\end{array}$$

(106)

The inversion formula of the JTPs allows one to convert the last formula into the following one:

$$\begin{array}{cc}\hfill D^p{\varphi }_k(x)=& \sum _{r=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{H}_{r,k}{(-1)}^{k-p}{2}^{2(k-p-2r)}{(1+k-p-2r)}_p\times \hfill \\ \hfill & \left(\sum _{t=0}^{k-2r-p}{\displaystyle \frac{a^{2(-k+p+2r+t)}{(1+k-p-2r-2t)}_{2t}}{(2t)!}}{J}_{2k-4r-2p-2t}^a(x)\right.\hfill \\ \hfill & \left.-\sum _{t=0}^{k-2r-p-1}{\displaystyle \frac{a^{1-2k+2p+4r+2t}{(k-p-2(r+t))}_{1+2t}}{(2t+1)!}}{J}_{2k-4r-2p-2t-1}^a(x)\right).\hfill \end{array}$$

(107)

Some tedious computations lead to the formula in (105). □

As an application to Theorem 11, the derivatives of the ultraspherical polynomials can be expressed in terms of the JTPs as in the following corollary.

Corollary 7. 

Consider the two non-negative integers k and p with $k\ge p$. The derivatives of the ultraspherical polynomials can be expressed in terms of $J_k^a(x)$ as follows:

$$\begin{array}{cc}\hfill D^p{C}_k^{(\gamma )}(x)=& \sum _{t=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{Q}_{1,t,k,p}{J}_{2k-2p-4t}^a(x)+\sum _{t=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋}{Q}_{2,t,k,p}{J}_{2k-2p-4t-1}^a(x)\hfill \\ \hfill & +\sum _{t=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋-1}{Q}_{3,t,k,p}{J}_{2k-2p-4t-2}^a(x)+\sum _{t=0}^{⌊{\displaystyle \frac{k-p}{2}}⌋-1}{Q}_{4,t,k,p}{J}_{2k-2p-4t-3}^a(x),\hfill \end{array}$$

(108)

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Q_{1,t,k,p}=& {\displaystyle \frac{{(-1)}^{k-p}{2}^{-1+3k-2p+2\gamma }{a}^{-2k+2p+4t}k!\Gamma \left(\frac{1}{2}+\gamma \right)\Gamma (k+\gamma )}{\sqrt{\pi }(k-p-4t)!(4t)!\Gamma (k+2\gamma )}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-t,{\displaystyle \frac{1}{4}}-t,{\displaystyle \frac{1}{2}}-t,{\displaystyle \frac{3}{4}}-t\\ {\displaystyle \frac{1}{2}}+{\displaystyle \frac{k}{2}}-{\displaystyle \frac{p}{2}}-2t,1+{\displaystyle \frac{k}{2}}-{\displaystyle \frac{p}{2}}-2t,1-k-\gamma \end{array}|1\right)\hfill \end{array}\\ \hfill \begin{array}{cc}\hfill Q_{2,t,k,p}=& {\displaystyle \frac{{(-1)}^{1+k-p}{2}^{-1+3k-2p+2\gamma }{a}^{1-2k+2p+4t}k!\Gamma \left(\frac{1}{2}+\gamma \right)\Gamma (k+\gamma )}{\sqrt{\pi }(k-p-4t-1)!(4t+1)!\Gamma (k+2\gamma )}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-t,-{\displaystyle \frac{1}{4}}-t,{\displaystyle \frac{1}{4}}-t,{\displaystyle \frac{1}{2}}-t\\ {\displaystyle \frac{k}{2}}-{\displaystyle \frac{p}{2}}-2t,{\displaystyle \frac{1}{2}}+{\displaystyle \frac{k}{2}}-{\displaystyle \frac{p}{2}}-2t,1-k-\gamma \end{array}|1\right)\hfill \end{array}\\ \hfill \begin{array}{cc}\hfill Q_{3,t,k,p}=& {\displaystyle \frac{{(-1)}^{k-p}{2}^{-1+3k-2p+2\gamma }{a}^{2-2k+2p+4t}k!\Gamma \left(\frac{1}{2}+\gamma \right)\Gamma (k+\gamma )}{\sqrt{\pi }(k-p-4t-2)!(4t+2)!\Gamma (k+2\gamma )}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-t,-{\displaystyle \frac{1}{2}}-t,-{\displaystyle \frac{1}{4}}-t,{\displaystyle \frac{1}{4}}-t\\ -{\displaystyle \frac{1}{2}}+{\displaystyle \frac{k}{2}}-{\displaystyle \frac{p}{2}}-2t,{\displaystyle \frac{k}{2}}-{\displaystyle \frac{p}{2}}-2t,1-k-\gamma \end{array}|1\right)\hfill \end{array}\\ \hfill \begin{array}{cc}\hfill Q_{4,t,k,p}=& {\displaystyle \frac{{(-1)}^{1+k-p}{2}^{-1+3k-2p+2\gamma }{a}^{3-2k+2p+4t}k!\Gamma \left(\frac{1}{2}+\gamma \right)\Gamma (k+\gamma )}{\sqrt{\pi }(k-p-4t-3)!(4t+3))!\Gamma (k+2\gamma )}}\times \hfill \\ \hfill & {}_{4}F_3\left(\begin{array}{c}-t,-{\displaystyle \frac{3}{4}}-t,-{\displaystyle \frac{1}{2}}-t,-{\displaystyle \frac{1}{4}}-t\\ -1+{\displaystyle \frac{k}{2}}-{\displaystyle \frac{p}{2}}-2t,-{\displaystyle \frac{1}{2}}+{\displaystyle \frac{k}{2}}-{\displaystyle \frac{p}{2}}-2t,1-k-\gamma \end{array}|1\right)\hfill \end{array}\end{array}$$

Proof. 

Direct application of Theorem 11 considering the power form coefficients $C_k^{(\gamma )}(x)$ in the first row and second column of Table 3. □

7. Some Definite Integrals

This section is confined to deducing closed formulas for some definite integrals of the Jacobsthal polynomials, their moments, and their linearization formula.

Corollary 8. 

The following integral formula holds:

$${\displaystyle {\int }_0^1{J}_i^a(x)dx=\frac{4}{i+2}\left(1-2^{-i-1}\right).}$$

(109)

Proof. 

The following two formulas can be proved:

$$\begin{array}{cc}\hfill {\int }_0^1{J}_{2k}^a(x)dx=& {\displaystyle \frac{4^{-k}{a}^{2k}\left(2^{2k+1}-1\right)}{k+1}},\hfill \end{array}$$

(110)

$$\begin{array}{cc}\hfill {\int }_0^1{J}_{2k+1}^a(x)dx=& {\displaystyle \frac{4^{-k}{a}^{2k+1}\left(4^{k+1}-1\right)}{2k+3}}.\hfill \end{array}$$

(111)

The proof of the two formulas is similar. We prove Formula (111). The proof is based on the connection formula with Bernoulli polynomials that can be deduced from (83) by setting $q=0$. Thus, integrating both sides from 0 to 1, we obtain

$$\begin{array}{cc}\hfill {\int }_0^1{J}_{2k+1}^a(x)dx=& a^{2k+1}\sum _{m=0}^k{\displaystyle \frac{4^{-2k+m}\left(-(2k+2)!\Gamma \left(\frac{1}{2}-k+m\right)+{(-1)}^{m+k}{4}^{k+1}\sqrt{\pi }(k+m+2)!\right)}{\sqrt{\pi }(2m+3)!(k-m)!}}\times \hfill \\ & {\int }_0^1{B}_{k-m}(x)dx.\hfill \end{array}$$

If we note the identity:

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

then it is easy to reduce the above formula to

$${\int }_0^1{J}_{2k+1}^a(x)dx={\displaystyle \frac{4^{-k}{a}^{2k+1}\left(4^{k+1}-1\right)}{2k+3}}.$$

This proves (111). Formula (110) can be similarly proved from the connection formula that can be obtained from (82) setting $q=0$. □

Corollary 9. 

For all non-negative integers m and k, one has

$${\int }_0^1{x}^m{J}_k^a(x)dx=2^{2m}{a}^i\sum _{r=0}^m{\displaystyle \frac{2{(-1)}^r\left(2-2^{-i-m-r}\right)\left(\genfrac{}{}{0pt}{}{m}{r}\right)}{2+i+m+r}}.$$

(112)

Proof. 

From the moment formula, it can be shown that

$${\int }_0^1{x}^m{J}_k^a(x)dx=2^{2m}\sum _{r=0}^m{(-1)}^r\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r}}\right)a^{-m-r}{\int }_0^1{J}_{i+m+r}^a(x)dx.$$

(113)

Based on Formula (109), the last formula can be converted into

$${\int }_0^1{x}^m{J}_k^a(x)dx=2^{2m}{a}^i\sum _{r=0}^m{\displaystyle \frac{2{(-1)}^r\left(2-2^{-i-m-r}\right)\left(\genfrac{}{}{0pt}{}{m}{r}\right)}{2+i+m+r}}.$$

(114)

This proves Corollary 9. □

Corollary 10. 

Consider any two non-negative integers r and s, the following linearization formula holds:

$${\int }_0^1{J}_r^a(x)J_s^a(x)dx=\sum _{m=0}^{⌊{\displaystyle \frac{r}{2}}⌋}{U}_{m,r}+\sum _{m=0}^{⌊{\displaystyle \frac{r}{2}}⌋-1}{\overline{U}}_{m,r},$$

(115)

where $U_{m,r}$ and ${\overline{U}}_{m,r}$ are given as follows:

$$\begin{array}{cc}\hfill U_{m,r}& =\left\{\begin{array}{cc}{H}_{m,r},\hfill & \mathit{if}\mathit{r}\mathit{is}\mathit{even},\hfill \\ Z_{m,r},\hfill & \mathit{if}\mathit{r}\mathit{is}\mathit{odd},\hfill \end{array}\right.\hfill \end{array}$$

(116)

$$\begin{array}{cc}\hfill {\overline{U}}_{m,r}& =\left\{\begin{array}{cc}{\overline{H}}_{m,r},\hfill & \mathit{if}\mathit{r}\mathit{is}\mathit{even},\hfill \\ {\overline{Z}}_{m,r},\hfill & \mathit{if}\mathit{r}\mathit{is}\mathit{odd}.\hfill \end{array}\right.\hfill \end{array}$$

(117)

with

$$\begin{array}{cc}\hfill H_{m,r}& ={\displaystyle \frac{\left(4-2^{1+2m-r-s}\right)a^{r+s}\sqrt{\pi }\left(\frac{r}{2}\right)!}{(2-2m+r+s)(2m)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-m,1+{\displaystyle \frac{r}{2}},{\displaystyle \frac{1}{2}}-m\\ {\displaystyle \frac{1}{2}},1-2m+{\displaystyle \frac{r}{2}}\end{array}|1\right),\hfill \end{array}$$

(118)

$$\begin{array}{cc}\hfill {\overline{H}}_{m,r}& =-{\displaystyle \frac{4\left(1-2^{2m-r-s}\right)a^{r+s}\sqrt{\pi }\left(\frac{r}{2}\right)!}{(1-2m+r+s)(2m+1)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-m,1+{\displaystyle \frac{r}{2}},-{\displaystyle \frac{1}{2}}-m\\ {\displaystyle \frac{1}{2}},-2m+{\displaystyle \frac{r}{2}}\end{array}|1\right),\hfill \end{array}$$

(119)

$$\begin{array}{cc}\hfill Z_{m,r}& ={\displaystyle \frac{2\left(1-2^{2m-r-s}\right)a^{r+s}\sqrt{\pi }\left(\frac{r+1}{2}\right)!}{(1-2m+r+s)(2m)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-m,{\displaystyle \frac{r+3}{2}},{\displaystyle \frac{1}{2}}-m\\ {\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}}(1-4m+r)\end{array}|1\right),\hfill \end{array}$$

(120)

$$\begin{array}{cc}\hfill {\overline{Z}}_{m,r}& =-{\displaystyle \frac{\left(4-2^{3+2m-r-s}\right)a^{r+s}\sqrt{\pi }\left(\frac{r+1}{2}\right)!}{2(-2m+r+s)(2m+1)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-m,{\displaystyle \frac{r+3}{2}},-{\displaystyle \frac{1}{2}}-m\\ {\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}}(-1-4m+r)\end{array}|1\right).\hfill \end{array}$$

(121)

Proof. 

Integrating both sides of the linearization formulas in (60), and (61) leads, respectively, to the following integral formulas:

$$\begin{array}{cc}\hfill {\int }_0^1{J}_{2r}^a(x)J_s^a(x)dx& =\sqrt{\pi }r!\left(\sum _{m=0}^r{\displaystyle \frac{a^{2m}}{(2m)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-m,1+r,{\displaystyle \frac{1}{2}}-m\\ {\displaystyle \frac{1}{2}},1+r-2m\end{array}|1\right)G_{s+2r-2m}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.-\sum _{m=0}^{r-1}{\displaystyle \frac{a^{2m+1}}{(2m+1)!}}{}_3\tilde{F}_2\left(\begin{array}{c}-m,1+r,-{\displaystyle \frac{1}{2}}-m\\ {\displaystyle \frac{1}{2}},r-2m\end{array}|1\right)G_{s+2r-2m-1}\right),\hfill \end{array}$$

(122)

$$\begin{array}{cc}\hfill {\int }_0^1{J}_{2r+1}^a(x)J_s^a(x)dx=& {\displaystyle \frac{1}{2}}\sqrt{\pi }(r+1)!\left(\sum _{m=0}^r{\displaystyle \frac{a^{2m+1}}{(2m)!}}{}_3\tilde{F}_2\left(\begin{array}{c}2+r,{\displaystyle \frac{1}{2}}-m,-m\\ {\displaystyle \frac{3}{2}},1+r-2m\end{array}|1\right)G_{s+2r-2m}\right.\hfill \end{array}$$

(123)

$$\begin{array}{cc}\hfill & -\left.\sum _{m=0}^{r-1}{\displaystyle \frac{a^{2+2m}}{(2m+1)!}}{}_3\tilde{F}_2\left(\begin{array}{c}2+r,-{\displaystyle \frac{1}{2}}-m,-m\\ {\displaystyle \frac{3}{2}},r-2m\end{array}|1\right)G_{s+2r-2m-1}\right),\hfill \end{array}$$

(124)

where

$${\displaystyle G_i={\int }_0^1{J}_i^a(x)dx=\frac{4}{i+2}\left(1-2^{-i-1}\right).}$$

The above two formulas can be merged to give Formula (115). □

Remark 10. 

The formula that evaluates the integrals ${\displaystyle {\int }_0^1{\left(J_r^a(x)\right)}^{2}dx}$ can be obtained as a special case of Formula (115).

Corollary 11. 

For the two positive integers i and j, the following integral formula holds:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^1{J}_{2i}^a(x)F_j^{A,B}(x)dx=& {\left(-{\displaystyle \frac{1}{4A}}\right)}^i{a}^{2i}\sum _{p=0}^i{(-B)}^p\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)\times \hfill \\ & {}_{4}F_3\left(\begin{array}{c}-p,{\displaystyle \frac{1}{2}}+{\displaystyle \frac{i}{2}},1+{\displaystyle \frac{i}{2}},-i+p\\ {\displaystyle \frac{1}{4}},{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)I_{i+j-2p}^{A,B}(x)\hfill \\ & +{(-1)}^{1+i}{2}^{1-2i}{a}^{2i}{A}^{1-i}(i+1)!\sum _{p=0}^{i-1}{\displaystyle \frac{{(-B)}^p}{p!(i-p-1)!}}\times \hfill \\ & {}_{4}F_3\left(\begin{array}{c}-p,1+{\displaystyle \frac{i}{2}},{\displaystyle \frac{3}{2}}+{\displaystyle \frac{i}{2}},1-i+p\\ {\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)I_{i+j-2p-1}^{A,B},\hfill \end{array}\end{array}$$

(125)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\int }_0^1{J}_{2i+1}^a(x)F_j^{A,B}(x)dx=& {\left(-{\displaystyle \frac{1}{4}}\right)}^i{a}^{1+2i}{A}^{-i}(i+1)\sum _{p=0}^i{(-B)}^p\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)\times \hfill \\ & {}_{4}F_3\left(\begin{array}{c}-p,1+{\displaystyle \frac{i}{2}},{\displaystyle \frac{3}{2}}+{\displaystyle \frac{i}{2}},-i+p\\ {\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{4}},{\displaystyle \frac{5}{4}}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)I_{i+j-2p}^{A,B}\hfill \\ & +{\displaystyle \frac{1}{3}}{(-1)}^{i+1}{2}^{1-2i}{a}^{1+2i}{A}^{1-i}(i+2)!\sum _{p=0}^{i-1}{\displaystyle \frac{{(-B)}^p}{(i-p-1)!p!}}\times \hfill \\ & {}_{4}F_3\left(\begin{array}{c}-p,{\displaystyle \frac{3}{2}}+{\displaystyle \frac{i}{2}},2+{\displaystyle \frac{i}{2}},1-i+p\\ {\displaystyle \frac{5}{4}},{\displaystyle \frac{3}{2}},{\displaystyle \frac{7}{4}}\end{array}|-{\displaystyle \frac{A^2}{4B}}\right)I_{i+j-2p-1}^{A,B},\hfill \end{array}\end{array}$$

(126)

where $I_j^{A,B}$ is given by [57]

$$I_j^{A,B}={\int }_0^1{F}_j^{A,B}(x)dx=\left\{\begin{array}{cc}{\displaystyle \frac{A^j}{j+1}}{}_{2}F_1\left(\begin{array}{c}-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{j}{2}},-{\displaystyle \frac{j}{2}}\\ -j\end{array}|-{\displaystyle \frac{4B}{A^2}}\right),\hfill & j\mathit{even},\hfill \\ -{\displaystyle \frac{2B^{\frac{j+1}{2}}}{A(j+1)}}+{\displaystyle \frac{A^j}{j+1}}{}_{2}F_1\left(\begin{array}{c}-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{j}{2}},-{\displaystyle \frac{j}{2}}\\ -j\end{array}|-{\displaystyle \frac{4B}{A^2}}\right),\hfill & j\mathit{odd}.\hfill \end{array}\right.$$

(127)

Proof. 

The above two integral formulas are direct consequences of the two linearization Formulas (65), and (66) along with Formula (127). □

8. Conclusions

This paper presented novel results on a certain class of the JTPs. Some new formulas concerning these polynomials were developed. Derivative expression formulas for these polynomials were established as combinations of the different polynomials. The inverse derivatives formulas were also developed. The pivotal formula was the expression of the shifted CPs of the third in terms of the generalized Jacobsthal polynomials, which led to a new inversion formula of these polynomials. Most of the formulas in this paper are new and may be employed in numerical analysis and approximation theory. In future work, we aim to investigate other generalized sequences of polynomials and utilize them to solve several differential equations.