$3nj$-symbols and identities for $q$-Bessel functions

The $6j$-symbols for representations of the $\mathrm{SU}(2)$ quantum group are given by Hahn-Exton $q$-Bessel functions. This interpretation leads to several summation identities for the $q$-Bessel functions. Multivariate $q$-Bessel functions are defined, which are shown to be limit cases of multivariate Askey-Wilson polynomials. The multivariate $q$-Bessel functions occur as $3nj$-symbols.


Introduction
It is well known that Wigner's 6 j-symbols for the SU(2) group are multiples of hypergeometric orthogonal polynomials called the Racah polynomials. Similarly, 6 j-symbols for the SU(2) quantum group can be expressed in terms of q-Racah polynomials, which are q-hypergeometric orthogonal polynomials. With this interpretation, properties of 6 j-symbols such as summation formulas and orthogonality relations lead to properties of specific families of orthogonal polynomials, see e.g., [21,22,Chaps. 8,14].
Note that, π φ (γβ) is a self-adjoint diagonal operator in the standard basis.
Remark 2.1 In this paper, we consider tensor products of π 0 . We could also consider the representation π φ 1 ⊗ π φ 2 , but this would not lead to more general results in this paper, because representation labels only occur in phase factors; see [8,§II.A]. The representation space of the tensor product representation is the Hilbert space completion of the algebraic tensor product of copies of 2 (N).

123
Using this, eigenvectors of π 12 (γ γ * ) can be computed (see [8] for details): for p ∈ Z and x ∈ N define where we assume e −n = 0 for n ≥ 1, then π 12 (γ γ * )e 12 x, p = q 2x e 12 x, p . The Clebsch-Gordan coefficients C x,m,n can be given explicitly in terms of Wall polynomials, see [12], which are defined by for n, x ∈ N. The second expression follows from applying transformation [3,III.8] with b → 0. Note that, for x ∈ N, the 2 ϕ 0 -series can be considered as a polynomial in q −n of degree x. This polynomial is (proportional to) an Al-Salam-Carlitz II polynomial. Let the functionp n (q x ; a; q) be defined bȳ p n (q x ; a; q) = (−1) n+x (aq) x−n (aq; q) ∞ (aq; q) n (q; q) n (q; q) x p n (q x ; a; q), (2.5) then from the orthogonality relation for the Wall polynomials and from completeness, we obtain the orthogonality relations x∈Np n (q x ; a; q)p m (q x ; a; q) = δ nm , n∈Np n (q x ; a; q)p n (q y ; a; q) = δ xy , for 0 < a < q −1 . The second relation corresponds to orthogonality relations for Al-Salam-Carlitz II polynomials. The coefficients C x,m,n , m, n ∈ N are defined by and they satisfy C x,n,m = C x,m,n , (2.6) which follows from the explicit expression as a 2 ϕ 1 -function. Furthermore, we define C x,m,n = 0 for m ∈ −N ≥1 or n ∈ −N ≥1 or x ∈ −N ≥1 .

6 j -symbols and q-Bessel functions
In [8], explicit expressions for the 6 j-symbols (and for more general coupling coefficients) have been found. It turns out that they are essentially q-Bessel functions. Here, we derive these results again using a more direct approach, and use this interpretation of the q-Bessel functions to obtain summation identities.

Proposition 3.1 The coefficients R have the following properties:
(i) Orthogonality relations: (ii) R x p 1 ,r 1 ; p 2 ,r 2 = R x p 1 +k,r 1 ; p 2 +k,r 2 for k ∈ Z. (iii) R x p 1 ,r 1 ; p 2 ,r 2 = R x+k p 1 ,r 1 ; p 2 ,r 2 for k ∈ Z ≥−x . (iv) For k, m, n ∈ N, Proof The coefficients R are matrix coefficients of a unitary operator, which leads to the orthogonality relations. The next two identities follow from the * -structure of A q . From β * = −qγ , we obtain 3 x, p 2 ∓1,r 2 , which implies (ii). Identity (iii) follows from α * = δ. Identity (iv) follows from the expansion by taking inner products with e n ⊗ e m ⊗ e k . The duality property follows from identity (iv).

q-Bessel functions
Define which is Jackson's third q-Bessel function (also known as the Hahn-Exton q-Bessel function), see e.g., [16]. Note that, is an entire function in B, so we may take ν to be a negative integer in (3.2); in this case, we have the identity see [16, (2.6)]. We will use the following generating function to identify the 6 j-symbols with q-Bessel functions.
Proof Write J ν as a 1 ϕ 1 -series, interchange the order of summation, and use summa- If t −1 q ν+1 ∈ q −N , the right-hand side in the Proposition 3.2 can be written in terms of a Wall polynomial, which gives the following special case.

Corollary 3.3 For n ∈ N,
Proof In Proposition 3.2 replace ν by ν − n, set t = q ν+1 , and use the transformation (which is a special case of [3, (III.4)]) and the Definition (2.4) of the Wall polynomials.
We are now in a position to show that the 6 j-symbols are essentially q-Bessel functions.
Proof We write out Proposition 3.1(iv) for m = k = 0, and we replace p 2 by − p 2 , then the result follows from Corollary 3.3.

Identities
Several classical identities for 6 j-symbols for SU(2) remain valid for our 6 j-symbols. By Proposition 3.4, these can be interpreted as identities for q-Bessel functions. First of all, the orthogonality relations for the 6 j-symbols from Proposition 3.1 are equivalent to the well-known q-Hankel orthogonality relations, see [16, (2.11)], for the q-Bessel functions J ν .
To derive other identities, it is convenient to represent eigenvectors of γ γ * as binary trees; see e.g., Van der Jeugt's lecture notes [20] for more details. We denote e 12 x,n 2 −n 1 = x n 1 n 2 where n 1 , n 2 , x ∈ Z. Equivalently, we can identify this tree with the Clebsch-Gordan coefficient C x,n 1 ,n 2 , similar as in [18]. The identity e 12 x, p = e 21 x,− p , which is equivalent to (2.6), is represented as where p = n 1 − n 2 . By coupling two of these, we can represent eigenvectors corresponding to threefold tensor products: where p 1 = n 1 + p 1 , p 2 = n 3 − p 2 , and r i jk = x − n i + n j − n k for i, j, k ∈ {1, 2, 3}. Now we can e.g., represent the identities e 1(23) x, p 1 ,r 123 = e 1(32) x, p 1 ,r 132 = e (23)1 x,− p 1 ,r 231 by x n 1 n 2 n 3 The transition (3.1) from e 1(23) x, p 1 ,r to e (12)3 x, p 2 ,r which involves a 6 j-symbol, which is equivalent to identity (ii) in Proposition 3.1 in terms of Clebsch-Gordan coefficients, is represented as where the coefficient R is given by Note that the transition from right to left involves exactly the same 6 j-symbol. To find identities for the 6 j-symbols, we can use the binary trees and identities for these trees as explained above, without referring to the underlying eigenvectors. We obtain the following identities, which can be considered as analogs of Racah's backcoupling identity, the Biedenharn-Elliot (or pentagon) identity, and the hexagon identity.
Theorem 3. 6 The following identities hold: where r i jk = x − n i + n j − n k . (ii) R , which in terms of q-Bessel functions is equivalent to the product formula where P, Q, R, ν, μ 1 , μ 2 ∈ Z and Here 'idem' means that the same expression is inserted but with the parameters interchanged as indicated.
Proof The first identity follows from The corresponding identity for q-Bessel functions is obtained by substituting

123
The third identity is

Remark 3.7 (i) The q-Hankel transform of a function
Identity (i) of Theorem 3.6 shows that the q-Hankel transform maps an orthogonal basis of q-Bessel functions to another orthogonal basis of q-Bessel functions, which implies a factorization of the q-Hankel transform: H r 123 = H r 312 H r 132 . (ii) Identity (ii), the product formula for q-Bessel functions, has appeared before in the literature; representation theoretic proofs are given by Koelink in [13, Corollary 6.5] and Kalnins et al. in [11, (3.20)]. A direct analytic proof is given by Koelink and Swarttouw in [14]. (iii) It is well known that the hexagon identity for classical 6 j-symbols can be interpreted as a quantum Yang-Baxter equation. Here, we obtain an infinitedimensional solution: for u, v ∈ Z, define a unitary operator R(u, v) where {e x | x ∈ Z} is the standard orthonormal basis for 2 (Z). Then the hexagon identity says that the operator R satisfies as an operator identity on 2 (Z) ⊗ 2 (Z) ⊗ 2 (Z).
polynomials introduced by Gasper and Rahman in [4]. In this section, we use the following notation.
provided the sum converges.
Proof The self-duality property follows directly from (4.1). The orthogonality relations follow by induction using the q-Hankel orthogonality relations from Theorem 3.5, which can be written as Define for k = 1, . . . , d + 1, the empty product being equal to 1. Note that J 123 We will show that For k = 1, (4.4) follows directly from (4.2). Now assume that (4.4) holds for a certain k, then by (4.2) and (4.3), which proves the orthogonality relations.
Next we show that the multivariate q-Bessel functions can be considered as limit cases of multivariate Askey-Wilson polynomials. The 1-variable Askey-Wilson polynomials are defined by (ab, ac, ad; q) n a n 4 ϕ 3 q −n , abcdq n−1 , ax, a/x ab, ac, ad ; q, q , which are polynomials in x +x −1 of degree n, and they are symmetric in the parameters a, b, c, d. Using notation as in [9], the multivariate Askey-Wilson polynomials are defined as follows. Let n = (n 1 , . . . , n d ) ∈ N d and x = (x 1 , . . . , x d ) ∈ (C × ) d , then the d-variable Askey-Wilson polynomials are defined by Proof First we substitute in (4.6) (recall, x d+1 = α d+2 ). The 4 ϕ 3 -part of the jth factor p n j is where the empty sum equals 0. Letting m → ∞, this function tends to Finally, we substitute and set ν 0 − j k=1 λ k = j for j = 0, . . . , d, then we have which we recognize as the 1 ϕ 1 -part of the jth factor of the multivariate q-Bessel function J ν (x, ), see (4.1).

3n j-symbols
Let k ∈ N ≥1 , and let r, s ∈ Z k , n ∈ Z k+2 . We define the 3nj-symbols R x;n r,s to be the coupling coefficients between two specific binary trees corresponding to (k + 2)-fold tensor product representations. We will use the following notation: The second identity follows from repeated application of the first identity.
From (3.4) it follows that R x,n r,s is essentially a multivariate q-Bessel function as defined by (4.1).
Note that, this corollary and Proposition 4.3 together give a representation theoretic proof of Theorem 4.1.
Our next goal is to prove a summation identity for the multivariate q-Bessel functions. Let us first mention that by interpreting a binary tree as a product of Clebsch-Gordan coefficients, the 3nj-symbols R x,n r,s satisfy, by definition, the formula where r 0 = x, r k+1 = n k+2 , s 0 = n 1 , s k+2 = x. The functions C x,r,n can be considered as multivariate Wall polynomials, which are q-analogs of Laguerre polynomials. In this light, (4.7) is a multivariate q-analog of an identity proved by Erdélyi [2] which states that the Hankel transform maps a product of two Laguerre polynomials to a product of two Laguerre polynomials. For the 3nj-symbols R x,n r,s , there exists a multivariate analog of the Biedenharn-Elliott identity. In terms of q-Bessel functions, this gives an expansion formula for k-variable q-Bessel functions in terms of (k − 1)-variable q-Bessel functions. The identity requires also another 3nj-symbol. For r, s ∈ Z k , n ∈ Z k+2 , x ∈ Z, let S x,n Note that, ŝ = s 1 · · · s k . This 3nj-symbol can of course also be considered as a multivariate q-Bessel function (see the following result), but it lacks the self-duality property. Let us first express S in terms of the 6 j-symbols. Proof We use the transition and where r j = (r 1 , . . . , r k− j−2 ) and n j = (n 2 , . . . , n k− j+1 ). We set s k+1 = x and r 0 = n 2 , then applying this transition successively on subtrees for j = 0, . . . , k − 1 gives Changing the index gives the stated expression for the coupling coefficient S.
The following identity is the multivariate analog of the Biedenharn-Elliott identity from Theorem 3.6, i.e., the k = 2 case gives back Theorem 3.6(ii).
Proof This follows from the transition Remark 4.8 It seems that there are no analogs for the 3nj-symbols R of identities (i) and (iii) of Theorem 3.6, but there does exist an analog of Theorem 3.6(i) involving only the 3nj-symbols S which may be of interest. This is obtained as follows. Let n ∈ Z k+2 . For j ∈ {1, 2, . . . , k + 1}, we define n j = (n k+3− j , . . . , n k+2 , n 1 , . . . , n k+2− j ). Furthermore, given a vector v, we denote (as in Theorem 4.7) by v the vector v without the first component, and we set n j = (n j ) . Consider the transition For k = 1, this gives back Theorem 3.6(i).
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