Sums of finite products of Pell polynomials in terms of hypergeometric functions

In this work, we study sums of finite products of Pell polynomials and express them in terms of some special orthogonal polynomials. Furthermore, each of the obtained expression is represented as linear combinations of classical polynomials involving hypergeometric functions by means of explicit computations.


Introduction
In the recent years, the problem of expressing sums of products of certain special polynomials in terms of other special polynomials has drawn the attention of many researchers and mathematicians due to which this field has seen an increasing interest [1][2][3]. There are many special polynomials out of which we deal with the Pell polynomials [4][5][6]. The Pell polynomials P k (t) are defined by the binary recurrence relation In this work, the following summations of finite products of Pell polynomials have been considered, given by where the summation runs over all nonnegative integers j 1 , j 2 , . . . , j r+1 , with j 1 + j 2 + · · · + j r+1 = n. The summation (2) is represented in terms of some orthogonal polynomials such as the Legendre polynomials ( P n (x) ), Jacobi polynomials ( P α,β n (x) ), Hermite polynomials (H n (x)) , Gegenbauer polynomials (C ( ) n (x)) , extended Laguerre polynomials (L α n (x)) , and Chebyshev polynomials [7] of first kind (T n (x)) , second kind (U n (x)) , third kind (V n (x)) and fourth kind (W n (x)) which are further represented as hypergeometric functions. The hypergeometric function, denoted by 2 F 1 (a, b; c; z), is basically a special function represented by a hypergeometric series, which involves many special functions as specific cases which has been discussed in the preliminaries section.
(2) j 1 +j 2 +···+j r+1 =n Page 2 of 20 Patra and Panda Journal of the Egyptian Mathematical Society (2022) 30:4 which involves the determination of the unknown coefficients d ab (k) in the expansion of product of the polynomials s a (t) and p b (t) in terms of arbitrary polynomial {q k (t)} k≥0 . If the two polynomials s a (t) and p a (t) are equal to q a (t) , then problem (3) is known as the standard linearization problem or Clebsch-Gordan-type. In other words, for s a (t) = p a (t) = q a (t), (3) converts to which is called the Clebsch-Gordan-type problem. Furthermore, if we take p b (t) = 1 , then (3) is called the connection problem expressed as In addition, if s a (t) = t n in (3), then it is known as the inversion problem.
In particular, the present work is motivated by the linearization problem and may be viewed as a generalization of the classical linearization problem as in (3). Apart from that, our work is also motivated by the convolution identity of Bernoulli polynomials B n (x) that yields the famous Miki's identity and Faber-Pandharipande-Zagier identity. In other words, it is possible to represent the sums of products of two Bernoulli polynomials as linear combinations of Bernoulli polynomials. The polynomials B n (x) are given by Now, if for k ≥ 2, then, from [8], can be derived from the Fourier expansion of δ k ( x ) , where x is the fractional part of any real number x expressed as �x� = x − [x] and H k are the harmonic numbers denoted by H k = k n=1 1 n . Furthermore, it is interesting to observe that for x = 1 2 , (8) gives the Faber-Pandharipande-Zagier identity [9] and for x = 0 , (8) gives a slight variant of Miki's identity [10][11][12]. In this article, the Pell polynomials are represented in terms of t n n! = te tx e t − 1 . linear combination of some of the orthogonal polynomials. The generating function of the Pell polynomials is given by First few terms of the Pell polynomials derived from (1) can be written as P 2 (t) = 2t , P 3 (t) = 4t 2 + 1 , P 4 (t) = 8t 3 + 4t , P 5 (t) = 16t 4 + 12t 2 + 1 , P 6 (t) = 32t 5 + 32t 3 + 6t, . . . . The first few terms of P n (t) are graphically depicted in Fig. 1.
Furthermore, the Pell polynomials are the natural extension of the Pell numbers P n defined by the same recurrence relation [6,13] The methodology used in this work is beneficial over other techniques in the literature due to the simple Fourier series expansions used here to determine the unknown coefficients involved in the classical linearization type problem unlike the other methods in [10,11] which are quite complex in their approach. The literature survey includes the work of Zhang [7], where he derived a new identity for the Chebyshev polynomials. In [14], the authors have shown an application of a p-adic convolution using a suitable polynomial. In a different work, Kim et al. [2] have studied regarding the sums of finite products of Chebyshev polynomials and Fibonacci polynomials. Apart from that, the sums of finite products of the Genocchi functions have also been studied by Kim et al. [8]. In two different works, Kim et al. tackled with the sums of finite products of Chebyshev and Lucas-balancing polynomials [15,16]. Furthermore, certain identities relating to the symmetry for the Euler polynomials are derived in [17]. In [18], a difference of sums of finite products have been tackled in case of Lucas-balancing polynomials. Moreover, the Appell polynomials were utilized to represent a family of associated sequences [19]. Apart from that, a new class of Bernoulli polynomials have been introduced by [20], related to polyexponential functions. The present article is organized in the following manner: First, in "Preliminaries" section, the preliminaries regarding the properties of several polynomials, gamma and beta functions have been discussed to be used later in the subsequent sections. Then, the explicit formulas of some special orthogonal polynomials are given in "Methods" section. Apart from that, "Results and discussions" section includes some propositions and lemmas to be used later. In addition, some theorem have been proved regarding the sums of finite products of Pell polynomials in subsequent section. The final section is devoted for the concluding remarks.

Preliminaries
Definition 1 (Rising factorial polynomials and falling factorial polynomials) The rising factorial polynomials t n 1 , for n 1 ≥ 1 are defined by [21] and the falling factorial polynomials (t) n 1 , for n 1 ≥ 1 are defined by [21] Furthermore, the rising factorial polynomials t n 1 and the falling factorial polynomials (t) n 1 satisfy the following properties given by the following lemma.

Proof
The proof of the lemma can be referred from [21].

Definition 2 (Beta function) Now the beta function B(x, y) is defined in terms of gamma function Ŵ(x) as
for Re(x), Re(y) > 0. and furthermore, a special case of Gauss hypergeometric function is the Chu-Vandermonde formula given by Note: Furthermore, there is a link between the Pell polynomials P n (x) and Chebyshev polynomials of second kind denoted by U n (x) . Before, proceeding to establish the connection, we need to define the Chebyshev polynomials of second kind. Hence, the first few terms of U n (x) are given by ... The first few terms of U n (x) are graphically depicted in Fig. 2.
In addition, the Chebyshev polynomials are explicitly given by the formula, for n ≥ 0, where 2 F 1 (a, b; c; x) can be referred from (13) and is given by the generating function

Lemma 2
The fundamental connection between the Chebyshev polynomials U n (x) of second kind and the Pell polynomials P n (x) is given by

Note:
The explicit expression for P n+1 (x) can be either referred from [5] or can also be viewed by virtue of the combination of (16) and (15) after proving Lemma 2.

Methods
be a polynomial of degree n and further let q(x) = n l=0 D l P l+1 (x) . Then,

Proof
By virtue of the orthogonality property of U n (x) , we have for m, n ≥ 0 . Combining (19) and the property (16), we have which represents the orthogonality relation for the Pell polynomials P n (x). Furthermore, the Rodrigues' formula of the Chebyshev polynomials of second kind U n (x) is given, for n ≥ 0 , by (15) U n (x) = (n + 1) 2 F 1 (−n, 2 + n; Again by combining (16) and (21), we get the Rodrigues' formula of the Pell polynomials as Now, by means of (20) and (22), we get the desired result.

Explicit formulas of special orthogonal polynomials
In this section, some explicit definitions of certain polynomials have been recalled which will be used in the subsequent sections. The explicit formula for the Chebyshev polynomials of the first kind (T n (x)) , those of the third kind (V n (x)) , those of the fourth kind (W n (x)) , Hermite polynomials (H n (x)) , generalized Laguerre polynomials (L α n (x)) , Legendre polynomials (P n (x)) , Gegenbauer polynomials (C ( ) n (x)) , and Jacobi polynomials (P (α,β) n (x)) . They are explicitly given by W n (x) =(2n + 1) 2 F 1 −n, n + 1; T n (x) = n 2 2 F 1 −n, n; Proposition 2 [1,2]. Let q(x) ∈ R[x] be a polynomial of degree n. Then, we have the following:

Proof
The proof of the theorem is by virtue of Theorem 1 of [1], Theorem 1 of [2] and (16).

Remark
The r-th derivative of the Pell polynomial P n+1 (x) is given by Furthermore, By default, we assume that throughout the remaining part of the paper.