Some Novel Formulas of the Telephone Polynomials Including New Definite and Indefinite Integrals

Abstract

In this article, we present new theoretical findings on specific polynomials that generalize the concept of telephone numbers, namely, Telephone polynomials (TelPs). Several new formulas are developed, including expressions for higher-order derivatives, repeated integrals, and moment formulas of TelPs. Moreover, we derive explicit connections between the derivatives of TelPs and the two classes of symmetric and non-symmetric polynomials, producing many formulas between these polynomials and several celebrated polynomials such as Hermite, Laguerre, Jacobi, Fibonacci, Lucas, Bernoulli, and Euler polynomials. The inverse formulas are also obtained, expressing the derivatives of well-known polynomial families in terms of TelPs. Furthermore, some novel linearization formulas (LFs) with some classes of polynomials are established. Finally, some new definite and indefinite integrals of TelPs are established using some of the developed relations.

1. Introduction

The use of special functions is crucial to several fields within the scientific community. The special polynomials can be classified into orthogonal polynomials and non-orthogonal polynomials. Some applications regarding special functions and orthogonal polynomials can be found in [1,2,3,4].

Non-orthogonal sequences of polynomials, such as Fibonacci, Lucas, Bernoulli, Euler, Pell, and others, are highly significant in many areas of the applied sciences. Many authors are interested in deriving new formulas regarding these sequences. Advances have been made in the introduction and extension of many polynomial sequences and in their application to practical problems. For example, the authors of [5] introduced a class of Jacobsthal-type polynomials and they derived new formulas regarding them. The authors of [6] developed algebraic properties of Leonardo polynomials and their associated numbers. The authors of [7] used the features of Fibonacci and Lucas polynomials to create new combined polynomials and some of their important formulas. The authors of [8] used a different approach for introducing the Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi polynomials. The authors of [9] looked at Müntz-type systems and developed some new recurrence schemes and identities. In [10], the authors investigated Müntz–Legendre polynomials and how well they could be used to approximate other polynomials. A probabilistic framework for degenerate Stirling polynomials of the second kind was presented in [11]. The main point was to discuss how these polynomials are related to combinatorial distributions and dynamical models. The authors of [12] proposed a more general form of Bell polynomials that helps us grasp their generalized forms, asymptotic properties, and how they may be employed in both functional and combinatorial settings. Other studies in polynomial sequences can be found in [13,14,15].

The contributions regarding special functions are not restricted to theoretical work; many also focused on employing these sequences in approximation theory and numerical analysis, and, in particular, on solving various differential equations. For example, the authors of [16] utilized Fibonacci polynomials to apply a wavelet approach to solve Emden–Fowler type equations. Pell–Lucas polynomials were utilized in [17] to handle the fractional COVID-19 model. Discrete Chebyshev polynomials were used to handle the fractional Klein–Gordon equation. The works of [18,19,20] provide more insights into the applications of different polynomial sequences.

When computing the involutions of a collection of n elements, Heinrich Rothe introduced the telephone numbers (TelNs) in 1800. It is feasible to define the sequence of TelNs as the number of ways to connect n subscribers in a telephone system in which each subscriber can be linked to two other subscribers, at most. Graph theorists also make use of the TelNs to determine the total number of matchings in an n-vertex graph. An extension of the TelNs to construct a sequence of polynomials, called Telelphone polynomials (TelPs), will be investigated theoretically in this study.

It is worth mentioning here that the investigated TelPs belong to the class of Appell sequences. Many recent contributions have focused on the approximation properties of positive linear operators constructed or enhanced by Appell-type polynomial systems. In particular, Szász-type operators and their variants associated with Appell polynomials have been extensively studied in [21,22,23]. In addition, the general theory of summation operators and polynomial solutions of multidimensional difference equations developed in [24] presents a comprehensive theoretical study for polynomial sequences with summation and operational properties.

Many areas would benefit greatly from the linearization formulas (LFs) and the connection formulas between various special functions [25]. Several contributions, both old and recent, have studied these problems for various polynomials. For some pioneers who investigated these problems, one can refer to [26,27,28]. For some other studies in this area, see [29,30,31,32,33]. More recently, some studies regarding these problems and their applications can be found in [34,35,36].

In mathematics and related disciplines, hypergeometric functions were both studied and used extensively. In fact, almost all basic functions can be expressed in terms of hypergeometric functions. Furthermore, hypergeometric functions with different arguments are often used to express the connection and linearization coefficients between special functions, and they can be applied in different contexts; for example, see [37,38].

It is worth mentioning that the motivation, advantages, and necessity of studying TelPs can be listed in the following points:

• The TelPs generalize the well-known numbers that arise in number theory, namely telephone numbers, so the new results of the polynomials, of course, give new results of their associated numbers.
• To the best of our knowledge, studies on TelPs are lacking in the literature. More precisely, the connections of these polynomials with the classical orthogonal polynomials are not found.
• The new formulas in this paper address this gap by providing a unified framework for obtaining the derivatives, inverse relations, linearization formulas, and definite/indefinite integrals involving TelPs.
• The theoretical results obtained in this paper provide a fundamental basis for the use of these polynomials in many applications in numerical analysis and approximation theory.

The main objectives of this paper are listed in the following points:

• Introducing some properties of TelPs, such as the second-order differential equation that is satisfied by them.
• Deriving new explicit expressions of the derivatives and repeated integral moments of these polynomials.
• Deriving new derivative expressions of TelPs as combinations of different polynomials and some of their inverse formulas.
• Establishing mixed linearization formulas of TelPs with some celebrated polynomials.
• Deducing some definite and weighted definite integral formulas based on some of the developed formulas.

The current paper can be structured as follows. The next section presents preliminaries and essential Formulas of TelPs, as well as an overview of celebrated polynomials. Derivatives and integrals of the moments of TelPs are established in Section 3. Different expressions for the derivatives of TelPs are presented in Section 4. Section 5 displays the inverse derivative formulas. In Section 6, some new LFs are developed. New definite and weighted definite integrals for some products involving TelPs are deduced in Section 7. Other closed-form expressions for specific indefinite integrals involving TelPs are given in Section 8. Section 9 introduces a summary of the newly developed formulas in the paper and also introduces some expected applications in numerical analysis to these formulas. Section 10 concludes with a few last thoughts.

2. Background and Key Expressions of Several Polynomials

This section gives an account of TelPs, an account of some symmetric polynomials (SPs), and non-symmetric polynomials (NSPs).

2.1. An Account of Certain Polynomials

Our results will cover certain special polynomials. The two classes of SPs and NSPs are denoted, respectively, by ${\varphi }_m\left(x\right)$ and ${\psi }_m\left(x\right)$. In addition, they can be expressed as

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

(1)

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

(2)

where $H_{r,s}$ and $F_{r,s}$ are known coefficients, and $\lfloor ·\rfloor$ denotes the floor function.

In addition, the inverse formulas for (1) and (2) may be expressed as

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

(3)

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

(4)

with known coefficients ${\overline{H}}_{\mathcal{l},m},{\overline{F}}_{\mathcal{l},m}$.

It is possible to define various well-known polynomials as in (1) and (2). For example, the normalized shifted Jacobi polynomials can be expressed as [5]

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

(5)

They are orthogonal due to the relation

$${\int }_0^1{(1-x)}^{\rho }{x}^{\gamma }{\tilde{R}}_m^{(\rho ,\gamma )}\left(x\right){\tilde{R}}_n^{(\rho ,\gamma )}\left(x\right)dx=\left\{\begin{array}{cc}0,\hfill & n\ne m,\hfill \\ {\tilde{h}}_m^{\rho ,\gamma },\hfill & n=m,\hfill \end{array}\right.$$

(6)

where

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

(7)

There are four types of Chebyshev polynomials involved in Jacobi polynomials, which we also point out. One can obtain all of these types from the following unified recurrence relation:

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

(8)

but differ in starting values.

The relation that gives their moment is

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

(9)

The Schröder polynomials, which were studied in [39], are also NSPs.

A pair of polynomials, namely, the generalized Fibonacci and Lucas polynomials, which are symmetric generalizations of Fibonacci and Lucas polynomials, are generated, respectively, via the following recurrence relations:

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

(10)

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

(11)

Furthermore, there are two other symmetric generalizations of Fibonacci and Pell polynomials, namely, convolved Fibonacci and convolved Pell polynomials. The recurrence relation satisfied by ${\mathcal{CF}}_j^{\mu }\left(x\right)$ is [40]

$$j{\mathcal{CF}}_j^{\mu }\left(x\right)-(\mu +j-1)x{\mathcal{CF}}_{j-1}^{\mu }\left(x\right)-(2\mu +j-2){\mathcal{CF}}_{j-2}^{\mu }\left(x\right)=0,j\ge 2,$$

(12)

with the starting values:

$${\mathcal{CF}}_0^{p,q,\mu }\left(x\right)=1,{\mathcal{CF}}_1^{p,q,\mu }\left(x\right)=\mu x,$$

while the recurrence relation satisfied by the convolved Pell polynomials ${\mathcal{CP}}_j^{\mu }\left(x\right)$ is [41]

$$iC{P}_i^{\nu }\left(x\right)-2(\nu +i-1)xC{P}_{i-1}^{\nu }\left(x\right)-(2\nu +i-2)C{P}_{i-1}^{\nu }\left(x\right)=0,i\ge 2,$$

(13)

with the starting values:

$$C{P}_0^{\nu }\left(x\right)=1,C{P}_1^{\nu }\left(x\right)=2\nu x.$$

Remark 1. 

Particularly, we give in Table 1, the series and inversion coefficients of the following NSPs:

Table 1. The series and inversion coefficients for some NSPs.

Polynomial	${\mathit{F}}_{\mathit{r},\mathit{s}}$	${\overline{\mathit{F}}}_{\mathit{r},\mathit{s}}$
$L_s^{\left(\alpha \right)}\left(x\right)$	${\displaystyle \frac{{(-1)}^{s-r}\left(\genfrac{}{}{0pt}{}{s}{r}\right)\mathrm{\Gamma }(s+\alpha +1)}{s!\mathrm{\Gamma }(s-r+\alpha +1)}}$	${\displaystyle \frac{{(-1)}^{s-r}s!\mathrm{\Gamma }(1+s+\alpha )}{r!\mathrm{\Gamma }(1+s-r+\alpha )}}$
${\tilde{R}}_s^{(\rho ,\gamma )}\left(x\right)$	${\displaystyle \frac{{(-1)}^{r}s!\mathrm{\Gamma }(1+\rho ){(1+\gamma )}_s{(1+\rho +\gamma )}_{2s-r}}{(s-r)!r!\mathrm{\Gamma }(1+s+\rho ){(1+\gamma )}_{s-r}{(1+\rho +\gamma )}_s}}$	${\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{s}{r}\right){(1+\rho )}_{s-r}{(1+s-r+\gamma )}_r}{{(2+2s-2r+\rho +\gamma )}_r{(1+s-r+\rho +\gamma )}_{s-r}}}$
$S{H}_s\left(x\right)$	${\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{s}{s-r}\right)\left(\genfrac{}{}{0pt}{}{2s-r}{s-r}\right)}{1+s-r}}$	${\displaystyle \frac{r!(r+1)!{(-1)}^s(1+2r-2s)}{(2r-s+1)!s!}}$
$B_s\left(x\right)$	${\displaystyle B_r\left(\genfrac{}{}{0pt}{}{s}{s-r}\right)}$	${\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{s+1}{s-r}\right)}{s+1}}$
$E_s\left(x\right)$	${\displaystyle \frac{(2-2^{2+r})B_{r+1}\left(\genfrac{}{}{0pt}{}{s+1}{r+1}\right)}{s+1}}$	${\displaystyle \frac{1}{2}{c}_r\left(\genfrac{}{}{0pt}{}{s}{s-r}\right)}$

• The generalized Laguerre polynomials.
• The shifted Jacobi polynomials ${\tilde{R}}_s^{(\rho ,\gamma )}\left(x\right)$.
• Schröder polynomials $S{H}_s\left(x\right)$.
• Bernoulli polynomials $B_s\left(x\right)$.
• Euler polynomials $E_s\left(x\right)$.

Remark 2. 

Particularly, we give in Table 2, the series and inversion coefficients of the following SPs:

Table 2. The series and inversion coefficients for some SPs.

Polynomial	${\mathit{H}}_{\mathit{r},\mathit{s}}$	${\overline{\mathit{H}}}_{\mathit{r},\mathit{s}}$
$C_s^{\left(\lambda \right)}\left(x\right)$	${\displaystyle \frac{{(-1)}^r{2}^{-1+s-2r}s!\mathrm{\Gamma }(s-r+\lambda )\mathrm{\Gamma }(1+2\lambda )}{(s-2r)!r!\mathrm{\Gamma }(1+\lambda )\mathrm{\Gamma }(s+2\lambda )}}$	${\displaystyle \frac{2^{-s+1}(s-2r+\lambda )s!\mathrm{\Gamma }(\lambda +1)\mathrm{\Gamma }(s-2r+2\lambda )}{(s-2r)!r!\mathrm{\Gamma }(2\lambda +1)\mathrm{\Gamma }(1+s-r+\lambda )}}$
$H_s\left(x\right)$	${\displaystyle \frac{{(-1)}^r{2}^{-2r+s}s!}{r!(s-2r)!}}$	${\displaystyle \frac{2^{-s}s!}{r!(s-2r)!}}$
$F_s^{a,b}\left(x\right)$	${\displaystyle \frac{a^{s-2r}{b}^r{(s-2r+1)}_r}{r!}}$	${\displaystyle \frac{{(-1)}^r(s-2r+1){(s-r+2)}_{r-1}{b}^r}{r!a^s}}$
$L_s^{c,d}\left(x\right)$	${\displaystyle \frac{{\xi }_{s-2r}{c}^{s-2r}{d}^{r}s{(1-2r+s)}_{r-1}}{r!}}$	${\displaystyle \frac{c_{s-2r}{(-1)}^r{c}^{-s}{d}^r{(1-r+s)}_r}{r!}}$
$C{F}_s^{\mu }\left(x\right)$	${\displaystyle \frac{{\left(\mu \right)}_{s-r}}{r!(s-2r)!}}$	${\displaystyle \frac{{(-1)}^r}{r!{\left(\mu \right)}_{s-2r}{(\mu +s-2r+1)}_r}}$
$C{P}_s^{\nu }\left(x\right)$	${\displaystyle \frac{{\left(\nu \right)}_{s-r}{2}^{s-2r}}{r!(s-2r)!}}$	${\displaystyle \frac{s!{(-1)}^r}{2^{s}r!{\left(\nu \right)}_{s-2r}{(\nu +s-2r+1)}_r}}$

• The ultraspherical polynomials.
• Hermite polynomials.
• $F_s^{a,b}$ defined in (10).
• $L_s^{c,d}$ defined in (11).
• The convolved Fibonacci polynomials.
• The convolved Pell polynomials.

Remark 3. 

The constants $c_r$ appearing in the last row of Table 1 are defined by

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

Remark 4. 

Note that the constants ${\xi }_r$, which appear in the fourth row of Table 2, are defined as

$${\xi }_r=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & r=0,\hfill \\ 1,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

(14)

2.2. An Overview of TelPs and Their Corresponding Polynomials

The following recurrence relation can generate the telephone numbers [42]:

$${\mathcal{T}}_n={\mathcal{T}}_{n-1}+(n-1){\mathcal{T}}_{n-2},{\mathcal{T}}_0={\mathcal{T}}_1=1.$$

(15)

In addition, they can be explicitly represented as

$${\mathcal{T}}_n=\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{\displaystyle \frac{n!}{2^{\mathcal{l}}\mathcal{l}!(n-2\mathcal{l})!}}.$$

At this point, we generalize the numbers ${\mathcal{T}}_n$ by defining a series of polynomials ${\mathcal{T}}_n\left(x\right)$. According to [43], ${\mathcal{T}}_n\left(x\right)$ of degree n is defined as follows:

$${\mathcal{T}}_n\left(x\right)=\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{\displaystyle \frac{n!}{2^{\mathcal{l}}\mathcal{l}!(n-2\mathcal{l})!}}{x}^{n-2\mathcal{l}}.$$

(16)

The authors of [43] proved that ${\mathcal{T}}_n\left(x\right)$ the following three-term recurrence relation:

$${\mathcal{T}}_n\left(x\right)=x{\mathcal{T}}_{n-1}\left(x\right)+(n-1){\mathcal{T}}_{n-2}\left(x\right),{\mathcal{T}}_0\left(x\right)=1,{\mathcal{T}}_1\left(x\right)=x.$$

(17)

In addition, in [43], it was proved that the inversion formula of (16) can be expressed in the form

$$x^i=\sum _{r=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^r{(1+i-2r)}_{2r}}{r!}}{\mathcal{T}}_{i-2r}\left(x\right),i\ge 1.$$

(18)

The following lemma states the second-order differential equation satisfied by TelPs.

Lemma 1. 

The set of TelPs ${\left\{{\mathcal{T}}_j\left(x\right)\right\}}_{n\ge 0}$ satisfies the following second-order differential equation:

$${\mathcal{T}}_n^{″}\left(x\right)+x{\mathcal{T}}_n^{\prime }\left(x\right)-n{\mathcal{T}}_n\left(x\right)=0.$$

(19)

Proof. 

Based on the power form representation of TelPs in (16), the expressions for the first and second derivatives are given as follows:

$$\begin{array}{cc}\hfill {\mathcal{T}}_n^{\prime }\left(x\right)& =n!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{\displaystyle \frac{2^{-\mathcal{l}}(n-2\mathcal{l})}{\mathcal{l}!(n-2\mathcal{l})!}}{x}^{n-2\mathcal{l}-1},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathcal{T}}_n^{″}\left(x\right)& =n!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{\displaystyle \frac{2^{-\mathcal{l}}(n-2\mathcal{l})(n-2\mathcal{l}-1)}{\mathcal{l}!(n-2\mathcal{l})!}}{x}^{n-2\mathcal{l}-2}.\hfill \end{array}$$

(21)

Substituting the expressions (16), (20), and (21) in into the left-hand side of (19) will lead to

$$\begin{array}{cc}\hfill & {\mathcal{T}}_n^{″}\left(x\right)+x{\mathcal{T}}_n^{\prime }\left(x\right)-n{\mathcal{T}}_n\left(x\right)=n!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{\displaystyle \frac{2^{-\mathcal{l}}(n-2\mathcal{l})(n-2\mathcal{l}-1)}{\mathcal{l}!(n-2\mathcal{l})!}}{x}^{n-2\mathcal{l}-2}\hfill \\ \hfill & +xn!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{\displaystyle \frac{2^{-\mathcal{l}}(n-2\mathcal{l})}{\mathcal{l}!(n-2\mathcal{l})!}}{x}^{n-2\mathcal{l}-1}-nn!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{n}{2}}⌋}{\displaystyle \frac{2^{-\mathcal{l}}}{\mathcal{l}!(n-2\mathcal{l})!}}{x}^{n-2\mathcal{l}},\hfill \end{array}$$

(22)

and this consequently leads to

$${\mathcal{T}}_n^{″}\left(x\right)+x{\mathcal{T}}_n^{\prime }\left(x\right)-n{\mathcal{T}}_n\left(x\right)=0.$$

This completes the proof. □

3. Derivatives and Integrals of the Moments of TelPs

This section is confined to presenting the derivatives and integrals of the moments of TelPs. We also deduce the following three formulas as particular formulas:

• The high-order derivatives formula of TelPs in terms of their original ones.
• The repeated integrals formula of TelPs as combinations of TelPs.
• The moment formula for TelPs.

3.1. Derivatives of the Moments of TelPs

Theorem 1. 

Let $r,m,$ and s be positive integers with $m+s\ge r$. The following derivatives of the moment formula is valid:

$$\begin{array}{cc}\hfill D^r\left(x^m{\mathcal{T}}_s\left(x\right)\right)=& (s+m)!\sum _{p=0}^{⌊{\displaystyle \frac{1}{2}}(s+m-r)⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^p}{p!(s+m-2p-r)!}}\hfill \\ \hfill & \times {}_{3}F_2\left(\begin{array}{c}-p,{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}},-{\displaystyle \frac{s}{2}}\\ -{\displaystyle \frac{s}{2}}-{\displaystyle \frac{m}{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{m}{2}}\end{array}|1\right){\mathcal{T}}_{s+m-r-2p}\left(x\right).\hfill \end{array}$$

(23)

Proof. 

The power form expression for ${\mathcal{T}}_s\left(x\right)$ aids in expressing $D^r\left(x^m{\mathcal{T}}_s\left(x\right)\right)$ as in the following formula:

$$\begin{array}{cc}\hfill D^r\left(x^m{\mathcal{T}}_s\left(x\right)\right)=& s!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{1}{2}}(s+m-r)⌋}{\displaystyle \frac{2^{-\mathcal{l}}{(1-2\mathcal{l}+m-r+s)}_r}{\mathcal{l}!(-2\mathcal{l}+s)!}}{x}^{s+m-2\mathcal{l}-r}.\hfill \end{array}$$

(24)

The inversion Formula (18) can be applied to the last expression to give the following formula:

$$\begin{array}{cc}\hfill D^r\left(x^m{\mathcal{T}}_s\left(x\right)\right)=& s!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{1}{2}}(s+m-r)⌋}{\displaystyle \frac{2^{-\mathcal{l}}{(1-2\mathcal{l}+m-r+s)}_r}{\mathcal{l}!(-2\mathcal{l}+s)!}}\times \hfill \\ \hfill & \sum _{t=0}^{⌊{\displaystyle \frac{1}{2}}(s+m-r)⌋-\mathcal{l}}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^t{(1-2\mathcal{l}+m-r+s-2t)}_{2t}}{t!}}{\mathcal{T}}_{s+m-2\mathcal{l}-r-2t}\left(x\right).\hfill \end{array}$$

(25)

Introducing the new summation index $p=\mathcal{l}+t$, and rearranging the resulting finite sums, the above expression can be written as

$$\begin{array}{cc}\hfill D^r\left(x^m{\mathcal{T}}_s\left(x\right)\right)=& s!\sum _{p=0}^{⌊{\displaystyle \frac{s+m-r}{2}}⌋}{\displaystyle \frac{2^{-p}}{(s+m-2p-r)!}}\hfill \\ \hfill & \times \sum _{\mathcal{l}=0}^p{\displaystyle \frac{{(-1)}^{p-\mathcal{l}}(m+s-2\mathcal{l})!}{\mathcal{l}!(p-\mathcal{l})!(s-2\mathcal{l})!}}{\mathcal{T}}_{s+m-r-2p}\left(x\right).\hfill \end{array}$$

(26)

We now use the following identities:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{(-1)}^{p-\mathcal{l}}}{(p-\mathcal{l})!}}& ={\displaystyle \frac{{(-1)}^p}{p!}}{(-p)}_{\mathcal{l}},\hfill \\ \hfill (m+s-2\mathcal{l})!& ={\displaystyle \frac{(s+m)!}{2^{2\mathcal{l}}{\left(-\frac{s+m}{2}\right)}_{\mathcal{l}}{\left(\frac{1}{2}-\frac{s+m}{2}\right)}_{\mathcal{l}}}},\hfill \\ \hfill (s-2\mathcal{l})!& ={\displaystyle \frac{s!}{2^{2\mathcal{l}}{\left(-\frac{s}{2}\right)}_{\mathcal{l}}{\left(\frac{1}{2}-\frac{s}{2}\right)}_{\mathcal{l}}}},\hfill \end{array}$$

to convert (26) into the following form:

$$\begin{array}{cc}\hfill D^r\left(x^m{\mathcal{T}}_s\left(x\right)\right)=& (s+m)!\sum _{p=0}^{⌊{\displaystyle \frac{s+m-r}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^p}{p!(s+m-2p-r)!}}\hfill \\ \hfill & \times \sum _{\mathcal{l}=0}^p{\displaystyle \frac{{(-p)}_{\mathcal{l}}{\left(\frac{1}{2}-\frac{s}{2}\right)}_{\mathcal{l}}{\left(-\frac{s}{2}\right)}_{\mathcal{l}}}{{\left(-\frac{s}{2}-\frac{m}{2}\right)}_{\mathcal{l}}{\left(\frac{1}{2}-\frac{s}{2}-\frac{m}{2}\right)}_{\mathcal{l}}\mathcal{l}!}}{\mathcal{T}}_{s+m-r-2p}\left(x\right),\hfill \end{array}$$

(27)

which can be written alternatively in the following hypergeometric form:

$$\begin{array}{cc}\hfill D^r\left(x^m{\mathcal{T}}_s\left(x\right)\right)=& (s+m)!\sum _{p=0}^{⌊{\displaystyle \frac{1}{2}}(s+m-r)⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^p}{p!(s+m-2p-r)!}}\hfill \\ \hfill & \times {}_{3}F_2\left(\begin{array}{c}-p,{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}},-{\displaystyle \frac{s}{2}}\\ -{\displaystyle \frac{s}{2}}-{\displaystyle \frac{m}{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{m}{2}}\end{array}|1\right){\mathcal{T}}_{s+m-r-2p}\left(x\right),\hfill \end{array}$$

which completes the proof. □

Remark 5. 

Two basic formulas regarding the polynomials ${\mathcal{T}}_s\left(x\right)$ can be obtained from (23) as special cases. We give them in the following two corollaries.

Corollary 1. 

Consider m and s to be positive integers. The following moment formula is valid:

$$\begin{array}{c}\hfill x^m{\mathcal{T}}_s\left(x\right)=(m+s)!\sum _{p=0}^{⌊{\displaystyle \frac{s+m}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^p}{p!(m-2p+s)!}}{}_{3}F_2\left(\begin{array}{c}-p,{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}},-{\displaystyle \frac{s}{2}}\\ -{\displaystyle \frac{m}{2}}-{\displaystyle \frac{s}{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{m}{2}}-{\displaystyle \frac{s}{2}}\end{array}|1\right){\mathcal{T}}_{s+m-2p}\left(x\right).\end{array}$$

(28)

Proof. 

Formula (28) can be obtained from (23) by setting $r=0$. □

Corollary 2. 

Consider r and s to be positive integers with $s\ge r$. The following moment formula is valid:

$$D^r{\mathcal{T}}_s\left(x\right)={\displaystyle \frac{s!}{(s-r)!}}{\mathcal{T}}_{s-r}\left(x\right).$$

(29)

Proof. 

Setting $m=0$ in (23) yields the following formula:

$$D^r{\mathcal{T}}_s\left(x\right)=s!\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^p}{p!(s-r-2p)!}}{}_{1}F_0(-p;;1){\mathcal{T}}_{s-r-2p}\left(x\right).$$

(30)

If we use the identity:

$${}_{1}F_0(-p;;1)=\left\{\begin{array}{cc}1,\hfill & p=0,\hfill \\ 0,\hfill & p>0,\hfill \end{array}\right.$$

then, Formula (30) takes the following simple form:

$$D^r{\mathcal{T}}_s\left(x\right)={\displaystyle \frac{s!}{(s-r)!}}{\mathcal{T}}_{s-r}\left(x\right).$$

(31)

This ends the proof. □

3.2. Repeated Integrals of the Moments of TelPs

This section is confined to the derivation of the repeated integrals of the moments of TelPs. The following theorem presents this result.

Theorem 2. 

For all positive integers $s,m$, and r, one has the following r repeated integrals formula for the moments of TelPs:

$$\begin{array}{cc}\hfill \stackrel{rtimes}{\overbrace{\int \int \dots \int }}{x}^m{\mathcal{T}}_s\left(x\right)dx=& (s+m)!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{m+s}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{\mathcal{l}}}{\mathcal{l}!(s-2\mathcal{l}+m+r)!}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}-\mathcal{l},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}},-{\displaystyle \frac{s}{2}}\\ -{\displaystyle \frac{s}{2}}-{\displaystyle \frac{m}{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{m}{2}}\end{array}|1\right){\mathcal{T}}_{s+m+r-2\mathcal{l}}\left(x\right)+{\rho }_{r-1}\left(x\right),\hfill \end{array}$$

(32)

where ${\rho }_{r-1}\left(x\right)$ is a polynomial of degree at most $(r-1)$.

Proof. 

If we integrate the power form expression in (16) r times, then we obtain

$$\begin{array}{c}\hfill \stackrel{rtimes}{\overbrace{\int \int \dots \int }}{x}^m{\mathcal{T}}_s\left(x\right)dx=s!\sum _{p=0}^{⌊{\displaystyle \frac{m+s}{2}}⌋}{\displaystyle \frac{2^{-p}}{p!(s-2p)!{(1+m-2p+s)}_r}}{x}^{s+m-2p+r}+{\rho }_{r-1}\left(x\right).\end{array}$$

(33)

Utilizing Formula (18) leads to converting (33) into the following form:

$$\begin{array}{cc}\hfill \stackrel{rtimes}{\overbrace{\int \int \dots \int }}{x}^m{\mathcal{T}}_s\left(x\right)dx=& s!\sum _{p=0}^{⌊{\displaystyle \frac{m+s}{2}}⌋}{\displaystyle \frac{2^{-p}}{p!(s-2p)!{(1+m-2p+s)}_r}}\times \hfill \\ \hfill & \sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{1}{2}}(s+m-2p+r)⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{\mathcal{l}}{(1-2\mathcal{l}+m-2p+r+s)}_{2\mathcal{l}}}{\mathcal{l}!}}{\mathcal{T}}_{s+m-2\mathcal{l}+r-2p}\left(x\right)+{\rho }_{r-1}\left(x\right),\hfill \end{array}$$

(34)

which can be written again in the form

$$\begin{array}{c}\hfill \stackrel{rtimes}{\overbrace{\int \int \dots \int }}{x}^m{\mathcal{T}}_s\left(x\right)dx=s!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{m+s}{2}}⌋}{\displaystyle \frac{2^{-\mathcal{l}}}{(m+r+s-2\mathcal{l})!}}\sum _{p=0}^{\mathcal{l}}{\displaystyle \frac{{(-1)}^{\mathcal{l}-p}(m-2p+s)!}{p!(\mathcal{l}-p)!(s-2p)!}}{\mathcal{T}}_{s+m+r-2\mathcal{l}}\left(x\right)+{\rho }_{r-1}\left(x\right).\end{array}$$

(35)

The last formula is equivalent to

$$\begin{array}{cc}\hfill \stackrel{rtimes}{\overbrace{\int \int \dots \int }}{x}^m{\mathcal{T}}_s\left(x\right)dx=& (s+m)!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{m+s}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{\mathcal{l}}}{\mathcal{l}!(s-2\mathcal{l}+m+r)!}}\times \hfill \\ \hfill & {}_{3}F_2\left(\begin{array}{c}-\mathcal{l},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}},-{\displaystyle \frac{s}{2}}\\ -{\displaystyle \frac{s}{2}}-{\displaystyle \frac{m}{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{m}{2}}\end{array}|1\right){\mathcal{T}}_{s+m+r-2\mathcal{l}}\left(x\right)+{\rho }_{r-1}\left(x\right).\hfill \end{array}$$

Thus, Formula (32) is proved. □

A formula for the repeated integrals of TelPs expressed as combinations of TelPs is presented in the following corollary.

Corollary 3. 

For any integer r, the following is the r times repeated integrals formula for TelPs:

$$\stackrel{rtimes}{\overbrace{\int \int \dots \int }}{\mathcal{T}}_s\left(x\right)dx={\displaystyle \frac{s!}{(s+r)!}}{\mathcal{T}}_{s+r}\left(x\right)+{\rho }_{r-1}\left(x\right),$$

(36)

where ${\rho }_{r-1}\left(x\right)$ is a polynomial of degree at most $(r-1)$.

Proof. 

If we let $m=0$ in Formula (32), then we obtain

$$\stackrel{rtimes}{\overbrace{\int \int \dots \int }}{\mathcal{T}}_s\left(x\right)dx=s!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{s}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{\mathcal{l}}{}_{1}F_0(-\mathcal{l};;1)}{\mathcal{l}!(s-2\mathcal{l}+r)!}}{\mathcal{T}}_{s+r-2\mathcal{l}}\left(x\right)+{\rho }_{r-1}\left(x\right),$$

(37)

where ${\rho }_{r-1}\left(x\right)$ is a polynomial of degree at most $(r-1)$.

Formula (37) can be easily reduced to the following equation:

$$\stackrel{rtimes}{\overbrace{\int \int \dots \int }}{\mathcal{T}}_s\left(x\right)dx={\displaystyle \frac{s!}{(s+r)!}}{\mathcal{T}}_{s+r}\left(x\right)+{\rho }_{r-1}\left(x\right).$$

(38)

This ends the proof. □

4. Expressions for the Derivatives of TelPs

This section establishes expressions for the derivatives of the TElPs in terms of different polynomials. We give two generalized formulas for the derivatives of TelPs as combinations of the following two sequences of polynomials:

• The symmetric polynomials that are expressed in (1).
• The non-symmetric polynomials that are expressed in (2).

4.1. Derivatives in Terms of SPs

This part gives an explicit formula for the derivatives of TelPs in terms of any symmetric polynomial.

Theorem 3. 

Let ${\varphi }_j\left(x\right)$ be SPs defined in (1). For the two positive integers s, and r with $s\ge r$, the following formula holds:

$$D^r{\mathcal{T}}_s\left(x\right)=s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}\sum _{i=0}^k{\displaystyle \frac{2^{-i}{\overline{H}}_{k-i,s-r-2i}}{i!(s-r-2i)!}}{\varphi }_{s-r-2k}\left(x\right),$$

(39)

where ${\overline{H}}_{\mathcal{l},m}$ are the coefficients appear in (3).

Proof. 

If the power form Formula (16) is differentiated r times, then we can write

$$D^r{\mathcal{T}}_s\left(x\right)=s!\sum _{i=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{2^{-i}}{i!(s-r-2i)!}}{x}^{s-2i-r}.$$

(40)

If the inversion formula of ${\varphi }_i\left(x\right)$ in (3) is applied, then the following formula can be obtained:

$$D^r{\mathcal{T}}_s\left(x\right)=s!\sum _{i=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{2^{-i}}{i!(s-r-2i)!}}\sum _{t=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋-i}{\overline{H}}_{t,s-2i-r}{\varphi }_{s-2i-r-2t}\left(x\right),$$

(41)

which can be written again in the following more convenient form:

$$D^r{\mathcal{T}}_s\left(x\right)=s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}\sum _{i=0}^k{\displaystyle \frac{2^{-i}{\overline{H}}_{k-i,s-r-2i}}{i!(s-r-2i)!}}{\varphi }_{s-r-2k}\left(x\right).$$

(42)

This ends the proof. □

As a consequence of Theorem 3, we can express the derivatives of TelPs in terms of any symmetric polynomial represented by (1). First, we will give a new simple expression for the derivatives of TelPs as combinations of Hermite polynomials, which is free of any hypergeometric functions.

Corollary 4. 

For the two positive integers s, and r with $s\ge r$, one has the following derivative expression holds:

$$D^r{\mathcal{T}}_s\left(x\right)=2^{r-s}s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{3^k}{k!(s-2k-r)!}}{H}_{s-r-2k}\left(x\right).$$

(43)

Proof. 

Using the inversion formula of the Hermite polynomials (see Table 1), it can be shown that

$$D^r{\mathcal{T}}_s\left(x\right)=s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{1}{(s-2k-r)!}}\sum _{i=0}^k{\displaystyle \frac{2^{-s+r+i}}{(k-i)!i!}}{H}_{s-r-2k}\left(x\right).$$

(44)

Regarding the sum: ${\displaystyle \sum _{i=0}^k\frac{2^{-s+r+i}}{(k-i)!i!}}$, we set

$$M_{k,r,s}=\sum _{i=0}^k{\displaystyle \frac{2^{r-s+i}}{(k-i)!i!}}.$$

One can demonstrate that $M_{k,r,s}$ satisfies the following recursive formula:

$$(k+1)M_{k+1,r,s}-3M_{k,r,s}=0,M_{0,r,s}=2^{r-s},$$

whose solution is given by

$$\sum _{i=0}^k{\displaystyle \frac{2^{r-s+i}}{(k-i)!i!}}={\displaystyle \frac{2^{r-s}{3}^k}{k!}},$$

hence, we may derive the following formula:

$$D^r{\mathcal{T}}_s\left(x\right)=2^{r-s}s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{3^k}{k!(s-2k-r)!}}{H}_{s-r-2k}\left(x\right).$$

This ends the proof. □

Example 1. 

As a concrete illustration of Formula (43), we present the following specific formula for the case that corresponds to $s=9,r=4$.

$$D^4{\mathcal{T}}_9\left(x\right)={\displaystyle \frac{189}{2}}{H}_5\left(x\right)+5670H_3\left(x\right)+51030H_1\left(x\right).$$

Some other expressions for the derivatives of TelPs as combinations of the SPs are displayed in the following corollary.

Corollary 5. 

For the two positive integers s, and r with $s\ge r$, one has

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& {\displaystyle \frac{\sqrt{\pi }s!}{\mathrm{\Gamma }\left(\frac{1}{2}+\lambda \right)}}\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{2^{1-s+k+r-2\lambda }(s-2k-r+\lambda )\mathrm{\Gamma }(s-2k-r+2\lambda )}{k!(s-2k-r)!}}\times \hfill \\ \hfill & {}_1\tilde{F}_1\left(-k;1+s-2k-r+\lambda ;-{\displaystyle \frac{1}{2}}\right)U_{s-r-2k}^{\left(\lambda \right)}\left(x\right),\hfill \end{array}\end{array}$$

(45)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^k{a}^{-s+2k+r}(1+s-2k-r)}{k!(s-k-r+1)!}}\times \hfill \\ \hfill & U\left(-k,2+s-2k-r,{\displaystyle \frac{2b}{a^2}}\right)F_{s-r-2k}^{a,b}\left(x\right),\hfill \end{array}\end{array}$$

(46)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{\xi }_{s-r-2k}{\left(-\frac{1}{2}\right)}^k{c}^{-s+2k+r}}{k!(s-k-r)!}}\times \hfill \\ \hfill & U\left(-k,1+s-2k-r,{\displaystyle \frac{2d}{c^2}}\right)L_{s-r-2k}^{c,d}\left(x\right),\hfill \end{array}\end{array}$$

(47)

where ${}_{1}F_1(a;b;z)$ is the Kummer confluent hypergeometric function of the first kind, and $U(a,b,z)$ is the confluent hypergeometric function of the second kind [44,45], and ${\xi }_k$ is as defined in (14).

Proof. 

Formulas (45)–(47) can be obtained by the application of Theorem 3 taking into consideration the inversion coefficients of the three polynomials $U_m^{\left(\lambda \right)}\left(x\right),F_m^{a,b}\left(x\right)$, and $L_m^{c,d}\left(x\right)$ that mentioned in Table 2. □

Corollary 6. 

Let $C{F}_s^{\mu }\left(x\right)$ and $C{P}_s^{\mu }\left(x\right)$ be the convolved Fibonacci and convolved Pell polynomials that are generated, respectively, by (12), and (13). The following derivative expressions hold:

$$\begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& \mathrm{\Gamma }\left(\mu \right)s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{2^{-k}(-2k+\mu -r+s){}_1\tilde{F}_1(-k;1-2k+\mu -r+s;2)}{k!}}{F}_{s-r-2k}^{\mu }\left(x\right),\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& s!\mathrm{\Gamma }\left(\mu \right)\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{2^{k+r-s}(-2k+\mu -r+s){}_1\tilde{F}_1\left(-k;1-2k+\mu -r+s;\frac{1}{2}\right)}{k!}}{P}_{s-r-2k}^{\nu }\left(x\right).\hfill \end{array}$$

(49)

Proof. 

The application of Theorem 3 can yield the above formulas, taking into consideration the inversion coefficients of the two convolved polynomials $C{F}_j^{\left(\mu \right)}\left(x\right)$, and $C{P}_j^{\nu }\left(x\right)$ that are mentioned in Table 2. □

4.2. Derivatives as Combinations of NSPs

This section derives a general formula for the derivatives of TelPs in terms of any NSPs that can be expressed in (2).

Theorem 4. 

For any non-symmetric polynomial ${\psi }_i\left(x\right)$ defined in (2), one has the following expression:

$$\begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)& =s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}\sum _{i=0}^k{\displaystyle \frac{2^{-i}{F}_{2k-2i,s-2i-r}}{i!(s-r-2i)!}}{\psi }_{s-r-2k}\left(x\right)\hfill \\ \hfill & +s!\sum _{k=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}\sum _{i=0}^k{\displaystyle \frac{2^{-i}{F}_{2k-2i+1,s-2i-r}}{i!(s-r-2i)!}}{\psi }_{s-r-2k-1}\left(x\right).\hfill \end{array}$$

(50)

Proof. 

First, we can write

$$D^r{\mathcal{T}}_s\left(x\right)=s!\sum _{i=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{2^{-i}}{i!(s-r-2i)!}}{x}^{s-2i-r}.$$

(51)

If the inversion formula of ${\psi }_i\left(x\right)$ in the form of (4) is applied, then the following formula can be obtained:

$$D^r{\mathcal{T}}_s\left(x\right)=s!\sum _{i=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{2^{-i}}{i!(s-r-2i)!}}\sum _{t=0}^{s-2i-r}{F}_{t,s-2i-r}{\psi }_{s-2i-r-t}\left(x\right).$$

(52)

The last formula can be expanded and rearranged to give

$$\begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)& =s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}\sum _{i=0}^k{\displaystyle \frac{2^{-i}{F}_{2k-2i,s-2i-r}}{i!(s-r-2i)!}}{\psi }_{s-r-2k}\left(x\right)\hfill \\ \hfill & +s!\sum _{k=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}\sum _{i=0}^k{\displaystyle \frac{2^{-i}{F}_{2k-2i+1,s-2i-r}}{i!(s-r-2i)!}}{\psi }_{s-r-2k-1}\left(x\right).\hfill \end{array}$$

(53)

This ends the proof. □

As a consequence of Theorem 4, we can express the derivatives of TelPs in terms of any non-symmetric polynomial that is represented as in (2).

Corollary 7. 

For the positive integers s and r, with $s\ge r$, the following derivative can be expressed using the generalized Laguerre polynomials:

$$\begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& s!\mathrm{\Gamma }(1+s-r+\alpha )\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{1}{\left(2k\right)!\mathrm{\Gamma }(1+s-2k-r+\alpha )}}\times \hfill \\ \hfill & {}_{2}F_2\left(\begin{array}{c}-k,{\displaystyle \frac{1}{2}}-k\\ -{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{\alpha }{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{\alpha }{2}}\end{array}|{\displaystyle \frac{1}{2}}\right)L_{s-r-2k}^{\left(\alpha \right)}\left(x\right)\hfill \\ \hfill & -s!\sum _{k=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{\displaystyle \frac{\mathrm{\Gamma }(1+s-r+\alpha )}{(2k+1)!\mathrm{\Gamma }(s-2k-r+\alpha )}}\times \hfill \\ \hfill & {}_{2}F_2\left(\begin{array}{c}-k,-{\displaystyle \frac{1}{2}}-k\\ -{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{\alpha }{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{\alpha }{2}}\end{array}|{\displaystyle \frac{1}{2}}\right)L_{s-r-2k-1}^{\left(\alpha \right)}\left(x\right).\hfill \end{array}$$

(54)

Proof. 

The above formula is derived using Theorem 4 and the inversion formula for $L_k^{\left(\alpha \right)}\left(x\right)$. □

Corollary 8. 

For the two positive integers s, and r with $s\ge r$, $D^r{\mathcal{T}}_s\left(x\right)$ can be expressed in terms of Schröder polynomials as

$$\begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& s!(s-r+1)!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{1+2s-4k-2r}{\left(2k\right)!(2s-2k-2r+1)!}}\times \hfill \\ \hfill & {}_{4}F_2\left(\begin{array}{c}-k,-{\displaystyle \frac{1}{2}}-k,-{\displaystyle \frac{1}{2}}-s+k+r,-s+k+r\\ -{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}},-{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}\end{array}|2\right)S{H}_{s-r-2k}\left(x\right)\hfill \\ \hfill & +s!(s-r+1)!\sum _{k=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{\displaystyle \frac{1-2s+4k+2r}{(2k+1)!(2s-2k-2r)!}}\times \hfill \\ \hfill & {}_{4}F_2\left(\begin{array}{c}-k,-s+k+r,-{\displaystyle \frac{1}{2}}-k,{\displaystyle \frac{1}{2}}-s+k+\\ -{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}},-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}\end{array}|2\right)S{H}_{s-r-2k-1}\left(x\right).\hfill \end{array}$$

(55)

Proof. 

Applying Theorem 4 while taking into account the inversion formula of Schröder polynomials in Table 1 allows one to deduce the formula. □

Corollary 9. 

For the two positive integers s, and r with $s\ge r$, $D^r{\mathcal{T}}_s\left(x\right)$ can be expressed in terms of Bernoulli and Euler polynomials:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{-k}U\left(-k,\frac{3}{2},-\frac{1}{2}\right)}{(2k+1)!(s-2k-r)!}}{B}_{s-r-2k}\left(x\right)\times \hfill \\ \hfill & +s!\sum _{k=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{\displaystyle \frac{2^{-1-k}\left(-1+{}_{1}F_1\left(-1-k;\frac{1}{2};-\frac{1}{2}\right)\right)}{(k+1)!(s-2k-r-1)!}}{B}_{s-r-2k-1}\left(x\right),\hfill \end{array}\end{array}$$

(56)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^r{\mathcal{T}}_s\left(x\right)=& s!\sum _{k=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{2^{-1-k}\left(1+{}_{1}F_1\left(-k;\frac{1}{2};-\frac{1}{2}\right)\right)}{k!(s-2k-r)!}}{E}_{s-r-2k}\left(x\right)\hfill \\ \hfill & +s!\sum _{k=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{\displaystyle \frac{{(-2)}^k(1+k)U\left(-k,\frac{3}{2},-\frac{1}{2}\right)}{(2k+2)!(s-2k-r-1)!}}{E}_{s-r-2k-1}\left(x\right).\hfill \end{array}\end{array}$$

(57)

Proof. 

The formulas can be derived by applying Theorem 4, taking into account the two inversion formulas of Bernoulli and Euler polynomials in Table 1. □

5. Inverse Formulas of the Derivatives

In this section, we write the expressions for the derivatives of symmetric and non-symmetric polynomials that are defined, respectively, in (1) and (2), as combinations of TelPs.

5.1. Derivatives of SPs in Terms of TelPs

This part aims to develop a general formula for the derivatives of the SPs with respect to TelPs. Furthermore, some specific formulas will also be derived.

Theorem 5. 

For all non-negative integers $s,r$, with $s\ge r$, one has

$$D^r{\varphi }_s\left(x\right)=\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}\sum _{i=0}^p{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{p-i}{(1+s-2p-r)}_{2p-2i}{(s-2i-r+1)}_r}{(p-i)!}}{H}_{i,s}{\mathcal{T}}_{s-r-2p}\left(x\right),$$

(58)

and $H_{i,s}$ is the power form coefficient that appears in the expression (1).

Proof. 

If we differentiate the expression in (1), then we obtain

$$D^r{\varphi }_s\left(x\right)=\sum _{i=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{H}_{i,s}{(s-2i-r+1)}_r{x}^{s-2i-r}.$$

(59)

The inverse formula of ${\mathcal{T}}_i\left(x\right)$ in (18) leads to the following formula:

$$D^r{\varphi }_s\left(x\right)=\sum _{i=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{(s-2i-r+1)}_r{H}_{i,s}\sum _{t=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋-i}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^t{(1+s-r-2i-2t)}_{2t}}{t!}}{\mathcal{T}}_{s-2i-r-2t}\left(x\right),$$

(60)

which can be transformed into

$$D^r{\varphi }_s\left(x\right)=\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}\sum _{i=0}^p{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{p-i}{(1+s-2p-r)}_{2p-2i}{(s-2i-r+1)}_r}{(p-i)!}}{H}_{i,s}{\mathcal{T}}_{s-r-2p}\left(x\right).$$

(61)

This completes the proof. □

Now, as an application of the previous theorem, explicit expressions for some SPs in terms of the TeLPs are found in the following corollary.

Corollary 10. 

The derivatives of the classical Hermite polynomials can be expressed as combinations of TelPs, as shown in the following formula:

$$D^r{H}_s\left(x\right)=s!\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{(-3)}^p{2}^{s-2p}}{p!(s-2p-r)!}}{\mathcal{T}}_{s-r-2p}\left(x\right).$$

(62)

Proof. 

By employing the analytic form of Hermite polynomials, we derive

$$D^r{H}_s\left(x\right)=s!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{(-1)}^{\mathcal{l}}{2}^{-2\mathcal{l}+s}{(1-2\mathcal{l}-r+s)}_r}{\mathcal{l}!(-2\mathcal{l}+s)!}}{x}^{s-2\mathcal{l}-r}.$$

(63)

If the inversion formula of TelPs is applied, then we obtain

$$\begin{array}{cc}\hfill D^r{H}_s\left(x\right)=& s!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{(-1)}^{\mathcal{l}}{2}^{-2\mathcal{l}+s}{(1-2\mathcal{l}-r+s)}_r}{\mathcal{l}!(-2\mathcal{l}+s)!}}\times \hfill \\ \hfill & \sum _{m=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋-\mathcal{l}}{\displaystyle \frac{{(-1)}^m{(1-2\mathcal{l}-r+s-2m)}_{2m}}{2^{m}m!}}{\mathcal{T}}_{s-2\mathcal{l}-r-2m}\left(x\right).\hfill \end{array}$$

(64)

Some computations convert the last formula to the following formula:

$$D^r{H}_s\left(x\right)=s!\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{1}{(-2p-r+s)!}}\sum _{\mathcal{l}=0}^p{\displaystyle \frac{{(-1)}^p{2}^{-\mathcal{l}-p+s}}{\mathcal{l}!(p-\mathcal{l})!}}{\mathcal{T}}_{s-r-2p}\left(x\right).$$

(65)

Regarding the sum: ${\displaystyle \sum _{\mathcal{l}=0}^p\frac{{(-1)}^p{2}^{-\mathcal{l}-p+s}}{\mathcal{l}!(p-\mathcal{l})!}}$, it can be computed using symbolic algebra. For this end, let

$${\displaystyle G_{p,s}=\sum _{\mathcal{l}=0}^p\frac{{(-1)}^p{2}^{-\mathcal{l}-p+s}}{\mathcal{l}!(p-\mathcal{l})!},}$$

the application of Zeilberger’s algorithm [46] produces the following recursive formula:

$$G_{p+1,s}+{\displaystyle \frac{3}{4(p+1)}}{G}_{p,s}=0,G_{0,s}=2^s.$$

(66)

An exact solution to the last recurrence relation is

$$G_{p,s}={\displaystyle \frac{{(-3)}^p{2}^{-2p+s}}{p!}},$$

and so, we can derive the following formula:

$$D^r{H}_s\left(x\right)=s!\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{(-3)}^p{2}^{s-2p}}{p!(s-2p-r)!}}{\mathcal{T}}_{s-r-2p}\left(x\right).$$

(67)

This ends the proof. □

Corollary 11. 

The following derivative formulas hold for every $s\ge r$:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D^r{U}_s^{\left(\lambda \right)}\left(x\right)=& {\displaystyle \frac{s!\mathrm{\Gamma }(s+\lambda )\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)}{\sqrt{\pi }\mathrm{\Gamma }(s+2\lambda )}}\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{(-1)}^p{2}^{-1+s-p+2\lambda }{}_{1}F_1\left(-p;1-s-\lambda ;\frac{1}{2}\right)}{p!(s-2p-r)!}}\times \hfill \\ \hfill & {\mathcal{T}}_{s-r-2p}\left(x\right),\hfill \end{array}\end{array}$$

(68)

$$\begin{array}{cc}\hfill D^r{F}_s^{a,b}\left(x\right)=& a^{s}s!\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^p{}_{1}F_1\left(-p;-s;-\frac{2b}{a^2}\right)}{p!(s-2p-r)!}}{\mathcal{T}}_{s-r-2p}\left(x\right),\hfill \end{array}$$

(69)

$$\begin{array}{cc}\hfill D^r{L}_s^{c,d}\left(x\right)=& c^{s}s!\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^p{}_{1}F_1\left(-p;1-s;-\frac{2d}{c^2}\right)}{p!(s-2p-r)!}}{\mathcal{T}}_{s-r-2p}\left(x\right).\hfill \end{array}$$

(70)

Proof. 

Formulas (68)–(70) are direct results of Theorem 5, taking into consideration the power form coefficients in Table 2. □

Corollary 12. 

Let $C{F}_s^{\mu }\left(x\right)$ and $C{P}_s^{\left(\nu \right)}\left(x\right)$ be the convolved Fibonacci and convolved Pell polynomials that are expressed, respectively, in (12), and (13). Their derivatives can be expressed in terms of TelPs as follows:

$$\begin{array}{cc}\hfill D^{r}C{F}_s^{\mu }\left(x\right)& ={\left(\mu \right)}_s\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^p{}_{1}F_1\left(-p;1-s-\mu ;-2\right)}{p!(s-2p-r)!}}{\mathcal{T}}_{s-r-2p}\left(x\right),\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill D^{r}C{P}_s^{\nu }\left(x\right)& ={\left(\nu \right)}_s\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{(-1)}^p{2}^{s-p}{}_{1}F_1\left(-p;1-s-\nu ;-\frac{1}{2}\right)}{p!(s-2p-r)!}}{\mathcal{T}}_{s-r-2p}\left(x\right).\hfill \end{array}$$

(72)

Proof. 

Formulas (71) and (72) are direct results of Theorem 5, taking into consideration the power form coefficients of the two convolved Fibonacci and Pell polynomials Table 2. □

5.2. Derivatives of NSPs in Terms of TelPs

This part is confined to presenting a general theorem in which we develop an expression that computes the high-order derivatives of NSPs as combinations of TelPs.

Theorem 6. 

For $s\ge r$, and for NSPs that are represented as in (2), one has the following derivative formula:

$$\begin{array}{cc}\hfill D^r{\psi }_s\left(x\right)=& \sum _{m=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}\sum _{p=0}^m{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{m-p}{(1+s-2m-r)}_{2m-2p}}{(m-p)!}}{F}_{2p,s}{\mathcal{T}}_{s-r-2m}\left(x\right)\hfill \\ \hfill & +\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}\sum _{p=0}^m{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{m-p}{(s-2m-r)}_{2m-2p}}{(m-p)!}}{F}_{2p+1,s}{\mathcal{T}}_{s-r-2m-1}\left(x\right),\hfill \end{array}$$

(73)

where $F_{p,s}$ are the power form coefficients that appear in (2).

Proof. 

We start with the analytic formula of the NSPs in (2), then insert the inversion formula of TelPs to obtain the following expression:

$$D^r{\psi }_s\left(x\right)=\sum _{p=0}^{s-r}{B}_{p,s}\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{1}{2}}(s-p-r)⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{\mathcal{l}}{(1+s-r-p-2\mathcal{l})}_{2\mathcal{l}}}{\mathcal{l}!}}{\mathcal{T}}_{s-p-r-2\mathcal{l}}\left(x\right).$$

(74)

Some computations lead to the following formula:

$$\begin{array}{cc}\hfill D^r{\psi }_s\left(x\right)=& \sum _{m=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}\sum _{p=0}^m{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{m-p}{(1+s-2m-r)}_{2m-2p}}{(m-p)!}}{F}_{2p,s}{\mathcal{T}}_{s-r-2m}\left(x\right)\hfill \\ \hfill & +\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}\sum _{p=0}^m{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^{m-p}{(s-2m-r)}_{2m-2p}}{(m-p)!}}{F}_{2p+1,s}{\mathcal{T}}_{s-r-2m-1}\left(x\right).\hfill \end{array}$$

(75)

This ends the proof. □

Corollary 13. 

For $s\ge r$, $D^r{L}_s^{\left(\alpha \right)}\left(x\right)$ may be expressed as combinations of TelPs in the following form:

$$\begin{array}{cc}\hfill D^r{L}_s^{\left(\alpha \right)}\left(x\right)=& \sum _{m=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{(-1)}^{s+m}{2}^{-m}{}_{3}F_1\left(\begin{array}{c}-m,-\frac{s}{2}-\frac{\alpha }{2},\frac{1}{2}-\frac{s}{2}-\frac{\alpha }{2}\\ \frac{1}{2}\end{array}|2\right)}{m!(s-2m-r)!}}{\mathcal{T}}_{s-r-2m}\left(x\right)\hfill \\ \hfill & +(s+\alpha )\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{\displaystyle \frac{{(-1)}^{1+s+m}{2}^{-m}{}_{3}F_1\left(\begin{array}{c}-m,\frac{1}{2}-\frac{s}{2}-\frac{\alpha }{2},1-\frac{s}{2}-\frac{\alpha }{2}\\ \frac{3}{2}\end{array}|2\right)}{m!(s-2m-r-1)!}}{\mathcal{T}}_{s-r-2m-1}\left(x\right).\hfill \end{array}$$

(76)

Proof. 

This is a direct consequence of Theorem 6. □

Corollary 14. 

For $s\ge r$, $D^r{R}_s^{(\alpha ,\beta )}\left(x\right)$ may be expressed as combinations of TelPs in the following form:

$$\begin{array}{cc}\hfill D^r{R}^{(\alpha ,\beta )}\left(x\right)=& {\displaystyle \frac{s!\mathrm{\Gamma }(1+\alpha )\mathrm{\Gamma }(1+2s+\alpha +\beta )}{\mathrm{\Gamma }(1+s+\alpha )\mathrm{\Gamma }(1+s+\alpha +\beta )}}\sum _{m=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{(-1)}^m{2}^{-m}}{m!(s-2m-r)!}}\times \hfill \\ \hfill & {}_{3}F_3\left(\begin{array}{c}-m,-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{\beta }{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{\beta }{2}}\\ {\displaystyle \frac{1}{2}},-s-{\displaystyle \frac{\alpha }{2}}-{\displaystyle \frac{\beta }{2}},{\displaystyle \frac{1}{2}}-s-{\displaystyle \frac{\alpha }{2}}-{\displaystyle \frac{\beta }{2}}\end{array}|{\displaystyle \frac{1}{2}}\right){\mathcal{T}}_{s-r-2m}\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{(s+\beta )s!\mathrm{\Gamma }(1+\alpha )\mathrm{\Gamma }(2s+\alpha +\beta )}{\mathrm{\Gamma }(1+s+\alpha )\mathrm{\Gamma }(1+s+\alpha +\beta )}}\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{\displaystyle \frac{{(-1)}^{m+1}{2}^{-m}}{m!(-2m-r+s-1)!}}\times \hfill \\ \hfill & {}_{3}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{\beta }{2}},1-{\displaystyle \frac{s}{2}}-{\displaystyle \frac{\beta }{2}}\\ {\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}}-s-{\displaystyle \frac{\alpha }{2}}-{\displaystyle \frac{\beta }{2}},1-s-{\displaystyle \frac{\alpha }{2}}-{\displaystyle \frac{\beta }{2}}\end{array}|{\displaystyle \frac{1}{2}}\right){\mathcal{T}}_{s-r-2m-1}\left(x\right).\hfill \end{array}$$

(77)

Proof. 

This is a direct consequence of Theorem 6. □

Corollary 15. 

For $s\ge r$, $D^{r}S{H}_s\left(x\right)$ may be expressed as combinations of TelPs in the following form:

$$\begin{array}{cc}\hfill D^{r}S{H}_s\left(x\right)=& {\displaystyle \frac{\left(2s\right)!}{(s+1)!}}\sum _{m=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^m}{m!(s-2m-r)!}}\times \hfill \\ \hfill & {}_{3}F_3\left(\begin{array}{c}-m,-{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}},-{\displaystyle \frac{s}{2}}\\ {\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}-s,-s\end{array}|{\displaystyle \frac{1}{2}}\right){\mathcal{T}}_{s-r-2m}\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{(2s-1)!}{s!}}\sum _{m=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{\displaystyle \frac{{\left(-\frac{1}{2}\right)}^m}{m!(s-2m-r-1)!}}\times \hfill \\ \hfill & {}_{3}F_3\left(\begin{array}{c}-m,{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}},-{\displaystyle \frac{s}{2}}\\ {\displaystyle \frac{3}{2}},{\displaystyle \frac{1}{2}}-s,1-s\end{array}|{\displaystyle \frac{1}{2}}\right){\mathcal{T}}_{s-r-2m-1}\left(x\right).\hfill \end{array}$$

(78)

Proof. 

This is a direct consequence of Theorem 6. □

6. LFs with Some Polynomials

This section presents some new LFs for TelPs with some polynomials. More precisely, the LFs for the Chebyshev polynomials, the generalized Fibonacci polynomials, and the generalized Lucas polynomials will be given. This section concentrates on developing some LFs of TelPs with some celebrated polynomials.

Theorem 7. 

Consider any two positive integers i and j. The following linearization formula (LF) holds:

$${\mathcal{T}}_i\left(x\right)C_j\left(x\right)=\sum _{p=0}^i{(-1)}^{-i+p}{2}^{-p}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)U\left(-i+p,1-i+2p,-{\displaystyle \frac{1}{2}}\right)C_{i+j-2p}\left(x\right).$$

(79)

Proof. 

Substituting by the power form representation of ${\mathcal{T}}_i\left(x\right)$ yields

$${\mathcal{T}}_i\left(x\right)C_j\left(x\right)=i!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{\displaystyle \frac{2^{-\mathcal{l}}}{(i-2\mathcal{l})!\mathcal{l}!}}{x}^{i-2\mathcal{l}}{C}_j\left(x\right).$$

(80)

If the moment formula of $C_j\left(x\right)$ given by (9) is inserted into (80), then it is possible to obtain

$${\mathcal{T}}_i\left(x\right)C_j\left(x\right)=i!\sum _{\mathcal{l}=0}^{⌊{\displaystyle \frac{i}{2}}⌋}{\displaystyle \frac{2^{-\mathcal{l}}}{(i-2\mathcal{l})!\mathcal{l}!}}\sum _{m=0}^{i-2\mathcal{l}}{2}^{-i+2\mathcal{l}}\left({\displaystyle \genfrac{}{}{0pt}{}{i-2\mathcal{l}}{m}}\right)C_{i+j-2\mathcal{l}-2m}\left(x\right),$$

(81)

which is equivalent to

$${\mathcal{T}}_i\left(x\right)C_j\left(x\right)=i!\sum _{p=0}^i\sum _{\mathcal{l}=0}^p{\displaystyle \frac{2^{-i+\mathcal{l}}\left(\genfrac{}{}{0pt}{}{i-2\mathcal{l}}{-\mathcal{l}+p}\right)}{(i-2\mathcal{l})!\mathcal{l}!}}{C}_{i+j-2p}\left(x\right),$$

(82)

which also gives

$${\mathcal{T}}_i\left(x\right)C_j\left(x\right)=\sum _{p=0}^i{(-1)}^{-i+p}{2}^{-p}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)U\left(-i+p,1-i+2p,-{\displaystyle \frac{1}{2}}\right)C_{i+j-2p}\left(x\right).$$

This proves Theorem 7. □

Example 2. 

As an application of the linearization formula (79), we give two specific linearization formulas of TelPs with Chebyshev polynomials of the second and third kinds.

$$\begin{array}{cc}\hfill {\mathcal{T}}_4\left(x\right)U_7\left(x\right)& ={\displaystyle \frac{1}{16}}\left(U_{11}\left(x\right)+28U_9\left(x\right)+102U_7\left(x\right)+28U_5\left(x\right)+U_3\left(x\right)\right),\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill {\mathcal{T}}_2\left(x\right)V_5\left(x\right)& ={\displaystyle \frac{1}{4}}\left(V_7\left(x\right)+6V_5\left(x\right)+V_3\left(x\right)\right).\hfill \end{array}$$

(84)

The following two formulas exhibit the LFs with the generalized Fibonacci and generalized Lucas polynomials.

Theorem 8. 

Consider any two positive integers i and j. The following LF holds:

$${\mathcal{T}}_i\left(x\right)F_j^{a,b}\left(x\right)=\sum _{p=0}^i{2}^{-i+p}{a}^{-i}{(-b)}^p{\left({\displaystyle \frac{b}{a^2}}\right)}^{-i+p}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)U\left(-i+p,1-i+2p,{\displaystyle \frac{2b}{a^2}}\right)F_{i+j-2p}^{a,b}\left(x\right).$$

(85)

Proof. 

Making use of the following moments formula [7]:

$$x^m{F}_j^{a,b}\left(x\right)=\sum _{r=0}^m\left({\displaystyle \genfrac{}{}{0pt}{}{m}{r}}\right)a^{-m}{(-b)}^r{F}_{j+m-2r}^{a,b}\left(x\right),$$

(86)

and following the same procedures as in the proof of Theorem 7, Formula (85) can be obtained. □

Theorem 9. 

Consider any two positive integers i and j. The following LF holds:

$${\mathcal{T}}_i\left(x\right)L_j^{c,d}\left(x\right)=\sum _{p=0}^i{2}^{-i+p}{c}^{-i}{(-d)}^p{\left({\displaystyle \frac{d}{c^2}}\right)}^{-i+p}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)U\left(-i+p,1-i+2p,{\displaystyle \frac{2d}{c^2}}\right)L_{i+j-2p}^{c,d}\left(x\right).$$

(87)

Proof. 

Similar to the proof of Theorem 7. □

Based on the two LFs in (85) and (87), some specific formulas can be deduced as special cases. The following corollaries present some of these formulas.

Corollary 16. 

For all non-negative integers i and j, the following four LFs hold:

$$\begin{array}{cc}\hfill {\mathcal{T}}_i\left(x\right)F_j\left(x\right)=& \sum _{p=0}^i{2}^{-i+p}{(-1)}^p\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)U\left(-i+p,1-i+2p,2\right)F_{i+j-2p}\left(x\right),\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill {\mathcal{T}}_i\left(x\right)L_j\left(x\right)=& \sum _{p=0}^i{2}^{-i+p}{(-1)}^p\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)U\left(-i+p,1-i+2p,2\right)L_{i+j-2p}\left(x\right),\hfill \end{array}$$

(89)

$$\begin{array}{cc}\hfill {\mathcal{T}}_i\left(x\right)P_j\left(x\right)=& \sum _{p=0}^i{2}^{-i+p}{2}^{-i}{(-1)}^p{\left({\displaystyle \frac{1}{4}}\right)}^{-i+p}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)U\left(-i+p,1-i+2p,{\displaystyle \frac{1}{2}}\right)P_{i+j-2p}\left(x\right),\hfill \end{array}$$

(90)

$$\begin{array}{cc}\hfill {\mathcal{T}}_i\left(x\right)Q_j\left(x\right)=& \sum _{p=0}^i{2}^{-i+p}{2}^{-i}{(-1)}^p{\left({\displaystyle \frac{1}{4}}\right)}^{-i+p}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)U\left(-i+p,1-i+2p,{\displaystyle \frac{1}{2}}\right)Q_{i+j-2p}\left(x\right).\hfill \end{array}$$

(91)

7. Evaluation of Some New Definite Integrals

This section is interested in obtaining closed forms for some new definite and weighted definite integrals based on some formulas developed in the previous sections.

Theorem 10. 

Consider the non-negative integers i and j. We have

$$\begin{array}{cc}\hfill {\int }_0^1{\mathcal{T}}_i\left(x\right)F_j^{a,b}\left(x\right)dx=& {\displaystyle a^{-i}\sum _{p=0}^i{2}^{-i+p}{(-b)}^p{\left(\frac{b}{a^2}\right)}^{-i+p}\left(\genfrac{}{}{0pt}{}{i}{p}\right)\times }\hfill \\ \hfill & U\left(-i+p,1-i+2p,{\displaystyle \frac{2b}{a^2}}\right)R_{i+j-2p},\hfill \end{array}$$

(92)

where $R_j$ is given by

$$R_j=\left\{\begin{array}{cc}{\displaystyle \frac{a^j{}_{2}F_1\left(\begin{array}{c}-\frac{1}{2}-\frac{j}{2},-\frac{j}{2}\\ -j\end{array}|-\frac{4b}{a^2}\right)}{j+1}},\hfill & \mathit{if}j\ge 0\mathit{and}\mathit{even},\hfill \\ -{\displaystyle \frac{2b^{\frac{1+j}{2}}}{a(j+1)}}+{\displaystyle \frac{a^j{}_{2}F_1\left(\begin{array}{c}-\frac{1}{2}-\frac{j}{2},-\frac{j}{2}\\ -j\end{array}|-\frac{4b}{a^2}\right)}{j+1}},\hfill & \mathit{if}j\ge 0\mathit{and}\mathit{odd},\hfill \\ {\displaystyle \frac{{(-1)}^{1+j}{a}^{-2-j}{b}^{1+j}{}_{2}F_1\left(\begin{array}{c}\frac{1+j}{2},\frac{2+j}{2}\\ 2+j\end{array}|-\frac{4b}{a^2}\right)}{1+j}},\hfill & \mathit{if}j<-1\mathit{and}\mathit{even},\hfill \\ {\displaystyle \frac{a^{-2-j}\left(2{(-1)}^j{a}^{1+j}{b}^{\frac{1+j}{2}}+{(-1)}^{1+j}{b}^{1+j}{}_{2}F_1\left(\begin{array}{c}\frac{1+j}{2},\frac{2+j}{2}\\ 2+j\end{array}|-\frac{4b}{a^2}\right)\right)}{1+j}},\hfill & \mathit{if}j<-1\mathit{and}\mathit{odd},\hfill \\ 0\hfill & \mathit{otherwise.}\hfill \end{array}\right.$$

(93)

Proof. 

Starting with the LF in (85), and then integrating over $[0,1]$ yields

$$\begin{array}{cc}\hfill {\int }_0^1{\mathcal{T}}_i\left(x\right)F_j^{a,b}\left(x\right)dx=& \sum _{p=0}^i{2}^{-i+p}{a}^{-i}{(-b)}^p{\left({\displaystyle \frac{b}{a^2}}\right)}^{-i+p}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)\times \hfill \\ \hfill & U\left(-i+p,1-i+2p,{\displaystyle \frac{2b}{a^2}}\right){\int }_0^1{F}_{i+j-2p}^{a,b}\left(x\right)dx.\hfill \end{array}$$

(94)

Based on the integral [47]

$${\int }_0^1{F}_j^{a,b}\left(x\right)=R_j,$$

(95)

where $R_j$ is given by (93), Formula (92) can be acquired. □

Theorem 11. 

For all non-negative integers i, and j, one has

$$\begin{array}{c}\hfill {\int }_0^1{\mathcal{T}}_i\left(x\right)L_j^{c,d}\left(x\right)dx=\sum _{p=0}^i{2}^{-i+p}{c}^{-i}{(-d)}^p{\left({\displaystyle \frac{d}{c^2}}\right)}^{-i+p}\left({\displaystyle \genfrac{}{}{0pt}{}{i}{p}}\right)\times \\ \hfill U\left(-i+p,1-i+2p,{\displaystyle \frac{2d}{c^2}}\right){\overline{G}}_{i+j-2p},\end{array}$$

(96)

where

$${\overline{G}}_j=\left\{\begin{array}{cc}{G}_j,\hfill & j\ge 0,\hfill \\ {(-d)}^j{G}_{-j},\hfill & j<0,\hfill \end{array}\right.$$

(97)

and

$$G_j=\left\{\begin{array}{cc}{\displaystyle \frac{c}{2}},\hfill & j=1,\hfill \\ {\displaystyle \frac{c^j{}_{2}F_1\left(\begin{array}{c}-\frac{1}{2}-\frac{j}{2},-\frac{j}{2}\\ 1-j\end{array}|-\frac{4d}{c^2}\right)}{{\xi }_j(1+j)}},\hfill & j\mathit{even},\hfill \\ {\displaystyle \frac{-4d^{\frac{1+j}{2}}j+c^{1+j}(j-1){}_{2}F_1\left(\begin{array}{c}-\frac{1}{2}-\frac{j}{2},-\frac{j}{2}\\ 1-j\end{array}|-\frac{4d}{c^2}\right)}{c(j^2-1)}},\hfill & j\mathit{odd}\mathit{and}j>2,\hfill \end{array}\right.$$

and the constants ${\xi }_j$ are given in (14).

Proof. 

The formula follows form the LF (87) along with the following integral formula for $L_j^{c,d}\left(x\right)$:

$${\int }_0^1{L}_j^{c,d}\left(x\right)dx={\overline{G}}_j,$$

(98)

and ${\overline{G}}_j$ is given in (97). □

Theorem 12. 

Let ${\psi }_j\left(x\right)$ be any NSPs that are orthogonal on $[a,b]$ regarding the weight function $w\left(x\right)$ with the following orthogonality relation:

$${\int }_a^{b}w\left(x\right){\psi }_j\left(x\right){\psi }_k\left(x\right)dx=\left\{\begin{array}{cc}0,\hfill & j\ne k,\hfill \\ h_k,\hfill & j=k,\hfill \end{array}\right.$$

(99)

and express the derivatives of TelPs using these polynomials, as

$$D^r{T}_s\left(x\right)=\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{A}_{p,s,r}{\psi }_{s-r-2p}\left(x\right)+\sum _{p=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{B}_{p,s,r}{\psi }_{s-r-2p-1}\left(x\right),$$

(100)

then the following integral formula is valid:

$${\int }_a^{b}w\left(x\right)D^r{\mathcal{T}}_s\left(x\right){\psi }_k\left(x\right)dx=\left\{\begin{array}{cc}{h}_k{A}_{{\displaystyle \frac{s-k-r}{2}},s,r}\hfill & \mathit{if}(s-k-r)\mathit{is}\mathit{even},\hfill \\ h_k{B}_{{\displaystyle \frac{s-k-r-1}{2}},s,r}\hfill & \mathit{if}(s-k-r)\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(101)

Proof. 

If we start with Formula (100), and multiply both sides by $w\left(x\right){\psi }_k\left(x\right)$, then we obtain

$$\begin{array}{cc}\hfill {\int }_a^{b}w\left(x\right)D^r{\mathcal{T}}_s\left(x\right){\psi }_k\left(x\right)dx=& \sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{A}_{p,s,r}{\int }_a^{b}w\left(x\right){\psi }_{s-r-2p}\left(x\right){\psi }_k\left(x\right)dx\hfill \\ \hfill & +\sum _{p=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{B}_{p,s,r}{\int }_a^{b}w\left(x\right){\psi }_{s-r-2p-1}\left(x\right){\psi }_k\left(x\right)dx.\hfill \end{array}$$

(102)

Using the orthogonality relation (99), we obtain

$$\begin{array}{cc}\hfill {\int }_a^{b}w\left(x\right)D^r{\mathcal{T}}_s\left(x\right){\psi }_k\left(x\right)dx& =h_k\left(\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{A}_{p,s,r}{\delta }_{s-r-2p,k}+\sum _{p=0}^{⌊{\displaystyle \frac{1}{2}}(s-r-1)⌋}{B}_{p,s,r}{\delta }_{s-r-2p-1,k}\right),\hfill \end{array}$$

(103)

which can be reduced to

$${\int }_a^{b}w\left(x\right)D^r{\mathcal{T}}_s\left(x\right){\psi }_k\left(x\right)dx=G_{s,k,r},$$

(104)

with

$$G_{s,k,r}=\left\{\begin{array}{cc}{h}_k{A}_{{\displaystyle \frac{s-k-r}{2}},s,r}\hfill & \mathrm{if}(s-k-r)\mathrm{is}\mathrm{even},\hfill \\ h_k{B}_{{\displaystyle \frac{s-k-r-1}{2}},s,r}\hfill & \mathrm{if}(s-k-r)\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

(105)

The proof is now complete. □

Remark 6. 

Based on the weighted integral formula in (101), several weighted integral formulas of the product of TelPs and some NSPs can be obtained. The following corollaries give some integral formulas.

Corollary 17. 

Let $L_k^{\left(\alpha \right)}\left(x\right)$ be the generalized Laguerre polynomials. The following integral formula holds:

$$\begin{array}{cc}\hfill & {\int }_0^{\infty }{e}^{-x}{x}^{\alpha }{D}^r{\varphi }_s\left(x\right)L_k^{\alpha }\left(x\right)dx=\left\{\begin{array}{cc}{M}_{s,k,r},\hfill & \mathit{if}(s-k-r)\mathit{is}\mathit{even},\hfill \\ -M_{s,k,r},\hfill & \mathit{if}(s-k-r)\mathit{is}\mathit{odd},\hfill \end{array}\right.\hfill \end{array}$$

(106)

where

$$\begin{array}{c}\hfill M_{s,k,r}={\displaystyle \frac{{(-1)}^{s-r}s!\mathrm{\Gamma }\left(\frac{5}{2}-r+s\right)}{k!(-k-r+s)!}}{}_{2}F_2\left(\begin{array}{c}{\displaystyle \frac{k}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{s}{2}},{\displaystyle \frac{1}{2}}+{\displaystyle \frac{k}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{s}{2}}\\ -{\displaystyle \frac{3}{4}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{s}{2}},-{\displaystyle \frac{1}{4}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{s}{2}}\end{array}|{\displaystyle \frac{1}{2}}\right).\end{array}$$

(107)

Proof. 

From Formula (76), $D^r{L}_s^{\left(\alpha \right)}\left(x\right)$ can be written in the form (100), with the following coefficients:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill A_{p,s,r}={\displaystyle \frac{{(-1)}^{s-r}s!\mathrm{\Gamma }(1+s-r+\alpha )}{\left(2p\right)!\mathrm{\Gamma }(1+s-2p-r+\alpha )}}{}_{2}F_2\left(\begin{array}{c}-p,{\displaystyle \frac{1}{2}}-p\\ -{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{\alpha }{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{\alpha }{2}}\end{array}|{\displaystyle \frac{1}{2}}\right),\end{array}\end{array}$$

(108)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill B_{p,s,r}={\displaystyle \frac{{(-1)}^{s-r+1}s!\mathrm{\Gamma }(1+s-r+\alpha )}{(2p+1)!\mathrm{\Gamma }(s-2p-r+\alpha )}}{}_{2}F_2\left(\begin{array}{c}-p,{\displaystyle \frac{1}{2}}-p\\ -{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{\alpha }{2}},{\displaystyle \frac{1}{2}}-{\displaystyle \frac{s}{2}}+{\displaystyle \frac{r}{2}}-{\displaystyle \frac{\alpha }{2}}\end{array}|{\displaystyle \frac{1}{2}}\right).\end{array}\end{array}$$

(109)

Noting that the orthogonality factor $h_k$ that appears in (105) is

$${\displaystyle h_k=\frac{\mathrm{\Gamma }(k+\alpha +1)}{k!},}$$

and making use of Formula (101), some computations lead to Formula (106). □

Theorem 13. 

Let ${\varphi }_j\left(x\right)$ be symmetric orthogonal polynomials on $[a,b]$ with the following orthogonality relation:

$${\int }_a^b\tilde{w}\left(x\right){\varphi }_j\left(x\right){\varphi }_k\left(x\right)dx=\left\{\begin{array}{cc}0,\hfill & j\ne k,\hfill \\ {\overline{h}}_k,\hfill & j=k,\hfill \end{array}\right.$$

(110)

and express the derivatives of TelPs using these polynomials as

$$D^r{T}_s\left(x\right)=\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{R}_{p,s,r}{\varphi }_{s-r-2p}\left(x\right),$$

(111)

then the following integral formula holds:

$${\int }_a^b\tilde{w}\left(x\right)D^r{\mathcal{T}}_s\left(x\right){\varphi }_k\left(x\right)dx=\left\{\begin{array}{cc}{\overline{h}}_k{R}_{{\displaystyle \frac{s-k-r}{2}},s,r}\hfill & \mathit{if}(s-k-r)\mathit{is}\mathit{even},\hfill \\ 0\hfill & \mathit{if}(s-k-r)\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(112)

Proof. 

Starting with Formula (111), multiply both sides by $\tilde{w}\left(x\right){\varphi }_k\left(x\right)$, and integrate over $[a,b]$ yields

$$\begin{array}{cc}\hfill {\int }_a^b\tilde{w}\left(x\right)D^r{\mathcal{T}}_s\left(x\right){\varphi }_k\left(x\right)dx=& \sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{R}_{p,s,r}{\int }_a^{b}w\left(x\right){\varphi }_{s-r-2p}\left(x\right){\varphi }_k\left(x\right)dx.\hfill \end{array}$$

(113)

Using the orthogonality relation (110), we obtain

$$\begin{array}{cc}\hfill {\int }_a^b\tilde{w}\left(x\right)D^r{\mathcal{T}}_s\left(x\right){\varphi }_k\left(x\right)dx=& {\overline{h}}_k\sum _{p=0}^{⌊{\displaystyle \frac{s-r}{2}}⌋}{A}_{p,s,r}{\delta }_{s-r-2p,k},\hfill \end{array}$$

(114)

which can be reduced into

$${\int }_a^b\tilde{w}\left(x\right)D^r{\mathcal{T}}_s\left(x\right){\varphi }_k\left(x\right)dx=G_{s,k,r},$$

(115)

with

$$G_{s,k,r}=\left\{\begin{array}{cc}{\overline{h}}_k{R}_{{\displaystyle \frac{s-k-r}{2}},s,r}\hfill & \mathrm{if}(s-k-r)\mathrm{is}\mathrm{even},\hfill \\ 0\hfill & \mathrm{if}(s-k-r)\mathrm{is}\mathrm{odd}.\hfill \end{array}\right.$$

(116)

The proof is now complete. □

Corollary 18. 

Consider the three positive integers $r, s, andk$ with $s\ge r$. The following integral formula holds:

$$\underset{-\infty }{\overset{\infty }{\int }}{e}^{-x^2}{D}^r{\mathcal{T}}_s\left(x\right)H_k\left(x\right)dx=\left\{\begin{array}{cc}{\displaystyle \frac{2^{-s+k+r}{3}^{\frac{1}{2}(s-k-r)}\sqrt{\pi }s!}{\left(\frac{1}{2}(s-k-r)\right)!},}\hfill & \mathit{if}(s-k-r)\mathit{is}\mathit{even},\hfill \\ 0,\hfill & \mathit{if}(s-k-r)\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(117)

In particular, we have

$$\underset{\infty }{\overset{\infty }{\int }}{e}^{-x^2}{\mathcal{T}}_s\left(x\right)H_k\left(x\right)dx=\left\{\begin{array}{cc}{\displaystyle \frac{2^{-s+k}·3^{\frac{1}{2}(s-k)}\sqrt{\pi }s!}{\left(\frac{1}{2}(s-k)\right)!},}\hfill & \mathit{if}(s-k)\mathit{is}\mathit{even},\hfill \\ 0,\hfill & \mathit{if}(s-k)\mathit{is}\mathit{odd}.\hfill \end{array}\right.$$

(118)

Proof. 

Formula (117) can be derived using Theorem 13, considering that the orthogonality factor ${\overline{h}}_k$ is

$${\overline{h}}_k=\sqrt{\pi }k!2^k.$$

Formula (118) is a special case of Formula (117) for $r=0$. □

Example 3. 

We present two specific weighted integral formulas of Formulas (117) and (118).

$$\begin{array}{cc}\hfill \underset{-\infty }{\overset{\infty }{\int }}{e}^{-x^2}{D}^4{\mathcal{T}}_{12}\left(x\right)H_6\left(x\right)dx& =359251200\sqrt{\pi },\hfill \end{array}$$

(119)

$$\begin{array}{cc}\hfill \underset{-\infty }{\overset{\infty }{\int }}{e}^{-x^2}{\mathcal{T}}_{15}\left(x\right)H_7\left(x\right)dx& =17239847625\sqrt{\pi }.\hfill \end{array}$$

(120)

Corollary 19. 

If $U_k^{\left(\lambda \right)}\left(x\right)$ are the ultraspherical polynomials, then the following integral formula holds:

$$\begin{array}{cc}\hfill & \underset{-1}{\overset{1}{\int }}{\left(1-x^2\right)}^{\lambda -{\displaystyle \frac{1}{2}}}{D}^r{\mathcal{T}}_s\left(x\right)U_k^{\left(\lambda \right)}\left(x\right)dx\hfill \\ \hfill & =\left\{\begin{array}{cc}{\displaystyle \frac{2^{\frac{1}{2}(-k+r-s)}\sqrt{\pi }s!\mathrm{\Gamma }\left(\frac{1}{2}+\lambda \right)}{\left(\frac{s-k-r}{2}\right)!}{}_{1}F_1\left(\frac{k+r-s}{2},1+k+\lambda ,-\frac{1}{2}\right),}\hfill & \mathit{if}(s-r-k)\mathit{is}\mathit{even},\hfill \\ 0,\hfill & \mathit{if}(s-r-k)\mathit{is}\mathit{odd}.\hfill \end{array}\right.\hfill \end{array}$$

(121)

Proof. 

A special case of Theorem 13. □

8. Introducing Some Indefinite Integrals Involving TelPs

Obtaining closed forms for the indefinite integrals involving special functions has drawn the attention of many authors. For example, Conway has contributed a series of papers [48,49,50,51], in which he derived new formulas for the indefinite integrals of various special functions. Some useful formulas of integrals can be found in [52,53,54].

This section introduces closed-form expressions for indefinite integrals involving special functions, including some TelPs. The formulas developed by Conway in [55] will be the suitable approach for deriving these integrals. First, we begin with this theorem

Theorem 14 

([55]). If we consider the second-order differential equation:

$$y^{″}\left(x\right)+P\left(x\right)y^{\prime }\left(x\right)+Q\left(x\right)y\left(x\right)=0,$$

and if we consider the two integrating factors

$$f\left(x\right)=e^{\int P\left(x\right),dx},\widehat{f}\left(x\right)=e^{\int {\displaystyle \frac{Q\left(x\right)}{P\left(x\right)}},dx},$$

then the two following indefinite integrals hold:

$$\begin{array}{cc}\hfill \int Q\left(x\right)f{\left(x\right)}^m{\left(y^{\prime }\left(x\right)\right)}^{m-1}y\left(x\right)dx& =-{\displaystyle \frac{f{\left(x\right)}^m{\left(y^{\prime }\left(x\right)\right)}^m}{m}},\hfill \end{array}$$

(122)

$$\begin{array}{cc}\hfill \int {\displaystyle \frac{1}{P\left(x\right)}}\widehat{f}{\left(x\right)}^m{\left(y\left(x\right)\right)}^{m-1}{y}^{″}\left(x\right)dx& =-{\displaystyle \frac{{\left(\widehat{f}\left(x\right)\right)}^m{\left(y\left(x\right)\right)}^m}{m}}.\hfill \end{array}$$

(123)

As a consequence of the previous theorem, some indefinite integrals involving TelPs can be derived. The following corollary presents these results.

Corollary 20. 

The two indefinite integral formulas below are valid:

$$\begin{array}{c}\hfill \int e^{{\displaystyle \frac{m{x}^2}{2}}}{\mathcal{T}}_{j-1}{\left(x\right)}^{m-1}{\mathcal{T}}_j\left(x\right)dx={\displaystyle \frac{e^{\frac{m{x}^2}{2}}{\mathcal{T}}_{j-1}{\left(x\right)}^m}{m}},\end{array}$$

(124)

$$\begin{array}{c}\hfill \int x^{-jm-1}{\mathcal{T}}_j{\left(x\right)}^{m-1}{\mathcal{T}}_{j-2}\left(x\right)dx={\displaystyle \frac{-x^{-mj}{\mathcal{T}}_j{\left(x\right)}^m}{j(j-1)m}}.\end{array}$$

(125)

Proof. 

To show (124), we first express the first-order derivative of TelPs in terms of their original polynomials. Based on Formula (29) for the case $r=1$, we have the following expression:

$${\mathcal{T}}_j^{\prime }\left(x\right)=j{\mathcal{T}}_{j-1}\left(x\right),$$

(126)

and after that make use of the differential equation of TelPs that was stated in Lemma 1 in the following form:

$${\mathcal{T}}_j^{″}\left(x\right)+x{\mathcal{T}}_j^{\prime }\left(x\right)-j{\mathcal{T}}_j\left(x\right)=0.$$

Now, we can apply the closed integral Formula (122) with the following choices:

$$y\left(x\right)={\mathcal{T}}_j\left(x\right),p\left(x\right)=x,q\left(x\right)=-j,f\left(x\right)=e^{{\displaystyle \frac{x^2}{2}}},$$

to obtain the following formula:

$$\int -j{e}^{{\displaystyle \frac{m{x}^2}{2}}}{\left(j{\mathcal{T}}_{j-1}\left(x\right)\right)}^{m-1}{\mathcal{T}}_j\left(x\right)dx=-{\displaystyle \frac{j^m{e}^{\frac{m{x}^2}{2}}{\mathcal{T}}_{j-1}{\left(x\right)}^m}{m}},$$

(127)

which results in

$$\int e^{{\displaystyle \frac{m{x}^2}{2}}}{\mathcal{T}}_{j-1}{\left(x\right)}^{m-1}{\mathcal{T}}_j\left(x\right)dx={\displaystyle \frac{e^{\frac{m{x}^2}{2}}{\mathcal{T}}_{j-1}{\left(x\right)}^m}{m}}.$$

(128)

To show (125), we apply Formula (123) with the following choices:

$$y\left(x\right)={\mathcal{T}}_j\left(x\right),p\left(x\right)=x,q\left(x\right)=-j,\widehat{f}\left(x\right)=x^{-j},$$

and make use of the following identity:

$${\displaystyle \frac{d^2}{d{x}^2}{\mathcal{T}}_j\left(x\right)=j(j-1){\mathcal{T}}_{j-2}\left(x\right),}$$

(129)

to obtain the following identity:

$$\int x^{-mj-1}{\mathcal{T}}_j{\left(x\right)}^{m-1}j(j-1){\mathcal{T}}_{j-2}\left(x\right)dx=-{\displaystyle \frac{x^{-mj}{\mathcal{T}}_j{\left(x\right)}^m}{m}},$$

(130)

which leads to

$$\int x^{-jm-1}{\mathcal{T}}_j{\left(x\right)}^{m-1}{\mathcal{T}}_{j-2}\left(x\right)dx={\displaystyle \frac{-x^{-mj}{\mathcal{T}}_j{\left(x\right)}^m}{j(j-1)m}}.$$

□

Corollary 21. 

The following two indefinite integral formulas hold:

$$\begin{array}{cc}\hfill & \int e^{{\displaystyle \frac{-x^2}{2}}}{\mathcal{T}}_{j+1}{\left(x\right)}^{m-1}{\mathcal{T}}_j\left(x\right)dx={\displaystyle \frac{{\mathcal{T}}_{j+1}{\left(x\right)}^m}{(j+1)m}},\hfill \end{array}$$

(131)

$$\begin{array}{cc}\hfill & \int e^{{\displaystyle \frac{x^2}{2}}}{x}^{m+jm-1}{\mathcal{T}}_{j+2}\left(x\right){\mathcal{T}}_j{\left(x\right)}^{m-1}dx={\displaystyle \frac{x^{jm+m}{e}^{\frac{m{x}^2}{2}}{\mathcal{T}}_j{\left(x\right)}^m}{m}}.\hfill \end{array}$$

(132)

Proof. 

Define the following function:

$${\mathrm{\Theta }}_j\left(x\right)=e^{{\displaystyle \frac{x^2}{2}}}{\mathcal{T}}_j\left(x\right).$$

(133)

Differentiating with respect to x gives

$${\mathrm{\Theta }}_j^{\prime }\left(x\right)=e^{{\displaystyle \frac{x^2}{2}}}x{\mathcal{T}}_j\left(x\right)+e^{{\displaystyle \frac{x^2}{2}}}{\mathcal{T}}_j^{\prime }\left(x\right)=e^{{\displaystyle \frac{x^2}{2}}}\left(x{\mathcal{T}}_j\left(x\right)+{\mathcal{T}}_j^{\prime }\left(x\right)\right).$$

Based on Formula (126), we obtain

$${\mathrm{\Theta }}_j^{\prime }\left(x\right)=e^{{\displaystyle \frac{x^2}{2}}}\left(x{\mathcal{T}}_j\left(x\right)+j{\mathcal{T}}_{j-1}\left(x\right)\right).$$

The application of the recursive formula of TelPs in (17) leads to

$${\mathrm{\Theta }}_j^{\prime }\left(x\right)=e^{{\displaystyle \frac{x^2}{2}}}{\mathcal{T}}_{j+1}\left(x\right).$$

(134)

Similarly, it can be shown that

$${\mathrm{\Theta }}_j^{″}\left(x\right)=e^{{\displaystyle \frac{x^2}{2}}}{\mathcal{T}}_{j+2}\left(x\right).$$

(135)

The two derivatives in (134) and (135) lead to the following second-order differential equation for ${\mathrm{\Theta }}_j\left(x\right)$:

$${\mathrm{\Theta }}_j^{″}\left(x\right)-x{\mathrm{\Theta }}_j^{\prime }\left(x\right)-(j+1){\mathrm{\Theta }}_j=0.$$

Now, the application of (122) and (123) leads to Formulas (131) and (132). □

9. Summary of the New Formulas with Expected Applications

This section introduces a summary of the new formulas established in this paper. In addition, some expected applications of the derived formulas are presented in this section.

9.1. Summary of the New Formulas

For convenience, we give a summary of the new formulas in this paper in a table. Table 3 summarizes these formulas with a brief description.

Table 3. Summary of the main results established in this paper.

No.	Description of the Formula	Reference
1	Second-order differential equation met by TelPs	Lemma 1
2	Derivatives of the moments of TelPs	Theorem 1
3	Moment formula of TelPs	Corollary 1
4	Explicit derivative formula of TelPs	Corollary 2
5	Repeated integrals of the moments formula of TelPs	Theorem 2
6	Repeated integrals of TelPs in terms of their original polynomials	Corollary 3
7	Derivatives of TelPs expressed in terms of symmetric polynomials	Theorem 3
8	Derivatives of TelPs in terms of specific symmetric polynomials	Corollaries 4–6
9	Derivatives of TelPs expressed as combinations of non-symmetric polynomials	Theorem 4
10	Derivatives of TelPs in terms of specific non-symmetric polynomials	Corollaries 7–9
11	Derivatives of symmetric polynomials in terms of TelPs	Theorem 5
12	Derivatives of specific symmetric polynomial families in terms of TelPs	Corollaries 10–12
13	General derivative formula of non-symmetric polynomials in terms of TelPs	Theorem 6
14	derivative formulas for specific non-symmetric polynomials in terms of TelPs	Corollaries 13–15
15	Linearization formula for the product of TelPs and Chebyshev polynomials	Theorem 7
16	Linearization formula for the product of TelPs and generalized Fibonacci and Lucas polynomials	Theorems 8 and 9
17	Some special linearization formulas	Corollary 16
18	New definite integral formulas involving products of the derivatives of TelPs and generalized Fibonacci and Lucas polynomials	Theorem 10 and 11
19	Weighted integral formulas for the product of TelPs with symmetric and non-symmetric polynomials	Theorem 12 and 13
20	Some specific weighted definite integral formulas	Corollary 17–19
21	Two new indefinite integral formulas involving products of TelPs	Corollary 20
22	Additional two indefinite integral formulas involving TelPs	Corollary 21

9.2. Some Expected Applications

In this part, we present numerical applications that can be developed using the theoretical background presented in this paper, employing TelPs. Some of the formulas presented in this paper may be useful for solving differential equations of various types using spectral methods. The following points are essential:

• The telephone polynomials can be taken as basis functions. The required formulas, such as the derivatives of the moments and the explicit derivative formulas, can be used.
• The simple formula for derivatives enables one to adopt a matrix approach using the operational matrix of derivatives, mainly when the collocation method is employed.
• The explicit formulas for repeated integrals help find the corresponding integral equations in certain types of differential equations. In addition, the integral formulas may help in the treatment of certain types of integral differential equations.
• The definite integral formulas derived in this paper may be helpful in the Galerkin and Petrov–Galerkin methods.

10. Concluding Remarks

In this article, we have developed several new formulas of TelPs. Explicit derivative and integral representations were established. The general derivative expressions for TelPs with different polynomials were given in closed forms; hence, specific derivative formulas of different polynomials as combinations of TelPs were deduced. LFs were derived with several polynomials, such as Chebyshev, generalized Fibonacci, and generalized Lucas polynomials. Some definite and indefinite integral formulas were established. Most of the formulas in this paper are novel and could be useful for special functions and applications. In future work, we aim to investigate some generalizations of the TelPs that were introduced in this paper and give their connections to various polynomials. In addition, from a practical point of view, another future work may focus on employing these polynomials and their generalized polynomials in the numerical and computational aspects. They can be used to solve differential and integral equations.