Dimeric and monomeric palladium(II) parent-amido (NH2) complexes: Synthesis, characterization, and reactivity

Article abstract of DOI:10.1016/j.crci.2015.05.007

The treatment of [PdL₃(NH₃)]OTf (L₃Combining double low line(PEt₃)₂(Ph) (1), (2,6-(Cy₂PCH₂)₂C₆H₃) (3)) with NaNH₂ in THF afforded dimeric and monomeric parent-amido palladium(II) complexes with bridging and terminal NH₂, respectively, anti-[Pd(PEt₃)(Ph)(1/4-NH₂)]₂ (2) and Pd(2,6-(Cy₂PCH₂)₂C₆H₃)(NH₂) (4). The dimeric complex 2 crystallizes in the space group P2₁/n with aCombining double low line13.228(2) ?, bCombining double low line18.132(2) ?, cCombining double low line24.745(2) ?, βCombining double low line101.41(1)°, and ZCombining double low line4. It has been found that there are two crystallographically independent molecules with Pd(1)-Pd(2) and Pd(3)-Pd(4) distances of 2.9594 (10) and 2.9401(9) ?, respectively. The monomeric amido complex 4 protonates from trace amounts of water to give the cationic ammine species [Pd(2,6-(Cy₂PCH₂)₂C₆H₃)(NH₃)]⁺. Complex 4 reacts with diphenyliodonium triflate ([Ph₂I]OTf) to give aniline complex [Pd(2,6-(Cy₂PCH₂)₂C₆H₃)(NH₂Ph)]OTf (5). Reaction of 4 with dialkyl acetylenedicarboxylate (DMAD, DEAD) yields diastereospecific palladium(II) vinyl derivative (Z)-(Pd(Cy₂PCH₂)₂C₆H₃)(CRCombining double low lineCR(NH₂)) (RCombining double low lineCO₂Me (6a), CO₂Et (6b)). Reacting complexes 6a and 6b with p-nitrophenol produces (Pd(Cy₂PCH₂)₂C₆H₃)(OC₆H₄-p-NO₂) (8) and cis-CHRCombining double low lineCR(NH₂), exclusively.

Full text of DOI:10.1016/j.crci.2015.05.007

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Journal of Approximation Theory 199 (2015) 1–12
www.elsevier.com/locate/jat
Full length article
Explicit matrix inverses for lower triangular matrices
with entries involving continuous q-ultraspherical
polynomials
Noud Aldenhoven
Radboud Universiteit, IMAPP, FNWI, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
Received 13 November 2014; received in revised form 27 May 2015; accepted 12 June 2015
Available online 23 June 2015
Communicated by Arno B.J. Kuijlaars
Abstract
For a one-parameter family of lower triangular matrices with entries involving continuous q-
ultraspherical polynomials we give an explicit lower triangular inverse matrix, with entries involving again
continuous q-ultraspherical functions. The matrices are q-analogues of results given by Cagliero and Koorn-
winder recently. The proofs are not q-analogues of the Cagliero–Koornwinder case, but are of a different
nature involving q-Racah polynomials. Some applications of these new formulas are given. Also the limit
β → 0 is studied and gives rise to continuous q-Hermite polynomials for 0 < q < 1 and q > 1.
c
⃝ 2015 Elsevier Inc. All rights reserved.
Keywords: Orthogonal polynomials; Continuous q-ultraspherical polynomials; q-Racah polynomials; q-analogues
1. Introduction
In [12] Koelink, van Pruijssen and Roma´n needed to invert a lower triangular matrix with en-
tries involving Gegenbauer (or ultraspherical) polynomials. The solution was given by Cagliero
and Koornwinder [5] in the wider context of a two-parameter family of lower triangular matrices
involving Jacobi polynomials. The inverse of this matrix is given in terms of Jacobi polynomials
E-mail address: n.aldenhoven@science.ru.nl.
http://dx.doi.org/10.1016/j.jat.2015.06.004
c
0021-9045/⃝ 2015 Elsevier Inc. All rights reserved.

2
N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
as well. Cagliero and Koornwinder [5] solved this problem using the Rodrigues formula for the
Jacobi polynomials and some variations on the product rule. Thereafter Koelink, de los R´ıos and
Roma´n [10] used the results of Cagliero and Koornwinder [5] with an extra free parameter.
In this paper we give a partial q-analogue of the result of Cagliero and Koornwinder [5]. In a
forthcoming paper [1], which is a quantum analogue of [11,12], the main Theorem 1.1 is used to
obtain an inverse of a lower triangular matrix with entries involving continuous q-ultraspherical
polynomials. Theorem 1.1 gives the inverse of this matrix in a more general situation. Theo-
rem 1.1 is the main result of this paper.
k
2
Theorem 1.1. Let β ∈ C \ {0}, β ̸= q for k ∈ Z. Define doubly infinite lower triangular
matrices Lβ(x) and Mβ(x) by
1
Lβ(x)m,n
Mβ(x)m,n
=
C
m−n(x; βqn|q), n ≤ m,
(β2q2n; q)m−n
β
m−nq(m−1)(m−n)
=
C
m−n(x; β−1q1−m|q), n ≤ m,
(β2qm+n−1; q)m−n
where m, n ∈ Z and Cm(x; β|q) are the continuous q-ultraspherical polynomials defined in
Section 2 for all β. Then Mβ(x) and Lβ(x) are each other’s inverse, i.e. Lβ(x)Mβ(x) = I =
Mβ(x)Lβ(x), where Im,n = δm,n is the identity.
The proof of Theorem 1.1 is given in Section 3.
Theorem 1.1 has a finite dimensional analogue, because the entries of Lβ Mβ only involve
finite sums of continuous q-ultraspherical polynomials. From Theorem 1.1 we have the following
corollary.
k
2
Corollary 1.2. For a non-negative integer N and β ∈ C \ {0} such that β ̸= q− for k =
0, 1, . . . , 2N − 2. Define lower triangular matrices Lβ(x) and Mβ(x)
1
Lβ(x)m,n
=
C
m−n(x; βqn|q), 0 ≤ n ≤ m ≤ N
(β2q2n; q)m−n
β
m−nq(m−1)(m−n)
Mβ(x)m,n
=
C
m−n(x; β−1q1−m|q), 0 ≤ n ≤ m ≤ N.
(β2qm+n−1; q)m−n
Then Mβ(x) and Lβ(x) are each others inverse, i.e. Lβ(x)Mβ(x) = I = Mβ(x)Lβ(x), where
I is the identity matrix.
The proof of Theorem 1.1 is not a straightforward q-analogue of the proof given by Cagliero
and Koornwinder [5]. The proof uses q-Racah polynomials and does not use Rodrigues formulas
or product rules of differentials which are the essential ingredients for the proof in [5]. In
particular, the q → 1 limit of the proof presented here gives an alternative proof of the special
case α = β of Cagliero and Koornwinder [5].
The coefficients of eikθ of products of two continuous q-ultraspherical polynomials are com-
puted and we express the coefficients in terms of terminating balanced basic hypergeometric
series 4φ3. For certain parameters this series transforms to a q-Racah polynomial. The orthogo-
nality relations of the q-Racah polynomials then lead to Theorem 1.1. The proof of Theorem 1.1,
for q → 1, gives an interesting new proof of [5, Theorem 4.1] in the special case α = β, showing
that the coefficients of eikθ of products of certain Gegenbauer polynomials are actually Racah

N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
3
polynomials. The entries of the matrix identity L(x)M(x) = I in [5, Theorem 4.1] correspond
to orthogonality relations of Racah polynomials, see Example 4.1.
In Section 5 we study matrices Lβ and Mβ for Theorem 1.1 for a suitable limit β → 0.
The entries of Lβ become continuous q-Hermite polynomials and the entries of Mβ converge to
continuous q−1-Hermite polynomials as β → 0.
We emphasise again that our proof is different than the proof of Cagliero and Koornwinder [5].
It is possible to extend the proof of Cagliero and Koornwinder [5] to a q-analogue for certain
polynomials in the q-Askey scheme [9]. For example [5, Lemma 5.1] has a q-analogue for the
q-derivative operator [6, Exercise 1.12]. Then with the use of Rodrigues’ formula and suitable
parameters for the orthogonal polynomials it is possible to find q-analogues for [5, (4.1), (4.2)].
The author was able to extend [5, (4.1), (4.2)] to the little q-Jacobi polynomials. However these
results involve different q-shifts in the x of the polynomials and do not seem to lead to a result
similar to Theorem 1.1 or [5, Theorem 4.1]. Also Cagliero and Koornwinder [5] were motivated
by [4,12] to extend their formulas to a two parameter family of Jacobi polynomials. We lack this
motivation and therefore decided not to include these results for the little q-Jacobi polynomials
in this paper. We did not extend the results to other families of polynomials.
2. Preliminaries
We recall some facts on basic hypergeometric series and related polynomials, see Gasper and
Rahman [6] and Koekoek, Lesky and Swarttouw [9]. We fix 0 < q < 1 and we follow notation
of [6].
For β ∈ C, the continuous q-ultraspherical polynomials are given by
n

(β; q)k(β; q)n−k
Cn(x; β|q) =
ei(n−2k)θ
,
x = cos(θ),
(2.1)
(q; q)k(q; q)n−k
k=0
see [6, Exercise 1.28] and [9, § 14.10]. Notice that the continuous q-ultraspherical polyno-
mials are defined for all β. A generating function for the continuous q-ultraspherical polyno-
mials is
∞
(βteiθ , βte−iθ ; q)

∞
∞
Cn(x; β|q)tn =
,
(2.2)
(teiθ , te−iθ ; q)
n=0
where |t| < 1 and x = cos(θ) ∈ [−1, 1], see [6, Exercise 1.29] and [9, (14.10.27)].
For α, β, γ, δ ∈ R such that qα = q−N , βδq = q−N or γ q = q−N , for N ∈ N, define the
q-Racah polynomials


q
−n, αβqn+1, q−x , γ δqx+1
αq, βδq, γ q
Rn(µ(x); α, β, γ, δ; q) = 4φ3
; q, q ,
(2.3)
where µ(x) = q−x + γ δqx+1 and n = 0, 1, . . . , N. If qα = q−N and β = 1 the q-Racah poly-
nomials are not orthogonal with respect to a positive measure. Still the q-Racah polynomials are
orthogonal
N
(q−N , γ δq; q)x (1 − γ δq2x+1
)

qNx Rm(µ(x))Rn(µ(x)) = δm,nhm(γ, δ; N),
x=0 (q, γ δqN+2; q)x (1 − γ δq)

4
N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
where Rm(µ(x)) = Rm(µ(x); q−N−1, 1, γ, δ; q) and hm(γ, δ; N) is given in [6, § 7.2] and
[9, § 14.2]. It follows that if n = 0 we have
N
(q−N , γ δq; q)x (1 − γ δq2x+1
)

qNx Rm(µ(x)) = δm,0h0(γ, δ; N),
(2.4)
x=0 (q, γ δqN+2; q)x (1 − γ δq)
where h0(γ, δ; N) = δN,0 if γ ̸= q−ℓ and δ ̸= q−m with ℓ, m = 1, 2, . . . , N.
Note that (2.4) can also be proved directly, also see [3]. To show this substitute (2.3) in (2.4)
so that
N
N
(q−N , γ δq; q)x (1 − γ δq2x+1
)
(q−m, qm−N , q−x , γ δqx+1; q)k
(q, q−N , δq, γ q; q)k


qNx
qk.
x=0 (q, γ δqN+2; q)x (1 − γ δq)
Then expand the left hand side of (2.4) in qx observing that it is a polynomial in qx of degree
N − k. Finally applying the summation formula [6, (II.21)] on the x-sum gives the right hand
side of (2.4).
k=0
Remark 2.1. One of the referees pointed out that if qα = q−N and β = 1 then from (2.3) it
follows that Rn = RN−n. Therefore for n > 1 N the polynomial Rn will have degree N −n < n.
2
So there can be no non-degenerate orthogonality. However, the system of polynomials Rn for
n ≤ 12 N can still be orthogonal with respect to positive weights.
Sears’ transformation formula, [6, (III.15) and (III.16)], for terminating balanced 4φ3 series is


q
−n, a, b, c
d, e, f
4φ3
; q, q


n (ea−1, f a−1; q)n
(e, f ; q)n
q
−n, a, db−1, dc−1
= a
4φ3
; q, q
(2.5)
(2.6)
d, aq1−n −1, aq1−n f −1
e


(a, ef (ab)−1, ef (ac)−1; q)n
q
−n, ea−1, f a−1, ef (abc)−1
=
4φ3
; q, q ,
(e, f, ef (abc)−1; q)n
ef (ab)−1, ef (ac)−1, q1−na−1
where abc = def qn−1
.
3. Proof of Theorem 1.1
The idea of the proof of Theorem 1.1 is to first expand a sum of products of continuous
q-ultraspherical polynomials in terms of eikθ , where x = cos(θ). We show that the coefficients
of eikθ are balanced basic hypergeometric series 4φ3. For the continuous q-ultraspherical poly-
nomials with parameters as in Theorem 1.1 we show that the coefficients of eikθ correspond to
the orthogonality relations for q-Racah polynomials. This proves the key Lemma 3.4 from which
Theorem 1.1 follows.
Lemma 3.1. Take n ∈ N. Let αk, βk and c(k) be constants for k = 0, 1, . . . , n. Then
n
n


c(k) Cn−k(x; αk|q)Ck(x; βk|q) =
d(p) ei(n−2p)θ
,
x = cos(θ),
(3.1)
k=0
p=0

N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
5
where d(p) is given by
n−p

(αk; q)p (αk; q)n−p−k (βk; q)k
(q; q)p (q; q)n−p−k (q; q)k
d(p) =
c(k)
k=0


q
−p, αkqn−p−k, q−k, βk
× 4φ3
; q, q2αk−1βk−1
q1−pαk−1, q1−kβk−1, qn−p−k+1
n

(αk; q)n−k (βk; q)p−n+k (βk; q)n−p
(q; q)n−k (q; q)p−n+k (q; q)n−p
+
c(k)
k=n−p+1


qk−n, q p−n, αk, βkqk+p−n
q1−n+kαk−1, qk+p−n+1, q1−n+pβk−1
× 4φ3
; q, q2αk−1βk−1
.
(3.2)
Proof. First expand the left hand side of (3.1) using (2.1), so that the left hand side of (3.1)
equals
n
n−k
k



(αk; q)s (αk; q)n−k−s
(q; q)s (q; q)n−k−s
(βk; q)t (βk; q)k−t
(q; q)t (q; q)k−t
c(k)
ei(n−2(s+t))θ
.
(3.3)
k=0
s=0
t=0
Now fix p = s + t and substitute s = p − t in (3.3) so that the coefficient of ei(n−2p)θ becomes
k∧p
n


(αk; q)p−t (αk; q)n−k−p+t (βk; q)t (βk; q)k−t
c(k)
.
(3.4)
(q; q)p−t (q; q)n−k−p+t (q; q)t (q; q)k−t
k=0
t=0∨(k+p−n)
For 0 ≤ k ≤ n−p so that k+p−n ≤ 0, the t-sum of (3.4) is, after simplifying the q-Pochhammer
symbols, the balanced terminating 4φ3
(αk; q)p (αk; q)n−p−k (βk; q)k
(q; q)p (q; q)n−p−k (q; q)k


q
−p, αkqn−p−k, q−k, βk
× 4φ3
; q, q2αk−1βk−1
.
(3.5)
q1−pαk−1, q1−kβk−1, qn−p−k+1
For n − p ≤ k ≤ n so that k + p − n ≥ 0 substitute t → t + k + p − n so that the t-sum of (3.4)
is, after simplifying the q-Pochhammer symbols, the balanced terminating 4φ3
(αk; q)n−k (βk; q)p−n+k (βk; q)n−p
(q; q)n−k (q; q)p−n+k (q; q)n−p


qk−n, q p−n, αk, βkqk+p−n
× 4φ3
; q, q2αk−1βk−1
.
(3.6)
q1−n+kαk−1, qk+p−n+1, q1−n+pβk−1
Combining (3.5) and (3.6) gives (3.2).
ꢀ
Remark 3.2. Since the continuous q-ultraspherical polynomials are polynomials in x the coeffi-
cients of ei(n−p)θ and eipθ of the left hand side of (3.1) must be equal. Therefore d(p) = d(n− p)
and (3.1) can be rewritten in terms of Chebyshev polynomials Tp of the first kind, see [9, § 9.8.2],
as follows:


n
n
2


c(k) Cn−k(x; αk|q)Ck(x; βk|q) =
(2 − δn,2p)d(p)Tn−2p(x).
k=0
p=0

6
N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
Remark 3.3. It is possible to write (3.2) uniformly
n

(αk; q)p (βk; q)k
(αk; q)
(qn−k−p+1; q)
∞
∞
d(p) =
c(k)
(q; q)p (q; q)k (αkqn−k−p; q)
(q; q)
∞
∞
k=0


q
−p, αkqn−p−k, q−k, βk
× 4φ3
; q, q2αk−1βk−1
.
q1−pαk−1, q1−kβk−1, qn−p−k+1
If 0 ≤ k ≤ n − p we have that (3.5) is equal to
(αk; q)p (βk; q)k
(αk; q)
(qn−k−p+1; q)
∞
∞
(q; q)p (q; q)k (αkqn−k−p; q)
(q; q)
∞
∞


q
−p, αkqn−p−k, q−k, βk
× 4φ3
; q, q2αk−1βk−1
.
(3.7)
q1−pαk−1, q1−kβk−1, qn−p−k+1
Use the convention
(q1−N ; q)
∞
= (q1−N+t ; q) ,
∞
(q1−N ; q)t
so that for n − p < k ≤ n
∞
∞
(q1−N ; q)
Ct
(qN+1; q)
CN+t


∞
∞
=
,
(q; q)
(q, q1−N ; q)t
(q; q)
(q, qN+1; q)t
∞
∞
t=0
t=0
where N ∈ N and Ct are arbitrary constants. Then for N = k+ p−n we have that (3.7) becomes
(αk; q)p (βk; q)k
(αk; q)
(q1+k+p−n; q)
∞
∞
(q; q)p (q; q)k (αkqn−k−p; q)
(q; q)
∞
∞
∞


(q−p, αkqn−k−p, βk, q−k; q)t+k+p−n

t+k+p−n
×
q2αk−1βk−1
(q, q1+k+p−n; q)t (q1−pαk−1, q1−kβk−1; q)t+k+p−n
t=0
(αk; q)p (βk; q)k
(αk; q)
∞
=
(q; q)p (q; q)k (αkqn−k−p; q)
∞


(q1+k+p−n; q) (q−p, αkqn−k−p, βk, q−k; q)k+p−n
k+p−n
∞
×
q2αk−1βk−1
(q; q)
∞
(q1−pαk−1, q1−kβk−1; q)k+p−n


qk−n, αk, βkqk+p−n, q p−n
× 4φ3
; q, q2αk−1βk−1
.
(3.8)
qk+p−n+1, qk−n+1αk−1, q1+p−nβk−1
Simplifying the q-Pochhammer symbols of (3.8) shows that (3.8) is equal to (3.6).
Lemma 3.4. For m, n ∈ Z such that n ≤ m, let β ∈ C such that β2 ̸= q−2m+1, q−2m+2, . . . ,
q
−2n. Then
m−n
(1 − β2q2n+2k−1
)

βkqk(k+n−1)
C
m−n−k(x; βqk|q)Ck(x; β−1q1−k−n|q)
(β2q2n+k−1; q)m−n+1
k=0
= δm,n
.
(3.9)

N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
7
Proof. Apply Lemma 3.1 with n, αk, βk specialised to m − n, qk+nβ, q1−k−nβ−1 so that in

particular αkβk = q for all k. Then the left hand side of (3.9) is m−n d(p)ei(m−n−2p)θ , where
p=0
x = cos(θ) and
m−n−p
(1 − β2q2n+2k−1
)

d(p) =
βkqk(k+n−1)
(β2q2n+k−1; q)m−n+1
k=0
(βqk+n; q)p (βqk+n; q)m−n−p−k (β−1q1−k−n; q)k
×
(q; q)p
(q; q)m−n−p−k
(q; q)k


q
−k, q−p, βqm−p, q−n−k+1β−1
× 4φ3
; q, q
βqn, q−p−n−k+1β−1, qm−n−k−p+1
m−n
(1 − β2q2n+2k−1
)

+
βkqk(k+n−1)
k=m−n−p+1 (β2q2n+k−1; q)m−n+1
(qk+nβ; q)m−n−k (q1−k−nβ−1; q)p−m+n+k (q1−k−nβ−1; q)m−n−p
×
(q; q)m−n−k
(q; q)p−m+n+k
(q; q)m−n−p


qk−m+n, q p−m+n, qk+nβ, q1+p−mβ−1
× 4φ3
; q, q .
(3.10)
q1−m
β
−1, qk+p−m+n+1, βq2n−m+p+k
We transform the basic hypergeometric series 4φ3 of (3.10). Apply Sears’ transformation formula
(2.5) to the first 4φ3 in (3.10) to see that the 4φ3 is equal to
(q−n−k+1β−1, qm−n−k+1; q)k
q−pk
(q−n−k−p+1β−1, qm−n−k−p+1; q)k


q
−k, q−p, qn−m+p, β2q2n+k−1
× 4φ3
; q, q .
(3.11)
βqn, βqn, qn−m
Apply Sears’ transformation formula (2.6) to the second 4φ3 in (3.10) in order to see that the 4φ3
is equal to
(q1+p−m
(βq2n−m+k+p, q1−m
β
−1, qn−m, βqn; q)m−n−p
β
−1, q−k; q)m−n−p


qn−m+p, β2q2n+k−1, q−p, q−k
× 4φ3
; q, q .
(3.12)
qn−m, βqn, βqn
The basic hypergeometric series of (3.11) and (3.12) can be written as the q-Racah polynomial
Rp(µ(k); qn−m−1, 1, βqn−1, βqn−1; q), see (2.3). Therefore (3.10) becomes, after simplifying
12 ℓ(ℓ−1)+rℓ
the q-Pochhammer symbols using (qr β−1; q)ℓ = (−1)ℓq
β
−ℓ(βq1−r−ℓ; q)ℓ repeat-
edly,
m−n
2
(βqn; q)p(βqn; q)m−n−p
(β q2n−1, qn−m; q)k (1 − β2q2n+2k−1
)

qk(m−n)
(β2q2n; q)m−n(q; q)p(q; q)m−n−p
(q, β2qn+m; q)k
(1 − β2q2n−1
)
k=0
× Rp(µ(k); qn−m−1, 1, βqn−1, βqn−1; q).
(3.13)

8
N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
The k-sum of (3.13) corresponds to the orthogonality relations (2.4) for the q-Racah polynomial.
Hence (3.13) becomes
(βqn; q)m−n
d(p) =
δp,0h0(βqn, βqn; m − n).
(βq2n; q)m−n(q; q)m−n
Since h0(βqn, βqn; m − n) = 0 if n < m and h0(βqn, βqn; m − n) = 1 if m = n, the result
follows.
ꢀ
Proof of Theorem 1.1. Multiplying the matrices Lβ and Mβ it is sufficient to evaluate the
entries of Lβ Mβ for m ≥ n. Hence
(Lβ(x)Mβ(x))m,n
m

β
k−nq(k−1)(k−n)
=
=
C
m−k(x; βqn|q)Ck−n(x; q1−kβ−1|q)
(β2q2k; q)m−k(β2qn+k−1; q)k−n
k=n
m−n
(1 − β2q2n+2k−1
)

βkqk(k+n−1)
C
m−n−k(x; βqk+n|q)Ck(x; q1−k−nβ−1|q).
(β2q2n+k−1; q)m−n+1
k=0
Applying Lemma 3.4 then yields the result.
ꢀ
4. Applications
Example 4.1. The limit q → 1 in the proof of Theorem 1.1 gives a new proof for [5, Theorem
4.1] for α = β. Lemma 3.1 gives
n
n


c(k)Cn(α−k)(x)Ck(β )(x) =
d(p)ei(n−2p)θ
,
x = cos(θ),
k
k
k=0
p=0
where Ck(α)(x) are the Gegenbauer polynomials, see [9, § 9.8.1], and
n−p

(αk)p (αk)n−p−k (βk)k
d(p) =
c(k)
p! (n − p − k)! k!
k=0


−p, αk + n − p − k, −k, βk
× 4 F3
; 2 − αk − βk
1 − p − αk, 1 − k − βk, n − p − k + 1
n

(αk)n−k (βk)p−n+k (βk)n−p
+
c(k)
(n − k)! (p − n + k)! (n − p)!
k=n−p+1


k − n, p − n, αk, βk + k + p − n
× 4 F3
; 2 − αk − βk
.
1 − n + k − αk, k + p − n + 1, 1 − n + p − βk
Then Lemma 3.4 yields, for 0 ≤ n ≤ m and α ∈ C, 2α ̸= −2m + 1, −2m + 2 . . . , −2n,
m−n

(2n + 2k + 2α − 1)
(α+k+n)
m−n−k
C
(x)Ck(1−k−n−α)(x) = δm,n
,
(2n + k + 2α − 1)m−n+1
k=0
which is the key equation to show [5, Theorem 4.1] for the case α = β.

N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
9
Example 4.2. The problem of finding an inverse of the matrix Lβ in Theorem 1.1 originally
arose in [1] where the finite dimensional lower triangular matrix
(q2; q2)m(q2; q2)2n+1
L(x)m,n = qm−n
C
m−n(x; q2n+2|q2),
(q2; q2)m+n+1(q2; q2)n
with 0 ≤ n ≤ m ≤ N, for arbitrary N ∈ N appears. Using Corollary 1.2 in base q2 with β = q2
after conjugation with a diagonal matrix we find that the inverse matrix is given by
(q2; q2)m(q2; q2)m+n
(L(x))
= q(2m+1)(m−n)
C
m−n(x; q−2m|q2),
−1
m,n
(q2; q2)2m(q2; q2)n
with 0 ≤ n ≤ m ≤ N. Note that the entries of L(x) and its inverse L(x)−1 are independent of
the size of N.
Example 4.3. From the continuous q-ultraspherical polynomial generating function (2.2) it
follows that
∞
(αteiθ , αte−iθ ; q) (αβteiθ , αβte−iθ ; q)

∞
∞
∞
∞
Cn(x; αβ|q)tn
=
=
(teiθ , te−iθ ; q)
(αteiθ , αte−iθ ; q)
n=0
∞

Cm(x; α|q)Cn(x; β|q)tm(αt)n.
m,n=0
Comparing the powers of t shows
n

Cn(x; αβ|q) =
αkCn−k(x; α|q)Ck(x; β|q).
k=0
Now take β = α−1, then (2.1) for β = 1 gives
n

δn,0
=
αkCn−k(x; α|q)Ck(x; α−1|q).
(4.1)
(4.2)
k=0
On the other hand from Lemma 3.1 it follows that
n
n


αkCn−k(x; α|q)Ck(x; α−1|q) =
d(p)ei(n−2p)θ
,
x = cos(θ).
k=0
p=0
Combining (4.1) and (4.2) it follows that d(p) = δn,0. Writing out the explicit expression of
d(p) gives for n > 0 the identity


n−p
(α; q)p (α; q)n−p−k (α−1; q)k
(q; q)p (q; q)n−p−k (q; q)k
q
−p, αqn−p−k, q−k, α−1
q1−p −1, αq1−k, qn−p−k+1

0 =
αk
4φ3
; q, q2
α
k=0
n
(α; q)n−k (α−1; q)p−n+k (α−1; q)n−p
αk

+
(q; q)n−k (q; q)p−n+k (q; q)n−p
k=n−p+1


qk−n, q p−n, α, qk+p−nα−1
× 4φ3
; q, q2
q1−n+k −1, qk+p−n+1, αq1−n+p
α
.

10
N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
In particular if p = 0
n
(α; q)n−k (α−1; q)k

αk
= 0.
(4.3)
(q; q)n−k (q; q)k
k=0
Remark that (4.3) also follows from the q-Chu–Vandermonde sum [6, (1.5.2)]. For p = 1 and
n → n + 1 we find


n
(α; q)n−k (α−1; q)k
(q; q)n−k (q; q)k
(1 − αqn−k)(1 − qk)
n (α−1; q)n
(q; q)n

αk
1 +
q1−k = α
.
(1 − αq1−k)(1 − qn+1−k
)
k=0
Remark that this result also follows from applying (4.3) twice.
5. Limit case β → 0
Define L0(x) and M0(x) by L0(x)m,n = limβ→0 Lβ(x)m,n and M0(x)m,n = limβ→0 Mβ
k
2
(x)m,n where the limit is taken over β ̸= q , where k ∈ Z. We show that the limits exist, that the
entries of L0(x) are given in terms of continuous q-Hermite polynomials and that the entries of
M0(x) are given in terms of continuous q−1-Hermite polynomials.
The continuous q-Hermite polynomials are given by
n

(q; q)n
H(x|q) =
ei(n−2k)θ
,
x = cos(θ),
(5.1)
(q; q)k(q; q)n−k
k=0
see [9, § 14.26]. The continuous q-Hermite polynomials are, apart from a different normalisation,
the special case β = 0 of the continuous q-ultraspherical polynomials
Hn(x|q)
Cn(x; 0|q) =
.
(5.2)
(5.3)
(q; q)n
The corresponding generating function for the continuous q-Hermite polynomials is
∞

Hn(x|q)
(q; q)n
1
tn =
|t| < 1, x = cos(θ),
(teiθ , te−iθ ; q)
∞
n=0
see [9, (14.26.11)].
The polynomials Hn(x|q−1) are called the continuous q−1-Hermite polynomials and are de-
fined by taking q → q−1 in (5.1), see [2]. The continuous q-Hermite polynomials are orthogonal
with respect to a positive measure on (−1, 1). However the continuous q−1-Hermite polynomi-
als are orthogonal on the imaginary axis and correspond to an indeterminate moment problem,
see [2,8].
Theorem 5.1. The doubly infinite lower triangular matrices L0(x) and M0(x) are given by
m−n(x|q−1
(q; q)m−n
)
m−n
H
m−n(x|q)
H
)
L0(x)m,n
=
,
M0(x)m,n = (−1)m−n
q
,
(
2
(q; q)m−n
where m, n ∈ Z so that n ≤ m. M0(x) and L0(x) are each others inverse, i.e. L0(x)M0(x) =
I = M0(x)L0(x), where Im,n = δm,n is the identity.

N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
11
Proof. With (5.2) we have for n ≤ m
1
H
m−n(x|q)
L0(x)m,n = lim Lβ(x)m,n = lim
C
m−n(x; βqn|q) =
.
2
2n
β→0
β→0
(q; q)m−n
(β q ; q)m−n
From (2.1) it follows that Cn(x; β|q) = (βq−1)nCn(x; β−1|q−1). Therefore write Mβ(x)m,n as
β
m−nq(m−1)(m−n)
(β−1q
)
−m m−nCm−n(x; βqm−1|q−1
)
(β2qm+n−1; q)m−n
q−(m−n)
=
C
m−n(x; βqm−1|q−1).
(β2qm+n−1; q)m−n
Upon taking the limit β → 0 and using (5.2) we find
H
m−n(x|q−1
)
M0(x)m,n = lim Mβ(x)m,n = q−(m−n)
(q−1; q−1
)
β→0
m−n
m−n
H
)
m−n(x|q−1
)
= (−1)m−n
q
.
(
2
(q; q)m−n
From Theorem 1.1 it follows that L0(x)M0(x) = I = M0(x)L0(x).
ꢀ
Corollary 5.2. For N ∈ N define lower triangular matrices L0(x) and M0(x)
m−n(x|q−1
(q; q)m−n
)
m−n
H
m−n(x|q)
H
L0(x)m,n
=
,
M0(x)m,n = (−1)m−n
q
,
(
)
2
(q; q)m−n
where 0 ≤ n ≤ m ≤ N. Then M0(x) and L0(x) are each others inverse, i.e. L0(x)M0(x) =
I = M0(x)L0(x), where I is the identity matrix.
Remark 5.3. Theorem 5.1 also follows from a generating function for the continuous q−1
-
Hermite polynomials. From [7, Theorem 21.2.1]
∞

n
Hn(x|q−1
(q; q)n
)
n
tn = (teiθ , te−iθ ; q) , |t| < 1, x = cos(θ).
(5.4)
(2)
(−1) q
∞
n=0
Combining (5.3) and (5.4) it follows that for |t| < 1

 

∞
∞
(teiθ , te−iθ ; q)
Hm(x|q)
Hn(x|q−1
(q; q)n
)


n
∞
∞
n
tm
(−1) q
tn
(2)
1 =
=
(teiθ , te−iθ ; q)
(q; q)m
m=0
n=0


p
∞
Hk(x|q−1
(q; q)k
)


k
H
p−k(x|q)
=
(−1) q
t p.
k
(2)
(q; q)p−k
p=0 k=0
Take p = m − n so that we have
m−n
Hk(x|q−1
(q; q)k
)

k
H
m−n−k(x|q)
k
(2)
(−1) q
= δm,n
.
(5.5)
(q; q)m−n−k
k=0
From (5.5) Theorem 5.1 also follows.

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N. Aldenhoven / Journal of Approximation Theory 199 (2015) 1–12
Acknowledgments
The author thanks Erik Koelink and Pablo Roma´n for many useful discussions. Parts of the
paper have been discussed with Tom Koornwinder, the author thanks him for his input.
The research of the author is supported by the Netherlands Organisation for Scientific
Research (NWO) under project number 613.001.005 and by the Belgian Interuniversity
Attraction Pole Dygest P07/18.
The author thanks the referees for their input, pointing out Remark 2.1 and deriving (3.12) in
an alternative way.
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