Expansions and Characterizations of Sieved Random Walk Polynomials
We consider random walk polynomial sequences (ₙ())ₙ∈ℕ₀ ⊆ ℝ[] given by recurrence relations ₀() = 1, ₁() = , ₙ() = (1−cₙ)ₙ₊₁()+cₙₙ₋₁(), ∈ ℕ with (cₙ)ₙ∈ℕ ⊆ (0, 1). For every ∈ ℕ, the -sieved polynomials (ₙ(; ))ₙ∈ℕ₀ arise from the recurrence coefficients c(; ):= cₙ/ₖ if | and c(; ):= 1/2 otherwise. A...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2023 |
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| Мова: | Англійська |
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Інститут математики НАН України
2023
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Expansions and Characterizations of Sieved Random Walk Polynomials. Stefan Kahler. SIGMA 19 (2023), 103, 18 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860227474021941248 |
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| author | Kahler, Stefan |
| author_facet | Kahler, Stefan |
| citation_txt | Expansions and Characterizations of Sieved Random Walk Polynomials. Stefan Kahler. SIGMA 19 (2023), 103, 18 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider random walk polynomial sequences (ₙ())ₙ∈ℕ₀ ⊆ ℝ[] given by recurrence relations ₀() = 1, ₁() = , ₙ() = (1−cₙ)ₙ₊₁()+cₙₙ₋₁(), ∈ ℕ with (cₙ)ₙ∈ℕ ⊆ (0, 1). For every ∈ ℕ, the -sieved polynomials (ₙ(; ))ₙ∈ℕ₀ arise from the recurrence coefficients c(; ):= cₙ/ₖ if | and c(; ):= 1/2 otherwise. A main objective of this paper is to study expansions in the Chebyshev basis {Tₙ(): n ∈ℕ₀}. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version Dₖ of the Askey-Wilson operator . It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative, and obtained from by letting approach a -th root of unity. However, for ≥ 2, the new operator Dₖ on ℝ[] has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for -sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator Aₖ.
|
| first_indexed | 2026-03-20T23:59:24Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 103, 18 pages
Expansions and Characterizations of Sieved Random
Walk Polynomials
Stefan KAHLER abc
a) Fachgruppe Mathematik, RWTH Aachen University,
Pontdriesch 14-16, 52062 Aachen, Germany
E-mail: kahler@mathematik.rwth-aachen.de
b) Lehrstuhl A für Mathematik, RWTH Aachen University, 52056 Aachen, Germany
c) Department of Mathematics, Chair for Mathematical Modelling,
Chair for Mathematical Modeling of Biological Systems, Technical University of Munich,
Boltzmannstr. 3, 85747 Garching b. München, Germany
Received July 03, 2023, in final form December 01, 2023; Published online December 22, 2023
https://doi.org/10.3842/SIGMA.2023.103
Abstract. We consider random walk polynomial sequences (Pn(x))n∈N0
⊆ R[x] given by
recurrence relations P0(x) = 1, P1(x) = x, xPn(x) = (1 − cn)Pn+1(x) + cnPn−1(x), n ∈ N
with (cn)n∈N ⊆ (0, 1). For every k ∈ N, the k-sieved polynomials (Pn(x; k))n∈N0 arise from
the recurrence coefficients c(n; k) := cn/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 {Tn(x) : n ∈ N0}.
As an application, we obtain explicit expansions for the sieved ultraspherical polynomials.
Moreover, we introduce and study a sieved version Dk of the Askey–Wilson operator Dq.
It is motivated by the sieved ultraspherical polynomials, a generalization of the classical
derivative and obtained from Dq by letting q approach a k-th root of unity. However,
for k ≥ 2 the new operator Dk on 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 Ak.
Key words: random walk polynomials; sieved polynomials; Askey–Wilson operator; averag-
ing operator; polynomial expansions; Fourier coefficients
2020 Mathematics Subject Classification: 42C05; 33C47; 42C10
1 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 |suppµ| = ∞ and suppµ ⊆ [−1, 1], and
let (Pn(x))n∈N0 ⊆ R[x] be the orthogonal polynomial sequence with respect to µ, normalized
by Pn(1) = 1, n ∈ N0. In the following, we call such a measure µ just an ‘orthogonalization
measure’, and we call (Pn(x))n∈N0 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.1 Moreover,
a sequence (Pn(x))n∈N0 ⊆ R[x] is an RWPS if and only if it is given by a recurrence relation of
1Standard results from the theory of orthogonal polynomials can be found in [5], for instance.
kahler@mathematik.rwth-aachen.de
https://doi.org/10.3842/SIGMA.2023.103
2 S. Kahler
the form P0(x) = 1 and
xPn(x) = anPn+1(x) + cnPn−1(x), n ∈ N0, (1.1)
where c0 := 0, (cn)n∈N ⊆ (0, 1) and an := 1− cn, n ∈ N0 [5, 15].2
We consider two related sequences: on the one hand, if (Pn(x))n∈N0 is an RWPS and k ∈ N
is fixed, then let (Pn(x; k))n∈N0 ⊆ R[x] denote the ‘k-sieved RWPS’ which corresponds to
(Pn(x))n∈N0 , i.e., (Pn(x; k))n∈N0 satisfies the recurrence relation P0(x; k) = 1,
xPn(x; k) = a(n; k)Pn+1(x; k) + c(n; k)Pn−1(x; k), n ∈ N0
with
c(n; k) :=
{
cn
k
, k|n,
1
2 , else,
and a(n; k) := 1 − c(n; k), n ∈ N0. Such sieved RWPS and related concepts have been studied
in a series of papers by Ismail et al. (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 (Pn(x))n∈N0 let (P ∗
n(x))n∈N0 ⊆ R[x] denote the polynomials
which are orthogonal with respect to dµ∗(x) :=
(
1−x2
)
dµ(x), normalized by P ∗
n(1) = 1, n ∈ N0.
3
Let h : N0 → (0,∞) be given by [15]
h(n) :=
1∫
RP
2
n(x) dµ(x)
=
1, n = 0,
n∏
j=1
aj−1
cj
, else.
Then one explicitly has (cf. [13])
P ∗
n(x) = C∗
n
Pn+2(x)− Pn(x)
1− x2
=
∑⌊n
2
⌋
k=0 h(n− 2k)Pn−2k(x)∑⌊n
2
⌋
k=0 h(n− 2k)
, n ∈ N0, (1.2)
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− x2
)
P ∗
n(x)Pk(x)dµ(x) =
∫
R
P ∗
n(x)Pk(x)dµ
∗(x) = 0,
n ∈ N, k ∈ {0, . . . , n−1}, which yields that
(
1−x2
)
P ∗
n(x) must be a linear combination of Pn(x)
and Pn+2(x); since
(
1−x2
)
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])
(
1− x2
) ⌊n
2
⌋∑
k=0
h(n− 2k)Pn−2k(x) = −cn+1cn+2h(n+ 2)[Pn+2(x)− Pn(x)], n ∈ N0.
2We make the convention that 0 times something undefined shall be 0.
3Note that µ∗ is no longer a probability measure.
Expansions and Characterizations of Sieved Random Walk Polynomials 3
Therefore, the second equality in (1.2) follows from the first. Moreover, we see that
C∗
n = −cn+1cn+2h(n+ 2)∑⌊n
2
⌋
k=0 h(n− 2k)
.
Via the Christoffel–Darboux formula (cf. [5]), one can show that C∗
n is also given by
C∗
n = − 2cn+1cn+2h(n+ 2)∑n
k=0 h(k) + cn+1h(n+ 1)
.
Recall that, given some q ∈ (0, 1), the Askey–Wilson operator Dq : R[x] → R[x] is defined by
linearity and the action [8, 10]
DqTn(x) =
q
n
2 − q−
n
2
√
q − 1√
q
Un−1(x), n ∈ N0, (1.3)
where U−1(x) := 0 and (Tn(x))n∈N0 , (Un(x))n∈N0 denote the sequences of Chebyshev polyno-
mials of the first and second kind, so Tn(cos(θ)) = cos(nθ), Un(cos(θ)) = sin((n + 1)θ)/ sin(θ),
T0(x) = U0(x) = 1, T1(x) = x, U1(x) = 2x,
xTn(x) =
1
2
Tn+1(x) +
1
2
Tn−1(x), xUn(x) =
1
2
Un+1(x) +
1
2
Un−1(x), n ∈ N
and
Un(x) = (n+ 1)T ∗
n(x), n ∈ N0. (1.4)
Note that (Tn(x))n∈N0 is the only RWPS which is invariant under sieving with arbitrary k. The
classical derivative d/dx is the limiting case q → 1 of Dq; more precisely, (1.3) is a q-extension
of the well-known relation
d
dx
Tn(x) = nUn−1(x), n ∈ N0. (1.5)
Relations (1.3), (1.4) and (1.5) can be interpreted in the following way: if Pn(x) = Tn(x),
n ∈ N0, then P ′
n(x) = P ′
n(1)P
∗
n−1(x) and DqPn(x) = DqPn(1)P
∗
n−1(x), n ∈ N. 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:
(i) Pn(x) = P
(1/(2c1)−3/2)
n (x), n ∈ N0,
(ii) P ′
n(x) = P ′
n(1)P
∗
n−1(x), n ∈ N.
[10, Theorem 5.2] gives a q-analogue:
Theorem 1.2 (Ismail–Obermaier 2011). Let q ∈ (0, 1), β ∈ (0, 1/
√
q) and A =
√
β/2+1/(2
√
β).
Moreover, let (Qn(x))n∈N0 be orthogonal with respect to a symmetric probability Borel measure µ
on R with |suppµ| = ∞ and suppµ ⊆ [−A,A]; furthermore, let (Qn(x))n∈N0 be normalized by
Qn(A) = 1(n ∈ N0), and let Q2(0) = −β(1− q)/
(
1− β2q
)
. Then the following are equivalent:
(i) Qn(x) = Pn(x;β|q), n ∈ N0,
(ii) (DqQn(x))n∈N is orthogonal with respect to
(
A2 − x2
)
dµ(x).
4 S. Kahler
In Theorems 1.1 and 1.2,
(
P
(α)
n (x)
)
n∈N0
and (Pn(x;β|q))n∈N0 denote the sequences of ultra-
spherical polynomials which correspond to α > −1 and continuous q-ultraspherical (Rogers)
polynomials which correspond to suitable q and β, respectively, normalized such that4
P (α)
n (1) = Pn(
√
β/2 + 1/(2
√
β);β|q) = 1, n ∈ N0.
Explicit formulas can be found in [8, 10, 14, 15]. At this stage, we just recall that
(
P
(α)
n (x)
)
n∈N0
is orthogonal with respect to the probability measure
dµ(x) = Γ(2α+ 2)/
(
22α+1Γ(α+ 1)2
)
·
(
1− x2
)α
χ(−1,1)(x)dx
and has the recurrence coefficients cn = n/(2n+ 2α+ 1), n ∈ N [15]. Furthermore, we note the
striking limit relation
lim
s→1
Pn
(
x; sαk+
k
2 |se
2πi
k
)
= P (α)
n (x; k), n ∈ N0 (1.6)
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− 1.5
A main purpose of the present paper is to study the interplay between the transitions
(Pn(x))n∈N0 −→ (Pn(x; k))n∈N0 , (Pn(x))n∈N0 −→ (P ∗
n(x))n∈N0 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” Dk : R[x] → R[x] via
DkTn(x) : = lim
s→1
(√
se
2πi
k
)n
−
(√
se
2πi
k
)−n
√
se
2πi
k − 1√
se
2πi
k
Un−1(x)
= Un−1
(∣∣∣cos(π
k
)∣∣∣)Un−1(x), n ∈ N0 (1.7)
and linear extension, it is a natural question to ask whether, in analogy to Theorems 1.1 and 1.2,
Dk characterizes the sieved ultraspherical polynomials
(
P
(α)
n (x; k)
)
n∈N0
.6 At first sight, this
might be a reasonable conjecture. However, observe that if k ≥ 2, then the kernel of Dk becomes
infinite-dimensional because Un−1(cos(π/k)) = sin((nπ)/k)/ sin(π/k) becomes zero for infinitely
many n ∈ N0 (whereas the operators d/dx = D1 and Dq 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 Dk does not lead to
characterizations of sieved ultraspherical polynomials but to characterizations of arbitrary k-
sieved RWPS.7 Moreover, we present a characterization which involves the eigenvectors of the
4The special case β = 1 is excluded in both the “standard” normalization of the continuous q-ultraspherical
polynomials and the original formulation of the cited result [10, Theorem 5.2]. However, Theorem 1.2 remains
valid with the definition Pn(x; 1|q) := Tn(x), n ∈ N0 (cf. [13]).
5Results which are cited from [13] can also be found in [12].
6In the definition of Dk and in the definition of Ak below,
√
. shall denote the principal value of the square
root.
7We say an RWPS (Pn(x))n∈N0 to be ‘k-sieved’ (without further specification) if it is k-sieved with respect to
some RWPS or, equivalently, if cn = 1/2 if k ̸ |n.
Expansions and Characterizations of Sieved Random Walk Polynomials 5
linear “sieved averaging operator” Ak : R[x] → R[x],
AkTn(x) : = lim
s→1
(√
se
2πi
k
)n
+
(√
se
2πi
k
)−n
2
Tn(x)
= Tn
(∣∣∣cos(π
k
)∣∣∣)Tn(x), n ∈ N0. (1.8)
The definition of Ak is motivated by (1.6) and the classical q-averaging operator Aq : R[x] → R[x]
[8, 10]
AqTn(x) =
q
n
2 + q−
n
2
2
Tn(x), n ∈ N0.
Aq is a q-analogue of the identity operator and appears in the product rule of the Askey–Wilson
operator. The characterization via Ak 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:
(i) Qn(x) = Pn(x;β|q), n ∈ N0,
(ii) the quotient∫
RAqQn+1(x)Qn−1(x)dµ(x)∫
RDqQn+1(x)Qn(x)dµ(x)
is independent of n ∈ N.
Again it turns out that the passage from Aq to Ak leads to characterizations of arbitrary k-
sieved RWPS. However, the information contained in a previously specific—(q-)ultraspherical—
underlying 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 (Pn(x))n∈N0 , the integral∫
R
AkPn+1(x)Pn−1(x)dµ(x)
vanishes if n ∈ N is a multiple of k. Our characterization results with respect to Dk and Ak
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.
2 Expansions of sieved polynomials in the Chebyshev basis
In this section, we study expansions of sieved polynomials in the Chebyshev basis {Tn(x) :
n ∈ N0}. 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 (Pn(x))n∈N0 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 ∈ N0, we define a mapping
rn : {0, . . . , ⌊n/2⌋} → R by the expansion
Pn(x) =
⌊n
2
⌋∑
j=0
rn(j)Tn−2j(x). (2.1)
6 S. Kahler
Moreover, let the mappings pn, qn : {0, . . . , ⌊n/2⌋} → R, n ∈ N0, be recursively defined by
p0(0) := 0, q0(0) := 1
and the coupled system of recursions
pn(j) :=
0, n even and j = n
2 ,
(2an − 1)qn−1(j) + pn−1(j − 1)
2an
, else,
(2.2)
qn(j) :=
pn−1
(n
2
− 1
)
, n even and j =
n
2
,
qn−1(j) + (2an − 1)pn−1(j − 1)
2an
, else
(2.3)
for n ∈ N and j ∈ {0, . . . , ⌊n/2⌋}, where we set
pn−1(−1) := 0, n ∈ N. (2.4)
The following theorem uses the sequences (pn)n∈N0 and (qn)n∈N0 to obtain the desired expansions
of the sieved polynomials (Pn(x; k))n∈N0 in the basis {Tn(x) : n ∈ N0}. Moreover, the theorem
provides a possibility to compute (pn)n∈N0 and (qn)n∈N0 directly from (rn)n∈N0 . To avoid case
differentiations, we define
q2n−1(n) := 0, n ∈ N. (2.5)
Theorem 2.1. For every k ∈ N, n ∈ N0 and i ∈ {0, . . . , k − 1}, one has
Pkn+i(x; k) =
⌊n
2
⌋∑
j=0
[pn(j)Tkn−2jk−i(x) + qn(j)Tkn−2jk+i(x)]. (2.6)
Moreover, one has
rn(j) = pn−1(j − 1) + qn−1(j), n ∈ N, j ∈
{
0, . . . ,
⌊n
2
⌋}
, (2.7)
qn(j) = rn(j)− pn(j), n ∈ N0, j ∈
{
0, . . . ,
⌊n
2
⌋}
, (2.8)
pn(j) =
j∑
i=0
[rn(i)− rn+1(i)], n ∈ N0, j ∈
{
0, . . . ,
⌊n
2
⌋}
. (2.9)
Concerning the expansions provided by Theorem 2.1, it is very remarkable that the coeffi-
cients of Tkn−2jk−i(x) and Tkn−2jk+i(x) do not rely on i, nor do they rely on k. Concerning well-
definedness in (2.6), note that the “polynomials” T−(k−1)(x), . . . , T−1(x) are not defined; how-
ever, they only occur for even n and together with a multiplication with pn(⌊n/2⌋) = pn(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 ∈ N0. It is clear from the recurrence relation for the Chebyshev polynomials of the first kind
that Pi(x; k) = Ti(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
Pkn−2(x; k) =
⌊n−1
2
⌋∑
j=0
[pn−1(j)Tkn−2jk−2k+2(x) + qn−1(j)Tkn−2jk−2(x)].
Expansions and Characterizations of Sieved Random Walk Polynomials 7
Due to (2.4) and (2.5), the latter equation can be rewritten as
Pkn−2(x; k) =
⌊n
2
⌋∑
j=0
[pn−1(j − 1)Tkn−2jk+2(x) + qn−1(j)Tkn−2jk−2(x)]. (2.10)
In the same way, we obtain
Pkn−1(x; k) =
⌊n
2
⌋∑
j=0
[pn−1(j − 1)Tkn−2jk+1(x) + qn−1(j)Tkn−2jk−1(x)]. (2.11)
We now use that Pkm(x; k) = Pm(Tk(x)), m ∈ N0 [7, Theorem 1, Section VI], which yields
Pkn(x; k) =
⌊n
2
⌋∑
j=0
rn(j)Tn−2j(Tk(x)) =
⌊n
2
⌋∑
j=0
rn(j)Tkn−2jk(x). (2.12)
Combining (2.10), (2.11) and (2.12) with the relation 2xPkn−1(x; k) = Pkn(x; k) + Pkn−2(x; k)
and using the recurrence relation for the Chebyshev polynomials of the first kind, we obtain
that rn(j) = pn−1(j − 1) + qn−1(j) for each j ∈ {0, . . . , ⌊n/2⌋}. Since
pn−1(j − 1) + qn−1(j) = pn(j) + qn(j)
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
Pkn(x; k) =
⌊n
2
⌋∑
j=0
[pn(j) + qn(j)]Tkn−2jk(x). (2.13)
We now combine (2.11) with (2.13), write
2xTkn−2jk(x) = Tkn−2jk+1(x) + T|kn−2jk−1|(x), j ∈
{
0, . . . ,
⌊n
2
⌋}
,
use (2.2), (2.3) and definition (2.5) again and obtain
2anPkn+1(x; k) = 2xPkn(x; k)− 2cnPkn−1(x; k)
=
⌊n
2
⌋∑
j=0
[pn(j) + qn(j)− 2cnpn−1(j − 1)]Tkn−2jk+1(x)
+
⌊n
2
⌋∑
j=0
[pn(j) + qn(j)− 2cnqn−1(j)]T|kn−2jk−1|(x)
= 2an
⌊n
2
⌋∑
j=0
[pn(j)Tkn−2jk−1(x) + qn(j)Tkn−2jk+1(x)].
Thus if k = 2, then (2.6) is shown. If k ≥ 3, we use induction on i to prove that
Pkn+i(x; k) =
⌊n
2
⌋∑
j=0
[pn(j)Tkn−2jk−i(x) + qn(j)Tkn−2jk+i(x)], i ∈ {0, . . . , k − 1} (2.14)
8 S. Kahler
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
Pkn+i+2(x; k) = 2xPkn+i+1(x; k)− Pkn+i(x; k)
=
⌊n
2
⌋∑
j=0
[pn(j)Tkn−2jk−i−2(x) + qn(j)Tkn−2jk+i+2(x)].
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 ∈ N0 and j ∈ {0, . . . , ⌊n/2⌋}. By (2.7) and (2.8), we have
rn(i)− rn+1(i) = pn(i)− pn(i− 1)
for all i ∈ {0, . . . , j}. Taking the sum from 0 to j and using definition (2.4), we get
j∑
i=0
[rn(i)− rn+1(i)] =
j∑
i=0
[pn(i)− pn(i− 1)] = pn(j)
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 {Tn(x) : n ∈ N0}:
Example 2.2 (sieved ultraspherical polynomials). Let Pn(x) = P
(α)
n (x), n ∈ N0, 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 (Tn(x))n∈N0 and is therefore trivial, so let α ̸= −1/2
from now on. Then (rn)n∈N0 is given by [8, Theorem 9.1.1]
rn(j) =
(
n
n
2
)(
α+ 1
2
)2
n
2
(2α+ 1)n
, n even and j = n
2 ,
2
(
n
j
)(
α+ 1
2
)
j
(
α+ 1
2
)
n−j
(2α+ 1)n
, else.
(2.15)
Applying (2.9), via induction on j we see that the relation between (pn)n∈N0 and (rn)n∈N0
becomes especially easy and reads
pn(j) =
0, n even and j = n
2 ,
2j + 2α+ 1
2n+ 4α+ 2
rn(j), else.
(2.16)
Theorem 2.1, (2.15) and (2.16) allow an explicit computation of the sieved ultraspherical poly-
nomials
(
P
(α)
n (x; k)
)
n∈N0
.
3 Characterizations via the sieved operators
Let (Pn(x))n∈N0 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
Dk (1.7) and Ak (1.8): for each n ∈ N0, we define mappings κn(·; k), αn(·; k) : N0 → R by the
projections
κn(j; k) :=
∫
R
DkPn(x)Pj(x)dµ(x), (3.1)
αn(j; k) :=
∫
R
AkPn(x)Pj(x)dµ(x). (3.2)
Expansions and Characterizations of Sieved Random Walk Polynomials 9
In other words, κn(·; k) and αn(·; k) correspond to the expansions
DkPn(x) =
n−1∑
j=0
κn(j; k)Pj(x)h(j), κn(j; k) = 0, j ≥ n, (3.3)
AkPn(x) =
n∑
j=0
αn(j; k)Pj(x)h(j), αn(j; k) = 0, j ≥ n+ 1. (3.4)
Due to the symmetry of (Pn(x))n∈N0 , we have κn(j; k) = 0 if n − j is even, and we have
αn(j; k) = 0 if n−j is odd. Furthermore, note that the functions κ0(·; k), . . . , κn+1(·; k), α0(·; k),
. . . , αn(·; k) and αn+1(·; k)|{0,...,n} are uniquely determined by the recurrence coefficients c1,
. . . , cn. For brevity, we define σ(·; k) : N → R,
σ(n; k) := κn(n− 1; k). (3.5)
We come to two further main results and give characterizations in terms of the sieved opera-
tors Ak and Dk. Theorem 3.1 is the sieved analogue to Theorem 1.3. Theorem 3.2 is the sieved
analogue to the Lasser–Obermaier result Theorem 1.1 and Ismail–Obermaier result Theorem 1.2.
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:
(i) (Pn(x))n∈N0 is k-sieved,
(ii) for each n ∈ N0, Pn(x) is an eigenvector of Ak,
(iii) αn+1(n− 1; k) = 0, n ∈ N.
If these equivalent conditions are satisfied, then Pn(| cos(π/k)|) = Tn(| cos(π/k)|) is the eigen-
value of Ak which corresponds to the eigenvector Pn(x), n ∈ N0.
Theorem 3.2. If k ≥ 2, then the following are equivalent:
(i) (Pn(x))n∈N0 is k-sieved,
(ii) DkPn(x) = DkPn(1)P
∗
n−1(x), n ∈ N,
(iii)
(
1− x2
)
DkPn(x) is orthogonal to P0(x), . . . , Pn−2(x), n ≥ 2,
(iv) one has
κn+2(n− 1; k) = σ(n+ 2; k), n ∈ N,
and for every n ∈ N there is an m ∈ {0, . . . , ⌊(n− 1)/2⌋} such that
κn+4(n− 1− 2m; k) = σ(n+ 4; k).
If k = 1, then (ii), (iii) and (iv) are equivalent to
(i′) Pn(x) = P
(1/(2c1)−3/2)
n (x), n ∈ N0.
The characterization provided by (iii) of the previous theorem has the advantage that it
is “stable” with respect to renormalization of the sequence (Pn(x))n∈N0 . The characterization
provided by (iv) is the strongest one, however, because the functions κn(·; k) (3.1) (3.3) have to
be considered just at some carefully chosen points.
10 S. Kahler
Note that the formal limits “D∞” and “A∞” are included in our investigations because they
coincide with D1 = d/dx and A1 = id.
Before coming to the proofs, we study some basic properties of Dk and Ak. We will make
use of the following well-known identities [1, formulas (22.7.25)–(22.7.28)]:(
2x2 − 2
)
Un−1(x) = Tn+1(x)− Tn−1(x), n ∈ N, (3.6)
2Tm(x)Un−1(x) = Um+n−1(x) + Un−m−1(x), m, n ∈ N0, m ≤ n, (3.7)
2Tm(x)Un−1(x) = Um+n−1(x)− Um−n−1(x), m, n ∈ N0, m ≥ n. (3.8)
The following lemma deals with the function σ(.; k) (3.5) and with special values of the functions
αn(.; 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
(i) αn(n; k) = Tn(| cos(π/k)|)/h(n), n ∈ N0,
(ii) the equation
αn(n− 2; k) =
Un−2
(∣∣cos (πk )∣∣) sin2 (πk ) (n− 4
∑n−1
j=1 aj−1cj
)
2cn−1cnh(n)
holds for all n ≥ 2,
(iii) σ(n; k) = Un−1(| cos(π/k)|)/(cnh(n)), n ∈ N.
Proof. (i) If one expands Pn(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
⌊n
2
⌋∑
j=1
rn(j)Tn−2j
(∣∣∣cos(π
k
)∣∣∣)Tn−2j(x)
= AkPn(x)− rn(0)Tn
(∣∣∣cos(π
k
)∣∣∣)Tn(x)
=
[
Tn
(∣∣∣cos(π
k
)∣∣∣) rn(1) + αn(n− 2; k)h(n− 2)rn−2(0)
]
Tn−2(x) +R(x)
for some R(x) ∈ R[x] with degR(x) ≤ n − 3; thus a comparison of the coefficients of Tn−2(x)
and (3.6) yield
αn(n− 2; k)h(n− 2)rn−2(0) =
[
Tn−2
(∣∣∣cos(π
k
)∣∣∣)− Tn
(∣∣∣cos(π
k
)∣∣∣)] rn(1)
= 2Un−2
(∣∣∣cos(π
k
)∣∣∣) sin2
(π
k
)
rn(1).
Finally, since
4an−2an−1rn(1) =
[
n− 4
n−1∑
j=1
aj−1cj
]
rn−2(0)
(see the proof of [10, Lemma 5.1]) and an−2an−1h(n− 2) = cn−1cnh(n), we obtain the assertion.
(iii) While on the one hand one has
DkPn(x) =
⌊n
2
⌋∑
j=0
rn(j)Un−2j−1
(∣∣∣cos(π
k
)∣∣∣)Un−2j−1(x)
Expansions and Characterizations of Sieved Random Walk Polynomials 11
by (1.7), on the other hand one has
DkPn(x) = κn(n− 1; k)h(n− 1)rn−1(0)Tn−1(x) +R(x)
=
κn(n− 1; k)h(n− 1)rn−1(0)
2− δn,1
Un−1(x) + S(x)
with polynomials R(x), S(x) ∈ R[x] with degR(x),degS(x) ≤ n− 2. Consequently, we have
rn(0)Un−1
(∣∣∣cos(π
k
)∣∣∣) =
κn(n− 1; k)h(n− 1)rn−1(0)
2− δn,1
,
and as obviously rn−1(0) = (2−δn,1)an−1rn(0) and an−1h(n−1) = cnh(n), the proof is complete
(note that, by definition, κn(n− 1; k) = σ(n; k)). ■
We also investigate the product rule for the sieved Askey–Wilson operator Dk. Its analogue
for Dq has the same structure (see [8, 10]).
Lemma 3.4. One has
Dk[P (x)Q(x)] = DkP (x)AkQ(x) + AkP (x)DkQ(x), P (x), Q(x) ∈ R[x].
Consequently,
anκn+1(j; k) + cnκn−1(j; k)
=
∣∣∣cos(π
k
)∣∣∣ [ajκn(j + 1; k) + cjκn(j − 1; k)] + αn(j; k), n, j ∈ N0. (3.9)
Proof. Due to linearity, it clearly suffices to establish that
Dk[Tm(x)Tn(x)] = DkTm(x)AkTn(x) + AkTm(x)DkTn(x), m, n ∈ N0.
This, however, can easily be seen from the equations (1.7), (1.8), (3.7) and (3.8) by the compu-
tation
DkTm(x)AkTn(x) + AkTm(x)DkTn(x)
= Tn
(∣∣∣cos(π
k
)∣∣∣)Um−1
(∣∣∣cos(π
k
)∣∣∣)Tn(x)Um−1(x)
+ Tm
(∣∣∣cos(π
k
)∣∣∣)Un−1
(∣∣∣cos(π
k
)∣∣∣)Tm(x)Un−1(x)
=
1
4
[
Um+n−1
(∣∣∣cos(π
k
)∣∣∣)− Un−m−1
(∣∣∣cos(π
k
)∣∣∣)] [Um+n−1(x)− Un−m−1(x)]
+
1
4
[
Um+n−1
(∣∣∣cos(π
k
)∣∣∣)+ Un−m−1
(∣∣∣cos(π
k
)∣∣∣)] [Um+n−1(x) + Un−m−1(x)]
=
1
2
Um+n−1
(∣∣∣cos(π
k
)∣∣∣)Um+n−1(x) +
1
2
Un−m−1
(∣∣∣cos(π
k
)∣∣∣)Un−m−1(x)
=
1
2
DkTm+n(x) +
1
2
DkTn−m(x) = Dk
[
1
2
Tm+n(x) +
1
2
Tn−m(x)
]
= Dk[Tm(x)Tn(x)]
for m ≤ n (the expansion Tm(x)Tn(x) = Tm+n(x)/2 + Tn−m(x)/2 is well known). Now let
n, j ∈ N0. Via (1.1), we compute
anDkPn+1(x) + cnDkPn−1(x) = Dk[xPn(x)]
= Dk[x]AkPn(x) + Ak[x]DkPn(x)
=
∣∣∣cos(π
k
)∣∣∣xDkPn(x) + AkPn(x);
multiplication with Pj(x), integration with respect to µ and the equations (3.1) and (3.2) yield
the second assertion. ■
12 S. Kahler
The recurrence relation (3.9) for (κn(·; k))n∈N0 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.
Proof of Theorem 3.1. The case k = 1 is trivial because A1 = id, so let k ≥ 2 from now on (in
particular, we then have | cos(π/k)| = cos(π/k)). “(i) ⇒ (ii)”: let n ∈ N0 and i ∈ {0, . . . , k− 1}.
Using Theorem 2.1 and (1.8), we have
Tkn+i
(
cos
(π
k
))
Pkn+i(x; k)−AkPkn+i(x; k)
=
⌊n
2
⌋∑
j=0
pn(j)
[
Tkn+i
(
cos
(π
k
))
− Tkn−2jk−i
(
cos
(π
k
))]
Tkn−2jk−i(x)
+
⌊n
2
⌋∑
j=0
qn(j)
[
Tkn+i
(
cos
(π
k
))
− Tkn−2jk+i
(
cos
(π
k
))]
Tkn−2jk+i(x).
For each j ∈ {0, . . . , ⌊n/2⌋} (except the case n even and j = n/2, but then pn(j) = 0), we
compute
Tkn+i
(
cos
(π
k
))
− Tkn−2jk−i
(
cos
(π
k
))
= cos
(
(kn+ i)π
k
)
− cos
(
(kn− 2jk − i)π
k
)
= −2 sin((n− j)π) sin
(
(jk + i)π
k
)
= 0.
In the same way, we have
Tkn+i
(
cos
(π
k
))
− Tkn−2jk+i
(
cos
(π
k
))
= 0
for all j ∈ {0, . . . , ⌊n/2⌋}. Hence, we see that Pkn+i(x; k) is an eigenvector of Ak which
corresponds to the eigenvalue Tkn+i(cos(π/k)). “(ii) ⇒ (iii)” is trivial from orthogonality.
“(iii) ⇒ (i)”: the condition and Lemma 3.3 yield
4
n∑
j=1
aj−1cj = n+ 1 if k ̸ |n ∈ N. (3.10)
It is immediate from (3.10) that c1 = 1/2. Let n ∈ N be such that k does not divide n + 1.
Moreover, assume that, for each j ∈ {1, . . . , n}, cj = 1/2 if j is not a multiple of k. Decomposing
n+ 1 = kl + i with unique l ∈ N0, i ∈ {1, . . . , k − 1}, on the one hand, we get
kl + i+ 1 = 4
kl+i∑
j=1
aj−1cj = 4
kl∑
j=1
aj−1cj + 4
kl+i∑
j=kl+1
aj−1cj = kl + 2ckl + 4
kl+i∑
j=kl+1
aj−1cj
from (3.10), which then simplifies to
ckl
2
+
kl+i∑
j=kl+1
aj−1cj =
i+ 1
4
. (3.11)
On the other hand, we compute
ckl
2
+
kl+i∑
j=kl+1
aj−1cj =
ckl
2
+ aklckl+1, i = 1,
i
4
+
ckl+i
2
, else.
(3.12)
Expansions and Characterizations of Sieved Random Walk Polynomials 13
Combining (3.11) and (3.12), we obtain that cn+1 = ckl+i = 1/2. This finishes the proof of
“(iii) ⇒ (i)”. Concerning the remaining assertion, it suffices to show that
Pn(cos(π/k); k) = Tn(cos(π/k)), n ∈ N0;
this can be seen as follows: let l ∈ N0. Since Tkl(cos(π/k)) = (−1)l, and since
Tkl−1
(
cos
(π
k
))
= (−1)l cos
(π
k
)
= Tkl+1
(
cos
(π
k
))
for l ̸= 0, we have
cos
(π
k
)
Tkl
(
cos
(π
k
))
= alTkl+1
(
cos
(π
k
))
+ clTkl−1
(
cos
(π
k
))
.
Consequently, the recurrence relations for the sequences
(Pn(cos(π/k); k))n∈N0 and (Tn(cos(π/k)))n∈N0
coincide. ■
Proof of Theorem 3.2. First, note that in view of (1.2) (iii) is an obvious reformulation of (ii).
Moreover, the case k = 1 is just Theorem 1.1 because D1 = d/dx (concerning (iv), we refer to
the proof given in [15]), so let k ≥ 2 from now on. “(i) ⇒ (ii)”: we know from Theorem 3.1
“(i) ⇒ (ii)” and Lemma 3.4 that
anκn+1(j; k) + cnκn−1(j; k)
= cos
(π
k
)
[ajκn(j + 1; k) + cjκn(j − 1; k)] + Tn
(
cos
(π
k
)) δn,j
h(n)
, n, j ∈ N0, (3.13)
and we now use induction on n to deduce that
κn(n− 1− 2j; k) =
Un−1
(
cos
(
π
k
))
cnh(n)
, j ∈
{
0, . . . ,
⌊
n− 1
2
⌋}
(3.14)
for each n ∈ N, which implies the assertion due to (1.2) and (3.3). It is obvious from Lemma 3.3
that (3.14) is satisfied for n ∈ {1, 2}, so let n ∈ N\{1} be arbitrary but fixed now and as-
sume (3.14) to hold for both n− 1 and n. If j ∈ {1, . . . , ⌊n/2⌋}, then (3.13) implies
cn+1h(n+ 1)κn+1(n− 2j; k) = cos
(π
k
) Un−1
(
cos
(
π
k
))
cn
− Un−2
(
cos
(π
k
)) an−1
cn−1
, (3.15)
and we distinguish two cases: if k ̸ |n, then (3.15) yields
cn+1h(n+ 1)κn+1(n− 2j; k) = 2 cos
(π
k
)
Un−1
(
cos
(π
k
))
− Un−2
(
cos
(π
k
))
because an−1 = cn−1 = 1/2 (if k ̸ | (n − 1)) or Un−2(cos(π/k)) = 0 (if k|(n − 1)), which, by the
recurrence relation for the Chebyshev polynomials of the second kind, simplifies to
cn+1h(n+ 1)κn+1(n− 2j; k) = Un
(
cos
(π
k
))
.
If k|n, however, we compute from (3.15)
cn+1h(n+ 1)κn+1(n− 2j; k) = −Un−2
(
cos
(π
k
))
= Un
(
cos
(π
k
))
.
14 S. Kahler
If one takes into account Lemma 3.3 for the remaining case “j = 0” again, the proof of the
direction “(i) ⇒ (ii)” is finished. “(ii) ⇒ (iv)” is trivial from (1.2). The direction “(iv) ⇒ (i)”
is more involved; we first deal with c1 and c2: due to the conditions κ3(0; k) = σ(3; k) and
κ4(1; k) = σ(4; k), a tedious but straightforward calculation based on Lemmas 3.3 and 3.4 yields
U2
(
cos
(π
k
))
[1− 4c1 + 4c1c2] = 3− 4c1 − 4c2 + 4c1c2 (3.16)
and
cos
(π
k
)
U2
(
cos
(π
k
))
[3− 4c1 − 8c2 + 4c1c2 + 8c2c3]
= cos
(π
k
)
[9− 12c1 − 12c2 − 8c3 + 12c1c2 + 16c2c3].
Combining these equations, we obtain
cos
(π
k
)
U2
(
cos
(π
k
))
[c1 − c2 − c1c2 + c2c3] = cos
(π
k
)
[−c3 + 2c2c3]. (3.17)
To get another equation for c1, c2, c3, we first note that Lemma 3.3 and (3.6) imply that
AkP4(x) = T4
(
cos
(π
k
))
P4(x)
+
[
T2
(
cos
(π
k
))
− T4
(
cos
(π
k
))] a1 − c3
a3
P2(x) + α4(0; k)
= T4
(
cos
(π
k
))
[P4(x)− 1]
+
[
T2
(
cos
(π
k
))
− T4
(
cos
(π
k
))] a1 − c3
a3
[P2(x)− 1] + AkP4(1),
which, since
P4(x) =
1
8a1a2a3
T4(x) +
a1 − c3
2a1a3
T2(x) +
8a1a2a3 − 4a1a2 + 4a2c3 − 1
8a1a2a3
due to (1.1), becomes
AkP4(x) = T4
(
cos
(π
k
))
P4(x) +
[
T2
(
cos
(π
k
))
− T4
(
cos
(π
k
))] a1 − c3
a3
P2(x)
+
[
T4
(
cos
(π
k
))
− 1
] [ 1
8a1a2a3
− 1
]
−
[
1− T2
(
cos
(π
k
))] a1 − c3
2a1a3
−
[
T2
(
cos
(π
k
))
− T4
(
cos
(π
k
))] a1 − c3
a3
(3.18)
after another tedious but straightforward calculation which uses (1.8). We then take into ac-
count that Lemma 3.4 and the conditions κ3(0; k) = σ(3; k), κ4(1; k) = σ(4; k) and κ5(0; k) =
κ5(2; k)(= σ(5; k)) yield α4(0; k) = α4(2; k), which can be rewritten as
U2
(
cos
(π
k
)) [
1− 8c1 + 8c21 + 16c1c2 − 16c21c2 − 8c1c
2
2 − 8c1c2c3 + 8c21c
2
2 + 8c1c
2
2c3
]
= 3− 4c1 − 4c2 − 4c3 + 4c1c2 + 8c1c3 + 4c2c3 − 8c1c2c3 (3.19)
as a consequence of (3.18) and the simplifications
T2
(
cos
(π
k
))
− T4
(
cos
(π
k
))
= 2U2
(
cos
(π
k
))
sin2
(π
k
)
,
T4
(
cos
(π
k
))
− 1 = −
[
2U2
(
cos
(π
k
))
+ 2
]
sin2
(π
k
)
,
1− T2
(
cos
(π
k
))
= 2 sin2
(π
k
)
.
Expansions and Characterizations of Sieved Random Walk Polynomials 15
Now, the combination of (3.16) with (3.19) gives
U2
(
cos
(π
k
)) [
c1 − 2c21 − 3c1c2 + 4c21c2 + 2c1c
2
2 + 2c1c2c3 − 2c21c
2
2 − 2c1c
2
2c3
]
= c3 − 2c1c3 − c2c3 + 2c1c2c3;
dividing this by 1− c2, we finally obtain
U2
(
cos
(π
k
)) [
c1 − 2c21 − 2c1c2 + 2c21c2 + 2c1c2c3
]
= c3 − 2c1c3. (3.20)
From now on, we consider the nonlinear system (3.16), (3.17), (3.20), which depends on k. If
k = 2, then (3.16) immediately gives c1 = 1/2 because U2(0) = −1. In the following, let k ≥ 3;
here, cos(π/k) ̸= 0 and (3.17) reduces to
U2
(
cos
(π
k
))
[c1 − c2 − c1c2 + c2c3] = −c3 + 2c2c3. (3.21)
Combining (3.16) and (3.20), we see that
[3− 4c1 − 4c2 + 4c1c2]
[
c1 − 2c21 − 2c1c2 + 2c21c2 + 2c1c2c3
]
= U2
(
cos
(π
k
))
[1− 4c1 + 4c1c2]
[
c1 − 2c21 − 2c1c2 + 2c21c2 + 2c1c2c3
]
= [1− 4c1 + 4c1c2][c3 − 2c1c3]. (3.22)
In the same way combining (3.16), (3.21) on the one hand and (3.20), (3.21) on the other hand,
we get
[3− 4c1 − 4c2 + 4c1c2][c1 − c2 − c1c2 + c2c3] = [1− 4c1 + 4c1c2][−c3 + 2c2c3],
[c3 − 2c1c3][c1 − c2 − c1c2 + c2c3] =
[
c1 − 2c21 − 2c1c2 + 2c21c2 + 2c1c2c3
]
[−c3 + 2c2c3],
which can be rewritten as[
1− 4c1 + c2 + 8c1c2 − 4c22 − 4c1c
2
2
]
c3
= −3c1 + 3c2 + 4c21 + 3c1c2 − 4c22 − 8c21c2 + 4c21c
2
2, (3.23)[
c2 − 4c1c
2
2
]
c3 = −2c1 + c2 + 4c21 + 3c1c2 − 8c21c2 − 4c1c
2
2 + 4c21c
2
2. (3.24)
Moreover, we can deduce from (3.23) and (3.24) that[
− 3c1 + 3c2 + 4c21 + 3c1c2 − 4c22 − 8c21c2 + 4c21c
2
2
][
c2 − 4c1c
2
2
]
=
[
1− 4c1 + c2 + 8c1c2 − 4c22 − 4c1c
2
2
]
c3
[
c2 − 4c1c
2
2
]
=
[
1− 4c1 + c2 + 8c1c2 − 4c22 − 4c1c
2
2
]
×
[
− 2c1 + c2 + 4c21 + 3c1c2 − 8c21c2 − 4c1c
2
2 + 4c21c
2
2
]
,
which now considerably simplifies to
[4c1c2 − 4c1 + 1][2c1c2 − 2c1 + c2][2c2 − 1][2c1 − 1] = 0.
If 4c1c2 − 4c1 + 1 = 0, it is immediate from (3.16) that c2 = 1/2. If 2c1c2 − 2c1 + c2 = 0,
then (3.16) reads U2(cos(π/k))[1−2c2] = 3−6c2, and we can conclude that c2 = 1/2, too
(
because
U2(cos(π/k)) = 4(cos(π/k))2−1 < 3
)
. If c2 = 1/2, then (3.24) reduces to (1−2c1)(1−c1−c3) = 0,
and if additionally c3 = 1 − c1, then (3.22) implies that c1 = 1/2. Finally, if c1 = 1/2,
then c2 = 1/2 because otherwise (3.16) would yield −1=U2(cos(π/k))=sin(3π/k)/ sin(π/k)≥0.
Therefore, allowing k ≥ 2 to be arbitrary again, we have seen that c1 = 1/2 in each possible
16 S. Kahler
case, and that c2 = 1/2 if k ̸= 2. Now let n ∈ N\{1} be arbitrary but fixed and assume that,
for all j ∈ {1, . . . , n}, cj = 1/2 if j is not a multiple of k. Lemma 3.4 and the conditions
κn+1(n− 2; k) = σ(n+ 1; k) and κn+2(n− 1; k) = σ(n+ 2; k) yield
an+1σ(n+ 2; k) + cn+1σ(n; k) = cos
(π
k
)
σ(n+ 1; k) + αn+1(n− 1; k). (3.25)
Since αn+1(n − 1; k) is uniquely determined by the recurrence coefficients c1, . . . , cn (which is
already a consequence of (3.4)), we have αn+1(n − 1; k) = 0 by Theorem 3.1 “(i) ⇒ (ii)” and
the induction hypothesis.8 Therefore, (3.25) simplifies to
an+1σ(n+ 2; k) + cn+1σ(n; k) = cos
(π
k
)
σ(n+ 1; k),
and then multiplication with ancnh(n), Lemma 3.3 and the recurrence relation for the Chebyshev
polynomials of the second kind imply that
cos
(π
k
)
Un
(
cos
(π
k
))
(1− 2cn+1)cn = Un−1
(
cos
(π
k
))
(1− 2cn)cn+1. (3.26)
If k ≥ 3 and if n+1 is not a multiple of k, then cos(π/k)Un(cos(π/k)) ̸= 0; consequently, (3.26)
tells cn+1 = 1/2 because Un−1(cos(π/k)) = 0 or cn = 1/2, depending on whether k divides n or
not. The case k = 2 is more involved—here, (3.26) does not allow for any conclusion. Instead, we
apply an idea which we similarly used in [13] (at that time for the operators Dq andAq; cf. partic-
ularly [13, Lemma 3.1, proof of Theorem 2.3]) and proceed as follows: by the conditions, there is
some m ∈ {0, . . . , ⌊(n− 2)/2⌋} such that κn+3(n− 2− 2m; 2) = σ(n+ 3; 2). Since c1, . . . , cn fix
the first n+2 polynomials P0(x), . . . , Pn+1(x), c1, . . . , cn yield unique λ0, . . . , λn+2, ν0, . . . , νn−1 ∈
R such that
A2[xPn+1(x)] = λ0 + x
n+1∑
j=0
λj+1Pj(x)
= λ0 + λ2c1 +
n∑
j=1
[λjaj−1 + λj+2cj+1]Pj(x)
+ λn+1anPn+1(x) + λn+2an+1Pn+2(x)
=
n−1∑
j=0
νjPj(x) + [λnan−1 + λn+2cn+1]Pn(x)
+ λn+1anPn+1(x) + λn+2an+1Pn+2(x).
Therefore, c1, . . . , cn determine the integral∫
R
A2[xPn+1(x)]Pn−2−2m(x) dµ(x) =
νn−2−2m
h(n− 2− 2m)
uniquely, so (1.1) and Theorem 3.1 “(i) ⇒ (ii)” imply that∫
R
A2[xPn+1(x)]Pn−2−2m(x) dµ(x) = 0
as a consequence of the induction hypothesis. Hence, using Theorem 3.1 “(i) ⇒ (ii)” again, (1.1)
and the fact that αn(n− 2− 2m; 2) is uniquely determined by c1, . . . , cn, too (use (3.4)), we see
that
0 = an+1αn+2(n− 2− 2m; 2) + cn+1αn(n− 2− 2m; 2) = an+1αn+2(n− 2− 2m; 2). (3.27)
8Such arguments will occur several times in our proof, so we give some more details at this stage: compare
(Pj(x))j∈N0 to the sequence
(
P̃j(x)
)
j∈N0
which is given by c̃j := cj for j ∈ {1, . . . , n} and c̃j := 1/2 otherwise.
Then
(
P̃j(x)
)
j∈N0
is k-sieved, so 0 = α̃n+1(n−1; k) = αn+1(n−1; k). We used similar ideas also in our paper [13].
Expansions and Characterizations of Sieved Random Walk Polynomials 17
Furthermore, we have
αn+2(n− 2− 2m; 2) = αn+2(n; 2), (3.28)
which can be seen as follows: on the one hand, since κn+1(n− 2− 2m; 2) is uniquely determined
by c1, . . . , cn (this is a consequence of (3.3)), one has κn+1(n− 2− 2m; 2) = σ(n+ 1; 2) by the
already established direction “(i) ⇒ (ii)”; hence, by Lemma 3.4 we have
an+2σ(n+ 3; 2) + cn+2σ(n+ 1; 2) = αn+2(n− 2− 2m; 2).
On the other hand, by Lemma 3.4 and the condition κn+3(n; 2) = σ(n+ 3; 2), we have
an+2σ(n+ 3; 2) + cn+2σ(n+ 1; 2) = αn+2(n; 2),
so (3.28) is established. Combining (3.28) with (3.27), we obtain that
αn+2(n; 2) = 0.
Now following the proof of Theorem 3.1 “(iii) ⇒ (i)”, we can conclude that cn+1 = 1/2 if n+ 1
is odd, which finishes the proof of Theorem 3.2. ■
Remark 3.5. We note that the implication “(iv) ⇒ (i)” of Theorem 3.2 remains valid if (iv) is
weakened in the following way:
� If k ∈ {1, 2}, it suffices to require that κ2n+1(2n − 2; k) = σ(2n + 1; k) for all n ∈ N and
that for every n ∈ N there is an m ∈ {0, . . . , n− 1} such that κ2n+3(2m; k) = σ(2n+3; k).
For k = 1 (which yields a characterization of the ultraspherical polynomials), this is
a consequence of [13, Theorem 2.1]. The case k = 2, however, is obvious from Theorem 3.2
because D2P2n(x) = 0 (n ∈ N0) by (1.7) without any further restriction.
� If k ≥ 3, one can drop the requirement “there is an m ∈ {0, . . . , ⌊(n− 1)/2⌋} such that
κn+4(n− 1− 2m; k) = σ(n+ 4; k)” for all n ≥ 2. This can be seen from the proof above.
Acknowledgements
The research was begun when the author worked at Technical University of Munich, and the
author gratefully acknowledges support from the graduate program TopMath of the ENB (Elite
Network of Bavaria) and the TopMath Graduate Center of TUM Graduate School at Technical
University of Munich. The research was continued and completed at RWTH Aachen University.
The author also thanks the referees for carefully reading the manuscript, as well as for their
valuable comments.
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1 Introduction
2 Expansions of sieved polynomials in the Chebyshev basis
3 Characterizations via the sieved operators
References
|
| id | nasplib_isofts_kiev_ua-123456789-212028 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T23:59:24Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kahler, Stefan 2026-01-23T10:08:18Z 2023 Expansions and Characterizations of Sieved Random Walk Polynomials. Stefan Kahler. SIGMA 19 (2023), 103, 18 pages 1815-0659 2020 Mathematics Subject Classification: 42C05; 33C47; 42C10 arXiv:2306.16411 https://nasplib.isofts.kiev.ua/handle/123456789/212028 https://doi.org/10.3842/SIGMA.2023.103 We consider random walk polynomial sequences (ₙ())ₙ∈ℕ₀ ⊆ ℝ[] given by recurrence relations ₀() = 1, ₁() = , ₙ() = (1−cₙ)ₙ₊₁()+cₙₙ₋₁(), ∈ ℕ with (cₙ)ₙ∈ℕ ⊆ (0, 1). For every ∈ ℕ, the -sieved polynomials (ₙ(; ))ₙ∈ℕ₀ arise from the recurrence coefficients c(; ):= cₙ/ₖ if | and c(; ):= 1/2 otherwise. A main objective of this paper is to study expansions in the Chebyshev basis {Tₙ(): n ∈ℕ₀}. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version Dₖ of the Askey-Wilson operator . It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative, and obtained from by letting approach a -th root of unity. However, for ≥ 2, the new operator Dₖ on ℝ[] has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for -sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator Aₖ. The research was begun when the author worked at the Technical University of Munich, and the author gratefully acknowledges support from the graduate program TopMath of the ENB (Elite Network of Bavaria) and the TopMath Graduate Center of TUM Graduate School at the Technical University of Munich. The research was continued and completed at RWTH Aachen University. The author also thanks the referees for carefully reading the manuscript, as well as for their valuable comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Expansions and Characterizations of Sieved Random Walk Polynomials Article published earlier |
| spellingShingle | Expansions and Characterizations of Sieved Random Walk Polynomials Kahler, Stefan |
| title | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_full | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_fullStr | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_full_unstemmed | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_short | Expansions and Characterizations of Sieved Random Walk Polynomials |
| title_sort | expansions and characterizations of sieved random walk polynomials |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212028 |
| work_keys_str_mv | AT kahlerstefan expansionsandcharacterizationsofsievedrandomwalkpolynomials |