Expansions and Characterizations of Sieved Random Walk Polynomials

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x),$ $n\in\mathbb{N}$ with $(c_n)_{n\in\mathbb{N}}\subseteq(0,1)$. For every $k\in\mathbb{N}$, the $k$-sieved polynomials $(P_n(x;k))_{n\in\mathbb{N}_0}$ arise from the recurrence coefficients $c(n;k):=c_{n/k}$ if $k|n$ and $c(n;k):=1/2$ otherwise. A main objective of this paper is to study expansions in the Chebyshev basis $\{T_n(x)\colon n\in\mathbb{N}_0\}$. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version $\mathrm{D}_k$ of the Askey-Wilson operator $\mathcal{D}_q$. It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative and obtained from $\mathcal{D}_q$ by letting $q$ approach a $k$-th root of unity. However, for $k\geq2$ the new operator $\mathrm{D}_k$ on $\mathbb{R}[x]$ has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for $k$-sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator $\mathrm{A}_k$.


Introduction
In the theory of orthogonal polynomials, it is an important problem to identify properties which characterize specific classes.The literature is extensive; [2] provides a valuable survey of such characterization results up to 1990.Our paper takes the newer contributions [10,13,15] as a starting point and deals with new characterizations of sieved random walk polynomials.Let µ be a symmetric probability Borel measure on R with |supp µ| = ∞ and supp µ ⊆ [−1, 1], and let (P n (x)) n∈N 0 ⊆ R[x] be the orthogonal polynomial sequence with respect to µ, normalized by P n (1) = 1, n ∈ N 0 .In the following, we call such a measure µ just an 'orthogonalization measure', and we call (P n (x)) n∈N 0 the (symmetric) 'random walk polynomial sequence' ('RWPS') with respect to µ.The relation to random walks is explained in [6,16], for instance.Under the assumptions made on the support, it is well known that if an RWPS is orthogonal with respect to two orthogonalization measures, then these measures must coincide. 1Moreover, a sequence (P n (x)) n∈N 0 ⊆ R[x] is an RWPS if and only if it is given by a recurrence relation of the form P 0 (x) = 1 and where c 0 := 0, (c n ) n∈N ⊆ (0, 1) and a n := 1 − c n , n ∈ N 0 [5,15]. 2 We consider two related sequences: on the one hand, if (P n (x)) n∈N 0 is an RWPS and k ∈ N is fixed, then let (P n (x; k)) n∈N 0 ⊆ R[x] denote the 'k-sieved RWPS' which corresponds to (P n (x)) n∈N 0 , i.e., (P n (x; k)) n∈N 0 satisfies the recurrence relation P 0 (x; k) = 1, (we particularly point out [9]).We refer to the seminal paper [7] of Geronimo and Van Assche which studies sieved polynomials via polynomial mappings (particularly to [7,Section VI]).Sieved polynomials are a very fruitful topic in the theory of orthogonal polynomials; more recent contributions in this context are [4,17], for instance.On the other hand, for an RWPS (P n (x)) n∈N 0 let (P * n (x)) n∈N 0 ⊆ R[x] denote the polynomials which are orthogonal with respect to dµ * (x) := 1−x 2 dµ(x), normalized by P * n (1) = 1, n ∈ N 0 . 3Let h : N 0 → (0, ∞) be given by [15] h(n) := 1 Then one explicitly has (cf.[13]) where C * n ∈ R\{0} depends on n but is independent of x.The first equality in (1.2) is an immediate consequence from the observation that R 1 − x 2 P * n (x)P k (x)dµ(x) = R P * n (x)P k (x)dµ * (x) = 0, n ∈ N, k ∈ {0, . . ., n − 1}, which yields that 1 − x 2 P * n (x) must be a linear combination of P n (x) and P n+2 (x); since 1−x 2 P * n (x) vanishes for x = 1, the occurring linearization coefficients must be equal up to sign.The second equality in (1.2) can be seen as follows: using (1.1), it is easy to see that (cf.[12]) Therefore, the second equality in (1.2) follows from the first.Moreover, we see that .
Via the Christoffel-Darboux formula (cf.[5]), one can show that C * n is also given by .
Recall that, given some q ∈ (0, 1), the Askey-Wilson operator D q : R[x] → R[x] is defined by linearity and the action [8, 10] where U −1 (x) := 0 and (T n (x)) n∈N 0 , (U n (x)) n∈N 0 denote the sequences of Chebyshev polynomials of the first and second kind, so T n (cos(θ)) = cos(nθ), U n (cos(θ)) = sin((n + 1)θ)/ sin(θ), Note that (T n (x)) n∈N 0 is the only RWPS which is invariant under sieving with arbitrary k.The classical derivative d/dx is the limiting case q → 1 of D q ; more precisely, (1.3) is a q-extension of the well-known relation Relations (1.3), (1.4) and (1.5) can be interpreted in the following way: if The question which RWPS share the first of these properties has been answered in [15, Lemma 1, Theorem 1] and involves the ultraspherical polynomials: Theorem 1.1 (Lasser-Obermaier 2008).The following are equivalent: [10, Theorem 5.2] gives a q-analogue: Then the following are equivalent:

S. Kahler
In Theorems 1.1 and 1.2, P n (x) n∈N 0 and (P n (x; β|q)) n∈N 0 denote the sequences of ultraspherical polynomials which correspond to α > −1 and continuous q-ultraspherical (Rogers) polynomials which correspond to suitable q and β, respectively, normalized such that4 Explicit formulas can be found in [8,10,14,15].At this stage, we just recall that P (α) n (x) n∈N 0 is orthogonal with respect to the probability measure and has the recurrence coefficients c n = n/(2n + 2α + 1), n ∈ N [15].Furthermore, we note the striking limit relation between the continuous q-ultraspherical polynomials and the sieved ultraspherical polynomials [3,Section 2].
In [13, Theorems 2.1 and 2.3], we sharpened the abovementioned Lasser-Obermaier result Theorem 1.1 and Ismail-Obermaier result Theorem 1.2 by showing that the characterizations remain valid if n in (ii) is replaced by 2n − 1. 5A main purpose of the present paper is to study the interplay between the transitions (P n (x)) n∈N 0 −→ (P n (x; k)) n∈N 0 , (P n (x)) n∈N 0 −→ (P * n (x)) n∈N 0 and a "sieved version" of the Askey-Wilson operator.The limit relation (1.6) motivates our research in the following way: if one defines a corresponding "sieved Askey-Wilson operator" and linear extension, it is a natural question to ask whether, in analogy to Theorems 1.1 and 1.2, D k characterizes the sieved ultraspherical polynomials P (α) n (x; k) n∈N 0 .6At first sight, this might be a reasonable conjecture.However, observe that if k ≥ 2, then the kernel of D k becomes infinite-dimensional because U n−1 (cos(π/k)) = sin((nπ)/k)/ sin(π/k) becomes zero for infinitely many n ∈ N 0 (whereas the operators d/dx = D 1 and D q have finite-dimensional kernels).This important property might give reason to expect an additional degree of freedom.The situation is also very different to results of Ismail and Simeonov [11] where Theorems 1.1 and 1.2 have been unified and extended to larger classes-but still for operators which reduce the polynomial degree by a fixed positive integer.In fact, the answer will depend on k.These results are given in Section 3 and rely on an expansion result which is provided (and applied to an explicit example) in Section 2. It turns out that as soon as k ≥ 2 the operator D k does not lead to characterizations of sieved ultraspherical polynomials but to characterizations of arbitrary ksieved RWPS. 7Moreover, we present a characterization which involves the eigenvectors of the linear "sieved averaging operator" A k The definition of A k is motivated by (1.6) and the classical q-averaging operator A q is a q-analogue of the identity operator and appears in the product rule of the Askey-Wilson operator.The characterization via A k will also be given in Section 3, and it will be motivated by our following result on continuous q-ultraspherical polynomials [13, Theorem 2.4]: Theorem 1.3.Under the conditions of Theorem 1.2 and the additional assumption that β ≤ 1, the following are equivalent: Again it turns out that the passage from A q to A k leads to characterizations of arbitrary ksieved RWPS.However, the information contained in a previously specific-(q-)ultrasphericalunderlying structure is lost due to additional degrees of freedom.Here, these additional degrees of freedom can be traced back to the following fact (which is not obvious and will be established in Section 3, too): for any RWPS (P n (x)) n∈N 0 , the integral Our characterization results with respect to D k and A k particularly prove a conjecture which we made in [12].
We remark that we used computer algebra systems (Maple) to find explicit formulas as in Example 2.2 below (which then can be verified by induction etc.), obtain factorizations of multivariate polynomials, get conjectures and so on.The final proofs can be understood without any computer usage, however.

Expansions of sieved polynomials in the Chebyshev basis
In this section, we study expansions of sieved polynomials in the Chebyshev basis {T n (x) : n ∈ N 0 }.Our result is suitable for explicit computations (see Example 2.2 below) and provides an important tool for the characterization results presented in Section 3.
Let (P n (x)) n∈N 0 be an RWPS as in Section 1, and let k ∈ N. We consider the connection coefficients to the Chebyshev polynomials of the first kind: for each n ∈ N 0 , we define a mapping r n : {0, . . ., ⌊n/2⌋} → R by the expansion

S. Kahler
Moreover, let the mappings p n , q n : {0, . . ., ⌊n/2⌋} → R, n ∈ N 0 , be recursively defined by p 0 (0) := 0, q 0 (0) := 1 and the coupled system of recursions for n ∈ N and j ∈ {0, . . ., ⌊n/2⌋}, where we set The following theorem uses the sequences (p n ) n∈N 0 and (q n ) n∈N 0 to obtain the desired expansions of the sieved polynomials (P n (x; k)) n∈N 0 in the basis {T n (x) : n ∈ N 0 }.Moreover, the theorem provides a possibility to compute (p n ) n∈N 0 and (q n ) n∈N 0 directly from (r n ) n∈N 0 .To avoid case differentiations, we define Theorem 2.1.For every k ∈ N, n ∈ N 0 and i ∈ {0, . . ., k − 1}, one has Moreover, one has (2.9) Concerning the expansions provided by Theorem 2.1, it is very remarkable that the coefficients of T kn−2jk−i (x) and T kn−2jk+i (x) do not rely on i, nor do they rely on k.Concerning welldefinedness in (2.6), note that the "polynomials" T −(k−1) (x), . . ., T −1 (x) are not defined; however, they only occur for even n and together with a multiplication with p n (⌊n/2⌋) = p n (n/2) = 0, and by our convention the product of 0 and these undefined polynomials is interpreted as 0. This convention will also be used in the following proof.
Proof of Theorem 2.1.Let k ≥ 2 first.We establish the expansion (2.6) via induction on n ∈ N 0 .It is clear from the recurrence relation for the Chebyshev polynomials of the first kind that P i (x; k) = T i (x) for all i ∈ {0, . . ., k}, so (2.6) is true for n = 0. Now let n ∈ N be arbitrary but fixed and assume the validity of (2.6) for n − 1.In particular, we then have Due to (2.4) and (2.5), the latter equation can be rewritten as (2.10) In the same way, we obtain (2.11) We now use that Section VI], which yields (2.12) Combining (2.10), (2.11) and (2.12) with the relation 2xP kn−1 (x; k) = P kn (x; k) + P kn−2 (x; k) and using the recurrence relation for the Chebyshev polynomials of the first kind, we obtain that r n (j) = p n−1 (j − 1) + q n−1 (j) for each j ∈ {0, . . ., ⌊n/2⌋}.Since for each j ∈ {0, . . ., ⌊n/2⌋} (which makes use of the recursions (2.2) and (2.3), as well as another use of definition (2.5)), we therefore get [p n (j) + q n (j)]T kn−2jk (x). (2.13) We now combine (2.11) with (2.13), write 3) and definition (2.5) again and obtain Thus if k = 2, then (2.6) is shown.If k ≥ 3, we use induction on i to prove that and have already shown the initial step i ∈ {0, 1}; we hence assume i ∈ {0, . . ., k − 3} to be arbitrary but fixed and (2.14) to hold for i, i + 1, and then calculate This finishes the proof of (2.6) for k ≥ 2, and we have simultaneously established (2.7) and (2.8).
(2.6) for the remaining case k = 1 is an immediate consequence of (2.8).Finally, (2.9) can be seen as follows: let n ∈ N 0 and j ∈ {0, . . ., ⌊n/2⌋}.By (2.7) and (2.8), we have for all i ∈ {0, . . ., j}.Taking the sum from 0 to j and using definition (2.4), we get as desired.■ We now apply Theorem 2.1 to the ultraspherical polynomials and obtain explicit expansions of the sieved ultraspherical polynomials with respect to the Chebyshev basis {T n (x) : n ∈ N 0 }: Example 2.2 (sieved ultraspherical polynomials).Let P n (x) = P (α) n (x), n ∈ N 0 , be the sequence of ultraspherical polynomials which corresponds to α > −1.The case α = −1/2 corresponds to the Chebyshev polynomials of the first kind (T n (x)) n∈N 0 and is therefore trivial, so let α ̸ = −1/2 from now on.Then (r n ) n∈N 0 is given by [8, Theorem 9.1.1] , n even and j = n 2 , 3 Characterizations via the sieved operators Let (P n (x)) n∈N 0 be an RWPS as in Section 1.Moreover, let k ∈ N again.Following [10,13,15] (q-and non-sieved analogues), we consider the Fourier coefficients which are associated with D k (1.7) and A k (1.8): for each n ∈ N 0 , we define mappings κ n (•; k), α n (•; k) : N 0 → R by the projections In other words, κ n (•; k) and α n (•; k) correspond to the expansions Due to the symmetry of (P n (x)) n∈N 0 , we have κ n (j; k) = 0 if n − j is even, and we have As soon as k ≥ 2 (and also for k = 1 in Theorem 3.1), we do not obtain characterizations of sieved ultraspherical polynomials (as one might expect due to comparison to the cited theorems) but characterizations of arbitrary k-sieved RWPS.
Theorem 3.1.If k ∈ N, then the following are equivalent: If these equivalent conditions are satisfied, then P n (| cos(π/k)|) = T n (| cos(π/k)|) is the eigenvalue of A k which corresponds to the eigenvector P n (x), n ∈ N 0 .
Theorem 3.2.If k ≥ 2, then the following are equivalent: and for every n ∈ N there is an m ∈ {0, . . ., ⌊(n − 1)/2⌋} such that If k = 1, then (ii), (iii) and (iv) are equivalent to The characterization provided by (iii) of the previous theorem has the advantage that it is "stable" with respect to renormalization of the sequence (P n (x)) n∈N 0 .The characterization provided by (iv) is the strongest one, however, because the functions κ n (•; k) (3.1) (3.3) have to be considered just at some carefully chosen points.
Note that the formal limits "D ∞ " and "A ∞ " are included in our investigations because they coincide with D 1 = d/dx and A 1 = id.
Before coming to the proofs, we study some basic properties of D k and A k .We will make use of the following well-known identities [1, formulas (22.7.25)-(22.7.28)]: The following lemma deals with the function σ(.; k) (3.5) and with special values of the functions α n (.; k) (3.2) (3.4).The analogous q-and non-sieved versions can be found in [10,15] with similar proofs.
Lemma 3.3.One has Proof .(i) If one expands P n (x) as in (2.1), this is obvious from the definitions (in particular, use (1.8)).
(ii) Using (1.8), (2.1) and (i), we have (iii) While on the one hand one has by (1.7), on the other hand one has Consequently, we have and as obviously r n−1 (0) = (2 − δ n,1 )a n−1 r n (0) and a n−1 h(n − 1) = c n h(n), the proof is complete (note that, by definition, κ n (n − 1; k) = σ(n; k)).■ We also investigate the product rule for the sieved Askey-Wilson operator D k .Its analogue for D q has the same structure (see [8,10]).Lemma 3.4.One has Consequently, Proof .Due to linearity, it clearly suffices to establish that This, however, can easily be seen from the equations (1.7), (1.8), (3.7) and (3.8) by the computation for m ≤ n (the expansion T m (x)T n (x) = T m+n (x)/2 + T n−m (x)/2 is well known).Now let n, j ∈ N 0 .Via (1.1), we compute multiplication with P j (x), integration with respect to µ and the equations (3.1) and (3.2) yield the second assertion.■ The recurrence relation (3.9) for (κ n (•; k)) n∈N 0 is the analogue to q-and non-sieved variants which can be found in [10,15].
We now come to the proofs of Theorems 3.1 and 3.2.
xP n (x; k) = a(n; k)P n+1 (x; k) + c(n; k)P n−1 (x; a(n; k) := 1 − c(n; k), n ∈ N 0 .Such sieved RWPS and related concepts have been studied in a series of papers by Ismail et al.