Studies in Sums of Finite Products of the Second, Third, and Fourth Kind Chebyshev Polynomials

: In this paper, we consider three sums of ﬁnite products of Chebyshev polynomials of two different kinds, namely sums of ﬁnite products of the second and third kind Chebyshev polynomials, those of the second and fourth kind Chebyshev polynomials, and those of the third and fourth kind Chebyshev polynomials. As a generalization of the classical linearization problem, we represent each of such sums of ﬁnite products as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials. These are done by explicit computations and the coefﬁcients involve terminating hypergeometric functions 2 F 1 , 1 F 1 , 2 F 2 , and 4 F 3 .


Introduction and Preliminaries
The classical linearization problem consists of determining the coefficients c n,m (k) in the expansion of the product of two polynomials q n (x) and r m (x) in terms of arbitrary polynomial sequence {p k (x)} k≥0 . See [1]. q n (x)r m (x) = n+m ∑ k=0 c nm (k)p k (x). (1) There are several special cases of this: If q n (x) = r n (x) = p n (x), then it is called either the standard linearization or a Clebsch-Gordan-type problem: If r m (x) = 1, then it is known as the connection problem: Let n, r, s be nonnegative integers with r + s ≥ 1. Here, as one motivation for the present research, we would like to generalize the linearization problem in (1) and consider the following three sums of finite products of Chebyshev polynomials of two different kinds: α n,r,s (x) = ∑ i 1 +···+i r +j 1 +···+j s =n U i 1 (x) · · · U i r (x)V j 1 (x) · · · V j s (x), (4) β n,r,s (x) = ∑ i 1 +···+i r +j 1 +···+j s =n U i 1 (x) · · · U i r (x)W j 1 (x) · · · W j s (x), (5) γ n,r,s (x) = ∑ i 1 +···+i r +j 1 +···+j s =n where U n (x), V n (x), and W n (x) are respectively Chebyshev polynomials of the second, third, and fourth kinds, and the sums are over all nonnegative integers, i 1 , . . . , i r , j 1 , . . . , j s with i 1 + · · · + i r + j 1 + · · · + j s = n. Then, we will represent each of the sums of finite products in Equations (4)- (6) as linear combinations of Hermite polynomials H n (x), generalized Laguerre polynomials L α n (x), Legendre polynomials P n (x), Gegenbauer polynomials C (λ) n (x), and Jacobi polynomials P (α,β) n (x) (see .
As another motivation for the present study, we would like to mention a convolution identity of Bernoulli polynomials that yields the famous Faber-Pandharipande-Zagier identity and Miki's identity. For this, let us first recall that the Bernoulli polynomials are given by Then, let us put In the Introduction of [2], we noted that the following identity can be derived from the Fourier expansion of E m ( x ). Here, x = x − [x] is the fractional part of x, for any real number x: where H m = m ∑ j=1 1 j are the harmonic numbers.
Furthermore, (7) can be modified to give Let x = 1 2 and x = 0 in (8) give respectively Faber-Pandharipande-Zagier identity (see [3]) and a slight variant of Miki's identity (see [4][5][6][7]). It is worth noting that our methods are very simple at the level of Fourier series expansions, whereas the other approaches in [4][5][6][7] use different methods from one another and are quite involved.
Before we state our main results in Theorems 1-3, we will fix notations that will be used throughout this paper and recall some basic facts about orthogonal polynomials that will be needed.
Let n be a nonnegative integer. Then, the falling factorial polynomials (x) n and the rising factorial polynomials x n are respectively given by x 0 = 1, The two factorial polynomials are related by Γ(x + 1) where Γ(x) is the gamma function. The hypergeometric function is defined by Next, we will recall some very basic facts about Chebyshev polynomials of the second kind U n (x), the third kind V n (x), and the fourth kind W n (x) (see [8,9]). In addition, we will state those facts about Hermite polynomials H n (x), extended Laguerre polynomials L α n (x), Legendre polynomials P n (x), Gegenbauer polynomials C (λ) n (x), and Jacobi polynomials P (α,β) n (x) (see [10][11][12][13][14]). We let the reader refer to the standard books [15][16][17] for further details on these family of orthogonal polynomials.
In terms of generating functions, the above mentioned orthogonal polynomials are given as follows: In terms of explicit expressions, they are given as follows: W n (x) = (2n + 1) 2 F 1 − n, n + 1;

Statements of Results
The following three theorems are the main results of this paper, all of which are new. Here, we note that we treat sums of finite products of Chebyshev polynomials of two different kinds, whereas all the results so far, except for [18], treated sums of finite products of some polynomials of single kind. Theorem 1. Let n, r, s be nonnegative integers with r + s ≥ 1. Then, we have the following identities: Here, (2n − 1)!! = (2n − 1)(2n − 3) · · · 1, for n ≥ 1, and (−1)!! = 1.

Theorem 2.
Let n, r, s be nonnegative integers with r + s ≥ 1. Then, we have the following representations: Theorem 3. Let n, r, s be nonnegative integers with r + s ≥ 1. Then, we have the following expressions: Before we move on to the next section, we would like to recall some of the related previous works. In [19], sums of finite products of Chebyshev polynomials of the first, third, and and fourth kinds were In addition, in [20], sums of finite products of Chebyshev polynomials of the second kind were expressed in terms of the same orthogonal polynomials. Here, we emphasize that, except for the paper [18], which considered the sums of finite products in (4)-(6) and represented them in terms of all kinds of Chebyshev polynomials, all of the results so far have treated sums of finite products of some polynomials of single type. For further details on these, we let the reader refer to the references in [19,20].

Proposition 2.
Let m, k be nonnegative integers. Then, we have the following: Lemmas 1 and 2 in the following can be shown by using (17)- (19) and were derived in [18]. However, for the sake of completeness and in view of its importance, we repeat the proof for Lemma 1. Lemma 2 can be proved analogously to Lemma 1.
From (25), we see that the rth derivative of U n (x) is given by from which we have Now, we are going to show Theorem 1. With α n,r,s (x) as in (4), we put Then, from (a) of Proposition 1, (48), and (52), and integration by parts k times, we have Here, we note from (a) of Proposition 2 that This shows (31) of Theorem 1. Next, we let Then, from (b) of Proposition 1, (48), and (52), integration by parts k times and proceeding just as in (54), we obtain From (57) and (58) and after some simplifications, we have Here, the innermost sum of (59) is equal to Combining (59) and (60), we get This completes the proof for (32) of Theorem 1. Let us put α n,r,s (x) = n ∑ k=0 C k,3 P k (x).
From (c) of Proposition 1, (48), and (52), integration by parts k times, (b) of Proposition 2 and after some simplifications, we obtain Combining (62) and (63), we have This shows (35) of Theorem 1 Let us set From (d) of Proposition 1, (48) and (52), integration by parts k times, (c) of Proposition 2 and after some simplifications, we have Combining (64) and (65), we obtain This shows (36) of Theorem 1. Let us let From (e) of Proposition 1, (48) and (52), integration by parts k times, (d) of Proposition 2 and after some simplifications, we get It can be seen that the innermost sum of (67) is equal to By making use of (66)-(68), we finally obtain This finishes up the proof for (35) of Theorem 1.

Proof of Theorem 3
Here, we will show only (45) and (47) in Theorem 3, while leaving (43), (44) and (46) as exercises to the reader. Lemma 3. Let n, r, s be nonnegative integers with r + s ≥ 1. The following identity holds: Proof. The identity in (69) is stated in [18] and can be deduced from (16) and (17). On the other hand, the identity in (70) follows from the elementary observation With γ n,r,s (x) as in (6), we put γ n,r,s (x) = n ∑ k=0 C k,3 P k (x).
orthogonal polynomials in terms of other orthogonal polynomials can be generalized to the cases of q-orthogonal polynomials.