Explicit matrix inverses for lower triangular matrices with entries involving continuous q-ultraspherical polynomials
aa r X i v : . [ m a t h . C A ] M a y Explicit matrix inverses for lower triangular matrices with entriesinvolving continuous q -ultraspherical polynomials Noud Aldenhoven ∗ Radboud University Nijmegen, FNWI, IMAPP,Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
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 involvingagain continuous q -ultraspherical functions. The matrices are q -analogues of results given by Caglieroand Koornwinder recently. The proofs are not q -analogues of the Cagliero-Koornwinder case, but are ofa different nature involving q -Racah polynomials. Some applications of these new formulas are given.Also the limit β → q -Hermite polynomials for 0 < q < q > In [11] Koelink, van Pruijssen and Rom´an needed to invert a lower triangular matrix with entries involvingGegenbauer (or ultraspherical) polynomials. The solution was given by Cagliero and Koornwinder [5] inthe wider context of a two-parameter family of lower triangular matrices involving Jacobi polynomials. Theinverse of this matrix is given in terms of Jacobi polynomials as well. Cagliero and Koornwinder [5] solvedthis problem using the Rodrigues formula for the Jacobi polynomials and some variations on the productrule. Thereafter Koelink, de los R´ıos and Rom´an [12] used the results of Cagliero and Koornwinder [5] withan extra free parameter.In this paper we give a partial q -analogue of the result of Cagliero and Koornwinder [5]. In a forthcomingpaper [1], which is a quantum analogue of [10, 11], the main Theorem 1.1 is used to obtain an inverse of alower triangular matrix with entries involving continuous q -ultraspherical polynomials. Theorem 1.1 givesthe inverse of this matrix in a more general situation. Theorem 1.1 is the main result of this paper. Theorem 1.1.
Let β ∈ C \{ } , β = q k for k ∈ Z . Define doubly infinite lower triangular matrices L β ( x )and M β ( x ) by L β ( x ) m,n = 1( β q n ; q ) m − n C m − n ( x ; βq n | q ) , n ≤ m,M β ( x ) m,n = β m − n q ( m − m − n ) ( β q m + n − ; q ) m − n C m − n ( x ; β − q − m | q ) , n ≤ m, where m, n ∈ Z and C m ( x ; β | q ) are the continuous q -ultraspherical polynomials defined in Section 2 for all β .Then M β ( x ) and L β ( x ) are each other’s inverse, i.e. L β ( x ) M β ( x ) = I = M β ( x ) L β ( x ), where I m,n = δ m,n isthe identity. ∗ Email address: [email protected] L β M β only involve finite sums ofcontinuous q -ultraspherical polynomials. From Theorem 1.1 we have to following corollary. Corollary 1.2.
For a non-negative integer N and β ∈ C \{ } such that β = q − k for k = 0 , , . . . , N − L β ( x ) and M β ( x ) L β ( x ) m,n = 1( β q n ; q ) m − n C m − n ( x ; βq n | q ) , ≤ n ≤ m ≤ NM β ( x ) m,n = β m − n q ( m − m − n ) ( β q m + n − ; q ) m − n C m − n ( x ; β − q − m | q ) , ≤ n ≤ m ≤ N. Then M β ( x ) and L β ( x ) are each others inverse, i.e. L β ( x ) M β ( x ) = I = M β ( x ) L β ( x ), where I is the identitymatrix.The proof of Theorem 1.1 is not a straightforward q -analogue of the proof given by Cagliero and Koorn-winder [5]. The proof uses q -Racah polynomials and does not use Rodrigues formulas or product rules ofdifferentials which are the essential ingredients for the proof in [5]. In particular, the q → α = β of Cagliero and Koornwinder [5].We compute the coefficients of e ikθ of products of two continuous q -ultraspherical polynomials and expressthe coefficients in terms of terminating balanced basic hypergeometric series φ . For certain parametersthis series transforms to a q -Racah polynomial. The orthogonality relations of the q -Racah polynomials thenlead 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 e ikθ of products of certain Gegenbauer polynomialsare actually Racah 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 L β and M β for Theorem 1.1 for a suitable limit β →
0. The entries of L β become continuous q -Hermite polynomials and the entries of M β converge to continuous q − -Hermitepolynomials as β → 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, Exercise1.12]. Then with the use of Rodrigues’ formula and suitable parameters for the orthogonal polynomials it ispossible 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 don’tseem to lead to a result similar to Theorem 1.1 or [5, Theorem 4.1]. Also Calgiero and Koornwinder [5] weremotivated by [4, 11] to extend their formulas to a two parameter family of Jacobi polynomials. We lackthis motivation and therefore decided not to include these results for the little q -Jacobi polynomials in thispaper. We didn’t extend the results to other families of polynomials. 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 < β ∈ C , the continuous q -ultraspherical polynomials are given by C n ( x ; β | q ) = n X k =0 ( β ; q ) k ( β ; q ) n − k ( q ; q ) k ( q ; q ) n − k e i ( n − k ) θ , x = cos( θ ) , (2.1)see [6, Exercise 1.28] and [9, § q -ultraspherical polynomials are defined for2ll β . A generating function for the continuous q -ultraspherical polynomials is ∞ X n =0 C n ( x ; β | q ) t n = ( βte iθ , βte − iθ ; q ) ∞ ( te iθ , te − iθ ; q ) ∞ | t | < , x = cos( θ ) ∈ [ − , , (2.2)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 γq = q − N , for N ∈ N , define the q -Racahpolynomials R n ( µ ( x ); α, β, γ, δ ; q ) = φ (cid:18) q − n , αβq n +1 , q − x , γδq x +1 αq, βδq, γq ; q, q (cid:19) , µ ( x ) = q − x + γδq x +1 , (2.3)where n = 0 , , . . . , N . If qα = q − N and β = 1 the q -Racah polynomials are not orthogonal with respect toa positive measure. Still the q -Racah polynomials are orthogonal N X x =0 ( q − N , γδq ; q ) x ( q, γδq N +2 ; q ) x (1 − γδq x +1 )(1 − γδq ) q Nx R m ( µ ( x ); q − N − , , γ, δ ; q ) R n ( µ ( x ); q − N − , , γ, δ ; q )= δ m,n h m ( γ, δ ; N ) , where h m ( γ, δ ; N ) is given in [6, § § n = 0 we have N X x =0 ( q − N , γδq ; q ) x ( q, γδq N +2 ; q ) x (1 − γδq x +1 )(1 − γδq ) q Nx R m ( µ ( x ); q − N − , , γ, δ ; q ) = δ m, h ( γ, δ ; N ) , (2.4)where h ( γ, δ ; N ) = δ N, if γ = q − ℓ and δ = q − m with ℓ, m = 1 , , . . . , 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 X x =0 ( q − N , γδq ; q ) x ( q, γδq N +2 ; q ) x (1 − γδq x +1 )(1 − γδq ) q Nx N X k =0 ( q − m , q m − N , q − x , γδq x +1 ; q ) k ( q, q − N , δq, γq ; q ) k q k . Then expand the left hand side of (2.4) in q x observing that it is a polynomial in q x of degree N − k . Finallyapplying the summation formula [6, (II.21)] on the x -sum gives the right hand side of (2.4). Remark 2.1.
One of the referees pointed out that if qα = q − N and β = 1 then from (2.3) it follows that R n = R N − n . Therefore for n > N the polynomial R n will have degree N − n < n . So there can be nonon-degenerate orthogonality. However, the system of polynomials R n for n ≤ N can still be orthogonalwith respect to positive weights.Sears’ transformation formula, [6, (III.15) & (III.16)], for terminating balanced φ series is φ (cid:18) q − n , a, b, cd, e, f ; q, q (cid:19) = a n ( ea − , f a − ; q ) n ( e, f ; q ) n φ (cid:18) q − n , a, db − , dc − d, aq − n e − , aq − n f − ; q, q (cid:19) (2.5)= ( a, ef ( ab ) − , ef ( ac ) − ; q ) n ( e, f, ef ( abc ) − ; q ) n φ (cid:18) q − n , ea − , f a − , ef ( abc ) − ef ( ab ) − , ef ( ac ) − , q − n a − ; q, q (cid:19) , (2.6)where abc = def q n − . The idea of the proof of Theorem 1.1 is to first expand a sum of products of continuous q -ultrasphericalpolynomials in terms of e ikθ , where x = cos( θ ). We show that the coefficients of e ikθ are balanced basichypergeometric series φ . For the continuous q -ultraspherical polynomials with parameters as in Theorem1.1 we show that the coefficients of e ikθ correspond to the orthogonality relations for q -Racah polynomials.This proves the key Lemma 3.4 from which Theorem 1.1 follows.3 emma 3.1. Take n ∈ N . Let α k , β k and c ( k ) be constants for k = 0 , , . . . , n . Then n X k =0 c ( k ) C n − k ( x ; α k | q ) C k ( x ; β k | q ) = n X p =0 d ( p ) e i ( n − p ) θ , x = cos( θ ) , (3.1)where d ( p ) is given by d ( p ) = n − p X k =0 c ( k ) ( α k ; q ) p ( q ; q ) p ( α k ; q ) n − p − k ( q ; q ) n − p − k ( β k ; q ) k ( q ; q ) k φ (cid:18) q − p , α k q n − p − k , q − k , β k q − p α − k , q − k β − k , q n − p − k +1 ; q, q α − k β − k (cid:19) + n X k = n − p +1 c ( k ) ( α k ; q ) n − k ( q ; q ) n − k ( β k ; q ) p − n + k ( q ; q ) p − n + k ( β k ; q ) n − p ( q ; q ) n − p φ (cid:18) q k − n , q p − n , α k , β k q k + p − n q − n + k α − k , q k + p − n +1 , q − n + p β − k ; q, q α − k β − k (cid:19) . (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 X k =0 c ( k ) n − k X s =0 ( α k ; q ) s ( q ; q ) s ( α k ; q ) n − k − s ( q ; q ) n − k − s k X t =0 ( β k ; q ) t ( q ; q ) t ( β k ; q ) k − t ( q ; q ) k − t e i ( n − s + t )) θ . (3.3)Now fix p = s + t and substitute s = p − t in (3.3) so that the coefficient of e i ( n − p ) θ becomes n X k =0 c ( k ) k ∧ p X t =0 ∨ ( k + p − n ) ( α k ; q ) p − t ( q ; q ) p − t ( α k ; q ) n − k − p + t ( q ; q ) n − k − p + t ( β k ; q ) t ( q ; q ) t ( β k ; q ) k − t ( q ; q ) k − t . (3.4)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 φ ( α k ; q ) p ( q ; q ) p ( α k ; q ) n − p − k ( q ; q ) n − p − k ( β k ; q ) k ( q ; q ) k φ (cid:18) q − p , α k q n − p − k , q − k , β k q − p α − k , q − k β − k , q n − p − k +1 ; q, q α − k β − k (cid:19) . (3.5)For n − p ≤ k ≤ n so that k + p − n ≥ t t + k + p − n so that the t -sum of (3.4) is, aftersimplifying the q -Pochhammer symbols, the balanced terminating φ ( α k ; q ) n − k ( q ; q ) n − k ( β k ; q ) p − n + k ( q ; q ) p − n + k ( β k ; q ) n − p ( q ; q ) n − p φ (cid:18) q k − n , q p − n , α k , β k q k + p − n q − n + k α − k , q k + p − n +1 , q − n + p β − k ; q, q α − k β − k (cid:19) . (3.6)Combining (3.5) and (3.6) gives (3.2). Remark 3.2.
Since the continuous q -ultraspherical polynomials are polynomials in x the coefficients of e i ( n − p ) θ and e ipθ of the left hand side of (3.1) must be equal. Therefore d ( p ) = d ( n − p ) and (3.1) can berewritten in terms of Chebychev polynomials T p of the first kind, see [9, § n X k =0 c ( k ) C n − k ( x ; α k | q ) C k ( x ; β k | q ) = [ n ] X p =0 (2 − δ n, p ) d ( p ) T n − p ( x ) . Remark 3.3.
It is possible to write (3.2) uniformly d ( p ) = n X k =0 c ( k ) ( α k ; q ) p ( q ; q ) p ( β k ; q ) k ( q ; q ) k ( α k ; q ) ∞ ( α k q n − k − p ; q ) ∞ ( q n − k − p +1 ; q ) ∞ ( q ; q ) ∞ × φ (cid:18) q − p , α k q n − p − k , q − k , β k q − p α − k , q − k β − k , q n − p − k +1 ; q, q α − k β − k (cid:19) .
4f 0 ≤ k ≤ n − p we have that (3.5) is equal to( α k ; q ) p ( q ; q ) p ( β k ; q ) k ( q ; q ) k ( α k ; q ) ∞ ( α k q n − k − p ; q ) ∞ ( q n − k − p +1 ; q ) ∞ ( q ; q ) ∞ φ (cid:18) q − p , α k q n − p − k , q − k , β k q − p α − k , q − k β − k , q n − p − k +1 ; q, q α − k β − k (cid:19) . (3.7)Use the convention ( q − N ; q ) ∞ ( q − N ; q ) t = ( q − N + t ; q ) ∞ , so that for n − p < k ≤ n ( q − N ; q ) ∞ ( q ; q ) ∞ ∞ X t =0 C t ( q, q − N ; q ) t = ( q N +1 ; q ) ∞ ( q ; q ) ∞ ∞ X t =0 C N + t ( q, q N +1 ; q ) t , where N ∈ N and C t are arbitrary constants. Then for N = k + p − n we have that (3.7) becomes( α k ; q ) p ( q ; q ) p ( β k ; q ) k ( q ; q ) k ( α k ; q ) ∞ ( α k q n − k − p ; q ) ∞ ( q k + p − n ; q ) ∞ ( q ; q ) ∞ × ∞ X t =0 ( q − p , α k q n − k − p , β k , q − k ; q ) t + k + p − n ( q, q k + p − n ; q ) t ( q − p α − k , q − k β − k ; q ) t + k + p − n (cid:0) q α − k β − k (cid:1) t + k + p − n = ( α k ; q ) p ( q ; q ) p ( β k ; q ) k ( q ; q ) k ( α k ; q ) ∞ ( α k q n − k − p ; q ) ∞ ( q k + p − n ; q ) ∞ ( q ; q ) ∞ ( q − p , α k q n − k − p , β k , q − k ; q ) k + p − n ( q − p α − k , q − k β − k ; q ) k + p − n (cid:0) q α − k β − k (cid:1) k + p − n × φ (cid:18) q k − n , α k , β k q k + p − n , q p − n q k + p − n +1 , q k − n +1 α − k , q p − n β − k ; q, q α − k β − k (cid:19) . (3.8)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 β = q − m +1 , q − m +2 , . . . , q − n . Then m − n X k =0 (1 − β q n +2 k − )( β q n + k − ; q ) m − n +1 β k q k ( k + n − C m − n − k ( x ; βq k | q ) C k ( x ; β − q − k − n | q ) = δ m,n . (3.9) Proof.
Apply Lemma 3.1 with n, α k , β k specialised to m − n, q k + n β, q − k − n β − so that in particular α k β k = q for all k . Then the left hand side of (3.9) is P m − np =0 d ( p ) e i ( m − n − p ) θ , where x = cos( θ ) and d ( p ) = m − n − p X k =0 (1 − β q n +2 k − )( β q n + k − ; q ) m − n +1 β k q k ( k + n − × ( βq k + n ; q ) p ( q ; q ) p ( βq k + n ; q ) m − n − p − k ( q ; q ) m − n − p − k ( β − q − k − n ; q ) k ( q ; q ) k × φ (cid:18) q − k , q − p , βq m − p , q − n − k +1 β − βq n , q − p − n − k +1 β − , q m − n − k − p +1 ; q, q (cid:19) + m − n X k = m − n − p +1 (1 − β q n +2 k − )( β q n + k − ; q ) m − n +1 β k q k ( k + n − × ( q k + n β ; q ) m − n − k ( q ; q ) m − n − k ( q − k − n β − ; q ) p − m + n + k ( q ; q ) p − m + n + k ( q − k − n β − ; q ) m − n − p ( q ; q ) m − n − p × φ (cid:18) q k − m + n , q p − m + n , q k + n β, q p − m β − q − m β − , q k + p − m + n +1 , βq n − m + p + k ; q, q (cid:19) . (3.10)5e transform the basic hypergeometric series φ of (3.10). Apply Sears’ transformation formula (2.5) tothe first φ in (3.10) to see that the φ is equal to( q − n − k +1 β − , q m − n − k +1 ; q ) k ( q − n − k − p +1 β − , q m − n − k − p +1 ; q ) k q − pk φ (cid:18) q − k , q − p , q n − m + p , β q n + k − βq n , βq n , q n − m ; q, q (cid:19) . (3.11)Apply Sears’ transformation formula (2.6) to the second φ in (3.10) in order to see that the φ is equal to( q p − m β − , q n − m , βq n ; q ) m − n − p ( βq n − m + k + p , q − m β − , q − k ; q ) m − n − p φ (cid:18) q n − m + p , β q n + k − , q − p , q − k q n − m , βq n , βq n ; q, q (cid:19) . (3.12)The φ of (3.11) and (3.12) can be written as the q -Racah polynomial R p ( µ ( k ); q n − m − , , βq n − , βq n − ; q ),see (2.3). Therefore (3.10) becomes, after simplifying the q -Pochhammer symbols using ( q r β − ; q ) ℓ =( − ℓ q ℓ ( ℓ − rℓ β − ℓ ( βq − r − ℓ ; q ) ℓ repeatedly,( βq n ; q ) p ( βq n ; q ) m − n − p ( β q n ; q ) m − n ( q ; q ) p ( q ; q ) m − n − p m − n X k =0 ( β q n − , q n − m ; q ) k ( q, β q n + m ; q ) k (1 − β q n +2 k − )(1 − β q n − ) q k ( m − n ) × R p ( µ ( k ); q n − m − , , βq n − , βq n − ; q ) . (3.13)The k -sum of (3.13) corresponds to the orthogonality relations (2.4) for the q -Racah polynomial. Hence(3.13) becomes d ( p ) = ( βq n ; q ) m − n ( βq n ; q ) m − n ( q ; q ) m − n δ p, h ( βq n , βq n ; m − n ) . Since h ( βq n , βq n ; m − n ) = 0 if n < m and h ( βq n , βq n ; m − n ) = 1 if m = n , the result follows. Proof of Theorem 1.1.
Multiplying the matrices L β and M β it is sufficient to evaluate the entries of L β M β for m ≥ n . Hence( L β ( x ) M β ( x )) m,n = m X k = n β k − n q ( k − k − n ) ( β q k ; q ) m − k ( β q n + k − ; q ) k − n C m − k ( x ; βq n | q ) C k − n ( x ; q − k β − | q )= m − n X k =0 (1 − β q n +2 k − )( β q n + k − ; q ) m − n +1 β k q k ( k + n − C m − n − k ( x ; βq k + n | q ) C k ( x ; q − k − n β − | q ) . Applying Lemma 3.4 then yields the result.
Example 4.1.
The limit q → α = β . Lemma 3.1 gives n X k =0 c ( k ) C ( α k ) n − k ( x ) C ( β k ) k ( x ) = n X p =0 d ( p ) e i ( n − p ) θ , x = cos( θ ) , where C ( α ) k ( x ) are the Gegenbauer polynomials, see [9, § d ( p ) = n − p X k =0 c ( k ) ( α k ) p p ! ( α k ) n − p − k ( n − p − k )! ( β k ) k k ! F (cid:20) − p, α k + n − p − k, − k, β k − p − α k , − k − β k , n − p − k + 1 ; 2 − α k − β k (cid:21) + n X k = n − p +1 c ( k ) ( α k ) n − k ( n − k )! ( β k ) p − n + k ( p − n + k )! ( β k ) n − p ( n − p )! × F (cid:20) k − n, p − n, α k , β k + k + p − n − n + k − α k , k + p − n + 1 , − n + p − β k ; 2 − α k − β k (cid:21) . ≤ n ≤ m and α ∈ C , 2 α = − m + 1 , − m + 2 . . . , − n , m − n X k =0 (2 n + 2 k + 2 α − n + k + 2 α − m − n +1 C ( α + k + n ) m − n − k ( x ) C (1 − k − n − α ) k ( x ) = δ m,n , which is the key equation to show [5, Theorem 4.1] for the case α = β . Example 4.2.
The problem of finding an inverse of the matrix L β in Theorem 1.1 originally arose in [1]where the finite dimensional lower triangular matrix L ( x ) m,n = q m − n ( q ; q ) m ( q ; q ) n +1 ( q ; q ) m + n +1 ( q ; q ) n C m − n ( x ; q n +2 | q ) , ≤ n ≤ m ≤ N, for arbitrary N ∈ N appears. Using Corollary 1.2 in base q with β = q after conjugation with a diagonalmatrix we find that the inverse matrix is given by( L ( x )) − m,n = q (2 m +1)( m − n ) ( q ; q ) m ( q ; q ) m + n ( q ; q ) m ( q ; q ) n C m − n ( x ; q − m | q ) , ≤ n ≤ m ≤ N. Note that the entries of L ( x ) and its inverse L ( x ) − are independent of the size of N . Example 4.3.
From the generating function (2.2) for the continuous q -ultraspherical polynomials it followsthat ∞ X n =0 C n ( x ; αβ | q ) t n = ( αte iθ , αte − iθ ; q ) ∞ ( te iθ , te − iθ ; q ) ∞ ( αβte iθ , αβte − iθ ; q ) ∞ ( αte iθ , αte − iθ ; q ) ∞ = ∞ X m,n =0 C m ( x ; α | q ) C n ( x ; β | q ) t m ( αt ) n . Comparing the powers of t shows C n ( x ; αβ | q ) = n X k =0 α k C n − k ( x ; α | q ) C k ( x ; β | q ) . Now take β = α − , then (2.1) for β = 1 gives δ n, = n X k =0 α k C n − k ( x ; α | q ) C k ( x ; α − | q ) . (4.1)On the other hand from Lemma 3.1 it follows that n X k =0 α k C n − k ( x ; α | q ) C k ( x ; α − | q ) = n X p =0 d ( p ) e i ( n − p ) θ , x = cos( θ ) . (4.2)Combining (4.1) and (4.2) it follows that d ( p ) = δ n, . Writing out the explicit expression of d ( p ) gives for n > n − p X k =0 α k ( α ; q ) p ( q ; q ) p ( α ; q ) n − p − k ( q ; q ) n − p − k ( α − ; q ) k ( q ; q ) k φ (cid:18) q − p , αq n − p − k , q − k , α − q − p α − , αq − k , q n − p − k +1 ; q, q (cid:19) + n X k = n − p +1 α k ( α ; q ) n − k ( q ; q ) n − k ( α − ; q ) p − n + k ( q ; q ) p − n + k ( α − ; q ) n − p ( q ; q ) n − p φ (cid:18) q k − n , q p − n , α, q k + p − n α − q − n + k α − , q k + p − n +1 , αq − n + p ; q, q (cid:19) . In particular if p = 0 n X k =0 α k ( α ; q ) n − k ( q ; q ) n − k ( α − ; q ) k ( q ; q ) k = 0 . (4.3)7emark that (4.3) also follows from the q -Chu-Vandermonde sum [6, (1.5.2)]. For p = 1 and n n + 1 wefind n X k =0 α k ( α ; q ) n − k ( q ; q ) n − k ( α − ; q ) k ( q ; q ) k (cid:18) − αq n − k )(1 − q k )(1 − αq − k )(1 − q n +1 − k ) q − k (cid:19) = α n ( α − ; q ) n ( q ; q ) n . Remark that this result also follows from applying (4.3) twice. β → Define L ( x ) and M ( x ) by L ( x ) m,n = lim β → L β ( x ) m,n and M ( x ) m,n = lim β → M β ( x ) m,n where thelimit is taken over β = q k , where k ∈ Z . We show that the limits exist, that the entries of L ( x ) are given interms of continuous q -Hermite polynomials and that the entries of M ( x ) are a given in terms of continuous q − -Hermite polynomials.The continuous q -Hermite polynomials are given by H ( x | q ) = n X k =0 ( q ; q ) n ( q ; q ) k ( q ; q ) n − k e i ( n − k ) θ , x = cos( θ ) , (5.1)see [9, § q -Hermite polynomials are, apart from a different normalisation, the specialcase β = 0 of the continuous q -ultraspherical polynomials C n ( x ; 0 | q ) = H n ( x | q )( q ; q ) n . (5.2)The corresponding generating function for the continuous q -Hermite polynomials is ∞ X n =0 H n ( x | q )( q ; q ) n t n = 1( te iθ , te − iθ ; q ) ∞ | t | < , x = cos( θ ) , (5.3)see [9, (14.26.11)].The polynomials H n ( x | q − ) are called the continuous q − -Hermite polynomials and are defined by taking q q − in (5.1), see [2]. The continuous q -Hermite polynomials are orthogonal with respect to a positivemeasure on ( − , q − -Hermite polynomials are orthogonal on the imaginary axisand correspond to an indeterminate moment problem, see [2] and [8]. Theorem 5.1.
The doubly infinite lower triangular matrices L ( x ) and M ( x ) are given by L ( x ) m,n = H m − n ( x | q )( q ; q ) m − n , M ( x ) m,n = ( − m − n q ( m − n ) H m − n ( x | q − )( q ; q ) m − n , n ≤ m, where m, n ∈ Z . M ( x ) and L ( x ) are each others inverse, i.e. L ( x ) M ( x ) = I = M ( x ) L ( x ), where I m,n = δ m,n is the identity. Proof.
With (5.2) we have for n ≤ mL ( x ) m,n = lim β → L β ( x ) m,n = lim β → β q n ; q ) m − n C m − n ( x ; βq n | q ) = H m − n ( x | q )( q ; q ) m − n . From (2.1) it follows that C n ( x ; β | q ) = ( βq − ) n C n ( x ; β − | q − ). Therefore write M β ( x ) m,n as β m − n q ( m − m − n ) ( β q m + n − ; q ) m − n ( β − q − m ) m − n C m − n ( x ; βq m − | q − ) = q − ( m − n ) ( β q m + n − ; q ) m − n C m − n ( x ; βq m − | q − ) . β → M ( x ) m,n = lim β → M β ( x ) m,n = q − ( m − n ) H m − n ( x | q − )( q − ; q − ) m − n = ( − m − n q ( m − n ) H m − n ( x | q − )( q ; q ) m − n . From Theorem 1.1 it follows that L ( x ) M ( x ) = I = M ( x ) L ( x ). Corollary 5.2.
For N ∈ N define lower triangular matrices L ( x ) and M ( x ) L ( x ) m,n = H m − n ( x | q )( q ; q ) m − n , M ( x ) m,n = ( − m − n q ( m − n ) H m − n ( x | q − )( q ; q ) m − n , ≤ n ≤ m ≤ N, Then M ( x ) and L ( x ) are each others inverse, i.e. L ( x ) M ( x ) = I = M ( x ) L ( x ), where I is the identitymatrix. Remark 5.3.
Theorem 5.1 also follows from a generating function for the continuous q − -Hermite polyno-mials. From [7, Theorem 21.2.1] ∞ X n =0 ( − n q ( n ) H n ( x | q − )( q ; q ) n t n = ( te iθ , te − iθ ; q ) ∞ , | t | < , x = cos( θ ) . (5.4)Combining (5.3) and (5.4) it follows that for | t | <
11 = ( te iθ , te − iθ ; q ) ∞ ( te iθ , te − iθ ; q ) ∞ = ∞ X m =0 H m ( x | q )( q ; q ) m t m ! ∞ X n =0 ( − n q ( n ) H n ( x | q − )( q ; q ) n t n ! = ∞ X p =0 p X k =0 H p − k ( x | q )( q ; q ) p − k ( − k q ( k ) H k ( x | q − )( q ; q ) k ! t p . Take p = m − n so that we have m − n X k =0 H m − n − k ( x | q )( q ; q ) m − n − k ( − k q ( k ) H k ( x | q − )( q ; q ) k = δ m,n . (5.5)From (5.5) Theorem 5.1 also follows. Acknowledgements
The author thanks Erik Koelink and Pablo Rom´an for many useful discussions. Parts of the paper havebeen 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 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 alter-native way.
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