Orthogonality of the Dickson polynomials of the (k+1)-th kind

We study the Dickson polynomials of the (k+1)-th kind over the field of complex numbers. We show that they are a family of co-recursive orthogonal polynomials with respect to a quasi-definite moment functional L_{k}. We find an integral representation for L_{k} and compute explicit expressions for all of its moments.


Introduction
Let n ∈ N, F q be a finite field and a ∈ F q . The Dickson polynomials D n (x; a), defined by [44, 9.6.1] D n (x; a) = ⌊ n 2 ⌋ j=0 n n − j n − j j (−a) j x n−2j , x ∈ F q where introduced by Leonard Eugene Dickson (1874Dickson ( -1954 in his 1896 Ph.D. thesis "The analytic representation of substitutions on a power of a prime number of letters, with a discussion of the linear group" [19], published in two parts in The Annals of Mathematics [20], [21]. The Dickson polynomials are the unique monic polynomials satisfying the functional equation [44, 9.6.3] D n y + a y ; a = y n + a y n , y ∈ F q 2 . See [37] for further algebraic and number theoretic properties of the Dickson polynomials. Let N 0 denote the set N ∪ {0} = 0, 1, 2, . . . . In [57], Wang and Yucas extended the Dickson polynomials to a family depending on a new parameter k ∈ N 0 , which they called Dickson polynomials of the (k + 1)-th kind. They defined them by with initial values D 0,k (x; a) = 2 − k, D 1,k (x; a) = x.
They also showed that the polynomials D n,k (x; a) satisfy the fundamental functional equation D n,k (y + a y ; a) = y n + a y n + k ay n − y 2 a y n y 2 − a , y = 0.
We clearly have D n,0 (x; a) = D n (x; a) (Dickson polynomials) and [44, 9.6.1] The polynomials D n,k (x; a) also satisfy the fundamental recurrence (see [57,Remark 2.5
The first few Dickson polynomials of the (k + 1)-th kind are They have zeros at as can be verified using a mathematical symbolic computation program such as Mathematica.
Remark 1. Note that the polynomials D n,k (x; a) are monic for n ≥ 1, but D 0,k (x; a) = 1 only for k = 1.
In this article, we study the polynomials D n,k (x; a) over the field of complex numbers, with a > 0 and k ∈ R. Our motivation is the three-term recurrence relation (4), which suggests that the Dickson polynomials of the (k + 1)-th kind form a family of orthogonal polynomials with respect to some linear functional L k . However, from (5) we see that for k > 2 the polynomials D n,k (x; a) may have a pair of purely imaginary roots. Also, the polynomials D 3,3 (x; a) and D 5, 5 2 (x; a) have a triple zero at x = 0. This implies that the linear functional L k is quasi-definite [16,Theorem 2.4.3], [27,Theorem 1].
The article is organized as follows: in Section 2, we derive some of the main properties of the Dickson polynomials of the (k + 1)-th kind, including different expressions, a hypergeometric representation, differential equations, and a generating function.
In Section 3, we present some basic results from the theory of orthogonal polynomials that we will need to find the linear functional L k . We define the co-recursive polynomials associated with a given family of orthogonal polynomials, and list some of their main properties. We also show that a family polynomials related to D n,k (x; a) are co-recursive polynomials associated with the Chebyshev polynomials of the second kind.
In Section 4 we apply the results of the previous sections to the Dickson polynomials of the (k +1)-th kind and obtain a representation for their moment functional L k . We also find explicit expressions for the moments of L k .
Finally, in Section 5 we summarize our results. In our hope that the results would be of interest to researchers outside the field of orthogonal polynomials and special functions, we have made the paper as self-contained as possible.

Properties of Dickson polynomials
We begin by checking the initial polynomial D 0,k (x; a). Since it is not clear from the definition (1) that D 0,k (x; a) = 2 − k, we consider even and odd degrees and obtain the following result. Proposition 2. The even and odd Dickson polynomials of the (k + 1)-th kind are given by and Proof. From (1), we have and switching the index to i = n − j, we get and (6) follows after using the symmetry of the binomial coefficients n k = n n − k .
Next, we will find a representation for D n,k (x; a) in terms of the generalized hypergeometric function where (u) j denotes the Pochhammer symbol (also called shifted or rising factorial) defined by [45, 5.2.4] (a) 0 = 1 Proposition 3. The Dickson polynomials of the (k + 1)-th kind admit the hypergeometric representation We also have We have c 0 = 1 and c j = 0 for j > n 2 . Using (9), we get Let k = 0. Then, and it follows that Thus, If k = 0, we see from (10) that and therefore Hence, Remark 4. A representation of D n,2 (x; a) in terms of associated Legendre functions of the first and second kinds [45, 14.3] was given in "A representation of the Dickson polynomials of the third kind by Legendre functions" by Neranga Fernando and Solomon Manukure (arXiv:1604.04682).
We can use the recurrence relation (4) to obtain a different representation for the polynomials D n,k (x; a).
√ a, the Dickson polynomials of the (k + 1)-th kind are given by where We also have, Proof. Let us assume that we can write for some function R(x, k, a). Using (14) in the recurrence (4), we obtain and therefore It follows that the general solution of (4) is given by Using the initial conditions (2) in (15), we get Thus, assuming that x = ±2 √ a, To verify (13), we replace it in the recurrence (4), and obtain Using (12), we can obtain a generating function for the polynomials D n,k (x; a).
Proposition 8. The ordinary generating function of the polynomials D n,k (x; a) is given by Proof. From (12), we have (as formal power series) Thus, and the result follows since Remark 9. The same generating function was obtained in [57, Lemma 2.6] using the recurrence (4).

Orthogonal polynomials
Let {µ n } be a sequence of complex numbers and L : C [x] → C be a linear functional defined by Then, L is called the moment functional determined by the moment sequence {µ n } . The number µ n is called the moment of order n.
A moment functional L is called positive-definite if L [r(x)] > 0 for every polynomial r(x) that is not identically zero and is non-negative for all real x. Otherwise, , with deg (P n ) = n is called an orthogonal polynomial sequence with respect to L provided that [13] L [P n P m ] = h n δ n,m , n, m ∈ N 0 , where h n = 0 and δ n,m is Kronecker's delta.
One of the fundamental properties of orthogonal polynomials is that they satisfy a three-term recurrence relation.
Theorem 10. Let L be a moment functional and let {P n } be the sequence of monic orthogonal polynomials associated with it. Then, there exist β n ∈ C and γ n ∈ C\{0} such that the polynomials P n (x) are a solution of the 3-term recurrence relation with initial conditions Proof. See [13, Theorem 4.1].
A linearly independent solution of (17) with initial conditions is given by the so-called associated orthogonal polynomials P * n (x) [13, 4.3]. Note that deg P * n (x) = n − 1. The converse of Theorem 10 is known as Favard's Theorem.
Theorem 11. Let {P n } be a sequence of polynomials satisfying the 3-term recurrence relation (17) with β n ∈ C and γ n ∈ C \ {0} . Then, there exists a unique linear functional L such that Proof. See [13,Theorem 4.4].
Remark 12. It follows from (2) and (4) that (at least for k = 2) {D n,k } is a sequence of monic (for n ≥ 1) orthogonal polynomials with respect to a moment functional L k 1 satisfying with h 0 (k) = 2 − k, h n (k) = a n , n ∈ N.
In Section 4 we will find a representation for the moment functional L k .
Proposition 13. Let L be a moment functional and {P n } be the sequence of monic orthogonal polynomials associated with it. Then, the following are equivalent: (a) All the moments of odd order are zero, Proposition 14. Let k = 2 and µ n (k) denote the moments of the linear functional defined by (21). Then, we have and The first few nonzero moments are Proof. From (22), we see that (23) follows.
Using (1), it is clear that and Proposition 13 gives µ 2n+1 (k) = 0, n = 0, 1, . . . . 1 In the remainder of the paper L k will denote the moment functional associated with the polynomials D n,k (x; a).
From (6) we have and therefore Remark 15. In Section 4 we will find a closed-form expression for µ 2n (k) .
The task of finding an explicit integral representation for the functional L is called a moment problem [1], [32], [49]. A moment functional L is called determinate if there exists a unique (up to an additive constant) distribution ψ (x) such that where the set Λ is called the support of the distribution ψ. Otherwise, L is called indeterminate [7], [54]. A criteria to decide if the moment functional L is determinate is due to Torsten Carleman [49, P 59]: If γ n > 0 and then L is determinate. Since for the Dickson polynomials we have γ n = a, it follows that the moment problem is determinate. One method to find a distribution function satisfying (26) is given by Markov's theorem.
Theorem 16. Let the moment functional L, supported on the set Λ ⊂ C, be determinate, {P n } be the monic orthogonal polynomials with respect to L and {P * n } be the associated polynomials. Then, where the convergence is uniform on compact subsets of C \ Λ.
The function is called the Stieltjes transform of L [55] and it has the asymptotic behavior [28, Section 12.9] 3.1. Co-recursive polynomials. Let {q n } be a sequence of polynomials satisfying (17) with initial conditions The polynomials q n (x) are called co-recursive with parameters (u, v). They were introduced by T. Chihara in [12], where he considered the case u = 1. See also [5], [36], [47], and [50]. Note that when u = 1 we could consider the polynomials q n (x) to be solutions of the perturbed 3-term recurrence relation Orthogonal polynomials that are solutions of 3-term recurrence relations with finite perturbations of the recurrence coefficients are called co-polynomials. See [11], [35], and [38]. Let {P n } and {P * n } be the linearly independent solutions of (17) with initial conditions (18) and (19). Then, we can represent q n (x) as a linear combination q n (x) = AP n (x) + BP * n (x) . Using (30), we see that Linear combinations of orthogonal polynomials have been studied by many authors. See [3], [33], [40], [41], and [42]. Note that the associated polynomials for both sequences are the same, i.e., q * n (x) = P * n (x) . Thus, we have q n (z) q * n (z) and assuming that the moment problem is determined, we can use Markov's theorem and obtain where S q (z) , S p (z) and L q , L p denote the Stieltjes transforms and linear functionals associated with the sequences {q n } and {P n } , respectively. Solving for S q (z) , we obtain .
If the coefficients in the 3-term recurrence relation (17) are constant, then we see that P * n (x) and P n−1 (x) satisfy the same recurrence, and have the same initial conditions. Therefore, they are identical P * n (x) = P n−1 (x) and we obtain Using Markov's theorem, we conclude that Assuming that L P [1] = 1, we find that S P (z) is the solution of the quadratic equation Multiplying (33) by γ, (34) by AQ, and subtracting, we get We conclude that Polynomial solutions of 3-term recurrence relations with constant coefficients xq n = q n+1 + βq n + γq n−1 , q 0 (x) = u, q 1 (x) = x − v, were analyzed in [15], where the authors concluded that the linear functional L q was of the form They only considered the case where u > 0 and all parameters (including the roots of f ) are real numbers. For the masses M 1 , M 2 they obtained The solution of (34) having the right asymptotic behavior, is and it follows that an integral representation of the lineal functional L P is given by Using the change of variables we obtain Therefore, it is enough to study the linear functional associated to the Chebyshev polynomials of the second kind, which we will define in the next subsection.

Chebyshev polynomials.
The monic Chebyshev polynomials of the second kind U n (x) are defined by [30, 9.8.36] U n (x) = 2 −n (n + 1) 2 F 1 −n, n + 2 They are a solution of the recurrence relation with initial conditions [30, 9.8.40] Note that The polynomials U n (x) satisfy the orthogonality relation [30, 9.8.38] The Stieltjes transforms of the linear functional L U is given by [55] S where here and in the rest of the paper √ : C → z ∈ C | − π 2 < arg (z) ≤ π 2 denotes the principal branch of the square root. Note that It is clear from the three-term recurrence relation (4) that the polynomials D n,k (x; a) are related to the polynomials U n (x). Let's introduce the scaled polynomials d n (x; k) defined by The polynomials d n (x; k) are a solution of the recurrence (37) satisfied by the polynomials U n (x), with initial conditions Therefore, we see that the scaled polynomials d n (x; k) are co-recursive polynomials (with respect to the Chebyshev polynomials of the second kind) with parameters u = 2 − k, v = 0. Using (31), we have in agreement with (6)- (7).
Remark 18. Stoll [51] studied second order recurrences with constant coefficients and general initial conditions and found polynomial decompositions in terms of Chebyshev polynomials.

Main results
In this section we find a representation for the linear functional L k defined by (21). Although this seems to be something already considered in [15], we have found that the range of parameters of the polynomials D n,k (x) requires a different analysis.
Let k = 2, l k [r] denote the linear functional satisfying and s(z; k) be it's Stieltjes transform. From (35) and (41), we have since v = β = 0. Our next objective is to represent the function s(z; k) as the Stieltjes transform of a distribution. We begin with a couple of lemmas.
Lemma 19. Let L be a linear functional with Stieltjes transform S (z) and Then, where we always assume that the functional L acts on the variable x.
we get , and the conclusion follows.
For the particular case of the linear functional L U defined by (40) with Stieltjes transform S U (z) given in (41) and because .
Note that since [55] 2 π we have and in particular Lemma 20. Let ω (k) be defined by where i 2 = −1. Then, Proof. The result follows immediately from the definition (50), since The function s(z; k) defined in (46) where χ (k) is the characteristic function defined by s c (z; k) is the continuous part of s(z; k) and s d (z; k) is the discrete part of s(z; k) Proof. Let k ∈ R \ {0, 1, 2} . From (46), we have Using (50), we get Hence, Since we know from Lemma 20 that ω (k) ∈ C \ [−1, 1], we can use (47) with b = ω and obtain , and therefore If k = 0, then we have from (46) and using (49), we get = s c (z; 0).
If k = 1, then we have from (46) and using (41) we get
Corollary 23. Let k = 2, the linear functional l k be defined by (45), the characteristic function χ (k) be defined by (52), and the function ω (k) de defined by 50.
Then, for all r (x) ∈ C [x] we have where l (c) and we assume that Proof. The result is a direct consequence of Theorem 21, since For k = 1, we see from (43) that d n (x; 1) = U n (x) , and therefore which agrees with (56) for k = 1 if we use (59).

Remark 24.
For k = 2, we see from (43) that d n (x; 2) = xU n−1 (x) , and therefore we can interpret l 2 as the linear functional defined for all polynomials r (x) such that r (x) = x 2 p (x) , p (x) ∈ C [x] . It follows that d n (x; 2) will be a family of orthogonal polynomials for n ≥ 1.

The Dickson polynomials.
We can now use the previous results to the Dickson polynomials of the (k + 1)-th kind D n,k (x; a), related to the scaled polynomials d n (x; k) by (42).
where l k is the linear functional defined by (45). Then, L k satisfies and for k = 2, its Stieltjes transform is given by Proof. Using (42) in (60), we have Therefore, (45) gives and L k [D n,k (x) D m,k (x)] = 2 √ a n+m 4 −n δ n,m , n, m ∈ N.
But since 2 √ a n+m 4 −n δ n,m = a n δ n,m , (61) follows. Using (46) in (60), we get Corollary 26. The linear functional L k defined by (60) is identical to the linear functional L k satisfying (21).
Next, we find a representation for the linear functional L k .
Theorem 27. Let k = 2 and L k be the linear functional defined by (21). Then, L k admits the representation where χ (k) was defined in (52), and L (d) with Ω (k) defined by Proof. Changing variables to t = 2 √ ax in the integral we obtain Therefore, from (57) we get Also, from (58) and (65) we have Therefore, using (56) in (60) we see that Although Theorem 27 seems to be valid only when k = 2, we can see that L 2 is well defined. (66) Proof. Lets assume that for some functions b 0 (k, a) and B (k, a) . Using (4), we have Using (2) Thus, and clearly It follows from the previous Lemma that L Thus, We can now extend Theorem 27 to all values of k.
Corollary 30. Let h n (k) be defined by (22) and χ (k) be defined by (52). Then, Remark 31. If we set k = 2 in (67), we obtain which seems to make no sense, since the integrand is singular at t = 0. However, if we use (44) we have and we can write (68) as This agrees with (40), since we have U −1 = 0 from (39) and h 0 (2) = 0 from (22), while for n, m ≥ 1 we know from (22) that a − n+m 2 h n (k) δ n,m = δ n,m .
Finally, we will use the function S (z; k, a) to find explicit expressions for the moments of L k . Proposition 32. Let L k be the linear functional defined by (21). Then, the moments of L k of even order µ 2n (k) = L k x 2n , are given by If k = 1, 2, we set y j = µ 2j , and obtain the recurrence As it is well known, the general solution of the initial value problem y n+1 = c n y n + g n , y n0 = y 0 , Thus, This agrees with (63), since When k = 1, we have from (63) µ 2n (1) = 1 2πa which can be verified directly. When k = 2, we can write (see Remark 31) µ 2n (2) = 1 2π

Conclusions
We have shown that the Dickson polynomials of the (k + 1)-th kind defined by where a > 0, k ∈ R, and h 0 (k) = 2 − k, h n (k) = a n , n = 1, 2, . . . . We hope that this work will outline some connections between finite fields and orthogonal polynomials, and that it would be of interest to researchers in both areas.
We also wish to acknowledge the hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), on the occasion of the Programme on "Algorithmic and Enumerative Combinatorics" held in October-November 2017.
Finally, we also wish to express our gratitude to the anonymous referees, who provided us with invaluable suggestions and comments that greatly improved our first draft of the paper.