Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC
aa r X i v : . [ m a t h . C A ] O c t Product formulas for a two-parameter family ofHeckman-Opdam hypergeometric functions of type BC
Michael VoitFakult¨at Mathematik, Technische Universit¨at DortmundVogelpothsweg 87, D-44221 Dortmund, Germanye-mail: [email protected] 6, 2018
Abstract
In this paper we present explicit product formulas for a continuous two-parameterfamily of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers C q ⊂ R q of type B . These formulas are related to continuous one-parameter families ofprobability-preserving convolution structures on C q × R . These convolutions on C q × R are constructed via product formulas for the spherical functions of the symmetric spaces U ( p, q ) / ( U ( p ) × SU ( q )) and associated double coset convolutions on C q × T with the torus T . We shall obtain positive product formulas for a restricted parameter set only, whilethe associated convolutions are always norm-decreasing.Our paper is related to recent positive product formulas of R¨osler for three series ofHeckman-Opdam hypergeometric functions of type BC as well as to classical productformulas for Jacobi functions of Koornwinder and Trimeche for rank q = 1. Key words: Hypergeometric functions associated with root systems, Heckman-Opdam the-ory, hypergroups, product formulas, Grassmann manifolds, spherical functions, signed hy-pergroups, Haar measure.AMS subject classification (2000): 33C67, 43A90, 43A62, 33C80.
It is well-known by the work of Heckman and Opdam ([H], [HS], [O1], [O2]) that the sphericalfunctions on the Grassmann manifolds of rank q ≥ F = R , C , H may beregarded as Heckman-Opdam hypergeometric functions F BC q of type BC on the closed Weylchambers C q := { t = ( t , · · · , t q ) ∈ R q : t ≥ t ≥ · · · ≥ t q ≥ } of type B . The associated product formulas for the spherical functions were stated by R¨osler[R3] for the dimensions p ≥ q in a form such that these formulas can be extended by someprinciple of analytic continuation to all parameters p ∈ R with p > q −
1. In this way, R¨osler[R3] obtained three continuous series of product formulas for F BC q (for F = R , C , H ) as wellas associated commutative, probability-preserving convolution algebras of measures on C q ,so-called commutative hypergroups. For the theory of hypergroups we refer to [J] and [BH].In this paper we start with the symmetric spaces U ( p, q ) / ( U ( p ) × SU ( q )) for F = C and p ≥ q . Here the spherical functions can be regarded as functions on C q × T with the torus1 := { z ∈ C : | z | = 1 } . These functions can be expressed in terms of K -spherical functions ofthe Hermitian symmetric spaces U ( p, q ) / ( U ( p ) × U ( q )) and thus also in terms of the functions F BC q depending on the integer multiplicity parameter p ≥ q and some spectral parameter l ∈ Z ; see [Sh], [HS]. Following [R3], we shall write down product formulas for the sphericalfunctions of U ( p, q ) / ( U ( p ) × SU ( q )) in Section 2 of this paper as product formulas on C q × T for integers p ≥ q . We then use these formulas in Section 3 to construct associated productformulas on the universal covering C q × R of C q × T for a class of functions which are definedin terms of the functions F BC q where the F BC q depend now on two continuous multiplicityparameters p ≥ q − l ∈ R . Here, the extension from integers p ≥ q and l ∈ Z to real numbers p > q − l ∈ R is carried out by Carleson’s theorem, a principle ofanalytic continuation. The degenerated limit case p = 2 q − C q × R for p ≥ q − C q × R , ∗ p ). We shall also derive the Haar measures ofthese hypergroups.The product formulas on C q × R and the associated hypergroup structures in Sections3 and 4 form the basis to derive a lot of further product formulas and convolution algebrasby taking suitable quotients. In particular, we immediately obtain extensions of the productformulas on C q × T for the spherical functions of U ( p, q ) / ( U ( p ) × SU ( q )) with integers p ≥ q to real parameters p ≥ q −
1. Moreover, we recover R¨osler’s formulas in [R3] on C q for F = C . More generally, we obtain explicit product formulas and convolution structures on C q for all hypergeometric functions F BC q with multiplicities k = ( k , k , k ) = ( p − q − l, / l, p ≥ q − l ∈ R . It will turn out in Section 5 that these formulaslead to probability-preserving convolutions and thus classical commutative hypergroups for | l | ≤ /q while for arbitrary l ∈ R the positivity of the product formulas remain open. Onthe other hand we shall see in Section 6 that the product formulas for F BC q lead for all l ∈ R at least to norm-decreasing convolution algebras which are associated with certain so-calledsigned hypergroup structures on C q . For this notion we refer to [R1], [RV], and referencescited there.We also notice that for rank q = 1, our results are closely related with the classicalwork of Flensted-Jensen and Koornwinder ([F], [FK], [K]) on Jacobi functions and to theconvolutions of Trimeche [T] on the space { z ∈ C : | z | ≥ } which is homeomorphic with[0 , ∞ [ × T .Before starting with the analysis of ( U ( p, q ) / ( U ( p ) × SU ( q )) in Section 2, we recapitulatesome notions and facts. For integers p > q ≥ G/K over F = R , C , H with G = SO ( p, q ) , SU ( p, q ) , Sp ( p, q )and the maximal compact subgroups K = SO ( p ) × SO ( q ) , S ( U ( p ) × U ( q )) , Sp ( p ) × Sp ( q ) , respectively. By the well-known KAK decomposition of G , a system of representatives ofthe K -double cosets on G is given by the matrices a t = I p − q t sinh t t cosh t (1.1)2ith t in the closed Weyl chamber C q where cosh t, sinh t are the q × q diagonal matricescosh t := diag (cosh t , . . . , cosh t q ) , sinh t := diag (sinh t , . . . , sinh t q ) . Therefore, continuous K -biinvariant functions on G are in a natural one-to-one correspon-dence with continuous functions on C q . Moreover, by the theory of Heckman and Opdam[H], [HS], [O1], [O2], in this way the spherical functions of ( G, K ), i.e., the continuous, K -biinvariant functions ϕ ∈ C ( G ) satisfying the product formula ϕ ( g ) ϕ ( h ) = Z K ϕ ( gkh ) dk ( g, h ∈ G, dk the normalized Haar measure of K ) , (1.2)are precisely the Heckman-Opdam hypergeometric functions t F ( λ, k ; t ) := F BC q ( λ, k ; t ) ( t ∈ C q )of type BC with λ ∈ C q and the multiplicity parameter k = ( k , k , k ) = ( d ( p − q ) / , ( d − / , d/ ± e i , ± e i , and 2( ± e i ± e j ) of the root system 2 · BC q in the notionof Heckman and Opdam. Here, d ∈ { , , } is the dimension of R , C , H over R ; see Remark2.3 of [H] and also [R3].For given q ≥ d , R¨osler [R3] derived the product formula (1.2) explicitly as aproduct formula for the corresponding F BC q on C q depending on p ≥ q such that thisformula can be extended to a product formula on C q for arbitrary real parameters p > q − p > q −
1, each of these productformulas gives rise to a commutative hypergroup structure on C q , i.e., a probability preservingcommutative Banach-*-algebra structure on the Banach space of all bounded signed Borelmeasures on C q with total variation norm; see [BH] and [J] for the theory of hypergroups.We now modify the approach of [R3] for F = C , i.e. d = 2, by considering K -sphericalfunctions according to Ch. I.5 of [HS] and [Sh] as follows: Take the Gelfand pair ( G, ˜ K ) :=( U ( p, q ) , U ( p ) × SU ( q )) for p ≥ q as well as the maximal compact subgroup K := U ( p ) × U ( q ) ⊂ K . Then G/K := U ( p, q ) / ( U ( p ) × U ( q )) ≡ SU ( p, q ) /S ( U ( p ) × U ( q ))is a Hermitian symmetric space. It is well-known that the usual spherical functions of ( G, ˜ K )(i.e., satisfying (1.2)) are in a natural way in a one-to-one correspondence with the so-called K -spherical functions on G of type l with l ∈ Z . Recapitulate that these K -spherical functionsof type l are defined as continuous functions ϕ ∈ C ( G ) with ϕ ( e ) = 1 satisfying twistedinvariance conditions as well as twisted product formulas associated with the characters χ l (cid:18)(cid:18) u v (cid:19)(cid:19) := (∆ v ) l ( l ∈ Z ) (1.3)of K , where ∆ v stands for the determinant of v . The twisted invariance condition reads as ϕ ( k gk ) = χ l ( k k ) − · ϕ ( g ) for g ∈ G, k , k ∈ K, (1.4)and the twisted product formula as ϕ ( g ) ϕ ( h ) = Z S ( U ( p ) × U ( q )) χ l ( k ) ϕ ( gkh ) dk ( g, h ∈ G ) . (1.5)3hese spherical functions ϕ of type l can be also characterized as eigenfunctions of somealgebra D ( χ l ) of SU ( p, q )-invariant differential operators in Section 5.1 of [HS] and [Sh] andcan be written down therefore explicitly in terms of F BC q . In particular, if G// ˜ K is identifiedwith C q × T , we conclude from Theorem 5.2.2 of [HS] that the spherical functions may beregarded as the functions( t, z ) z l q Y j =1 cosh l t j · F C q ( λ, k ( p, l ); t ) ( λ ∈ C q , l ∈ Z ) (1.6)with the multiplicities k ( p, l ) = ( k ( p, l ) , k ( p, l ) , k ( p, l )) = ( p − q − l, / l, . (1.7)We shall use this characterization in the next section to derive an explicit product formulafor these functions.It is a pleasure to thank Tom Koornwinder for some essential hints to details in themonograph [HS] as well Margit R¨osler for many fruitful discussions. ( U ( p, q ) , U ( p ) × SU ( q )) and their prod-uct formula In this section we derive an explicit product formula for the spherical functions for theGelfand pair ( G, ˜ K ) := ( U ( p, q ) , U ( p ) × SU ( q )). In fact, this is a Gelfand pair by standardcriteria, see e.g. Corollary 1.5.4 of [GV]. Moreover, this is also a direct consequence of theexplicit convolution (2.9) below. We first identify the double coset space G// ˜ K with thedirect product C q × T of the Weyl chamber C q and the torus T . This can be done similar towell known case of the Hermitian symmetric space G/K := U ( p, q ) / ( U ( p ) × U ( q )) where, bythe KAK -decomposition, a system of representatives of the K -double cosets in G is givenby the matrices a t of Eq. (1.1) with t ∈ C q . A system of representatives of the ˜ K -double cosets in G is given by the matrices a t,z = I p − q r q ( z ) − · cosh t sinh t t r q ( z ) · cosh t (2.1) for t ∈ C q and z ∈ T where the mapping r q : T → T is the q -th root on T with r q ( e it ) := e it/q with t ∈ [0 , π [ .Proof. We first check that each double coset has a representative of the form a t,z . In fact,by the well known KAK -decomposition of G , each g ∈ G has the form g = (cid:18) u v (cid:19) I p − q t sinh t t cosh t (cid:18) u v (cid:19) with t ∈ C q , u , u ∈ U ( p ), v , v ∈ U ( q ). The matrices v k ∈ U ( q ) ( k = 1 ,
2) can be writtenas v k = z k · ˜ v k with ˜ v k ∈ SU ( q ) and z k = r q (∆( v k )) ∈ T . Therefore, defining˜ u := u (cid:18) I p − q z I q (cid:19) and ˜ u := (cid:18) I p − q z I q (cid:19) u ,
4e obtain g = (cid:18) ˜ u
00 ˜ v (cid:19) I p − q z z ) − cosh t sinh t t z z cosh t (cid:18) ˜ u
00 ˜ v (cid:19) , (2.2)which shows that each double coset has a representative of the form˜ a t,z := I p − q z − cosh t sinh t t z cosh t (2.3)with z ∈ T . Moreover, this computation also shows that for all q -th roots of unity z ∈ T ,the matrices ˜ a t,z and ˜ a t,zz are contained in the same ˜ K -double coset, i.e., each double cosethas a representative of the form ˜ a t,z with z = e iθ , θ ∈ [0 , π/q [, as claimed.In order to show that the a t,z are contained in different double cosets for different ( t, z ),we briefly discuss how the parameters t, z of a double coset of a arbitrary group element g ∈ G can be constructed explicitly. This will be important also later on for the convolution.We write any g ∈ G in ( p × q )-block notation as g = (cid:18) A ( g ) B ( g ) C ( g ) D ( g ) (cid:19) . Moreover, in ( p × q )-block notation, we write a t,z = (cid:18) A t,z B t,z C t,z D t,z (cid:19) . Assume now that g ∈ G has the form (2.2) with z z = r q ( z ) with z = e iθ , θ ∈ [0 , π/q [.Then D ( g ) has the form D ( g ) = r q ( z ) · ˜ v cosh t ˜ v (2.4)with ˜ v , ˜ v ∈ SU ( q ). We now consider the singular spectrum σ sing : M q,q ( C ) → C q , σ sing ( a ) := p spec ( a ∗ a ) ∈ R q where the singular values are ordered by size. We also consider the map arg : C × → T , arg ( z ) := z/ | z | . Then, by (2.4), t = arcosh ( σ sing ( D ( g ))) in all components and z = arg (∆( D ( g ))) . (2.5)This completes the proof of the lemma.We proceed with the notations of the second part of the proof of the lemma and write thegeneral product formula (1.2) for spherical functions as a product formula on the parameterspace C q × T . For this, we take t, s ∈ C q and z , z ∈ T and evaluate the integral Z ˜ K f ( a t,z ka s,z ) dk with k = (cid:18) u v (cid:19) . We have a t,z ka s,z = (cid:18) ∗ ∗∗ C t,z u B s,z + D t,z v D s,z (cid:19) D ( a t,z k a s,z ) = C t,z u B s,z + D t,z v D s,z = (0 , sinh t ) u (cid:18) s (cid:19) + r q ( z z ) cosh t v cosh s. With the block matrix σ := (cid:18) I q (cid:19) ∈ M p,q ( C ) (2.6)this can be written as D ( a t,z k a s,w ) = sinh t σ ∗ uσ sinh s + r q ( z z ) cosh t v cosh s. Therefore, if we regard a ˜ K -biinvariant function f ∈ C ( G ) as a continuous function on C q × T , Z ˜ K f ( a t,z ka s,z ) dk == Z ˜ K f (arcosh ( σ sing ( D ( a t,z k a s,z ))) , arg (∆( D ( a t,z k a s,z ))))= Z U ( p ) Z SU ( q ) f (cid:16) arcosh ( σ sing (sinh t σ ∗ uσ sinh s + r q ( z z ) cosh t v cosh s )) ,arg (∆(sinh t σ ∗ uσ sinh s + r q ( z z ) cosh t v cosh s )) (cid:17) dv dw. Notice that σ ∗ uσ ∈ M q ( F ) is just the lower right q × q -block of σ and is contained in theball B q := { w ∈ M q,q ( C ) : w ∗ w ≤ I q } , where w ∗ w ≤ I q means that I q − w ∗ w is positive semidefinite. In order to reduce the U ( p )-integration, we use Lemma 2.1 of [R3] and obtain that for p ≥ q the integral above is equalto 1 κ p Z B q Z SU ( q ) f (cid:16) arcosh ( σ sing ( sinh t w sinh s + r q ( z z ) cosh t v cosh s )) , (2.7) arg (∆(sinh t w sinh s + r q ( z z ) cosh t v cosh s )) (cid:17) · ∆( I q − w ∗ w ) p − q dv dw. with κ p := Z B q ∆( I q − w ∗ w ) p − q dw, (2.8)and where dw means integration w.r.t. the Lebesgue measure. After substitution w r q ( z z ) w , we arrive at the following explicit product formula: Let p ≥ q . If a ˜ K -spherical function ϕ ∈ C ( G ) is regarded as a continuousfunction on C q × T as described above, then the associated product formula for sphericalfunctions ϕ has the following form on C q × T : ϕ ( t, z ) ϕ ( s, z ) = (2.9)= 1 κ p Z B q Z SU ( q ) ϕ (cid:16) arcosh ( σ sing ( sinh t w sinh s + cosh t v cosh s )) ,z z · arg (∆( sinh t w sinh s + cosh t v cosh s )) (cid:17) · ∆( I q − w ∗ w ) p − q dv dw. p ≤ q −
1, the integral over the ball B q in (2.9) does not exist and thathere κ p = ∞ . However, for p = 2 q −
1, a degenerated version of formula may be writtendown explicitly similar to other series of symmetric spaces of higher rank associated withmotion groups or Heisenberg groups; see Section 3 of [R2] and Remark 2.14 of [V2]. We shallpresent a degenerated version of (2.9) in the end of Section 3.In the case of rank q = 1, the space C q × T = [0 , ∞ [ × T may be identified with the exterior Z := { z ∈ C : | z | ≥ } of the unit disk via polar coordinates. The associated convolution(2.9) in the general case p ≥ q and in the degenerated case p = 2 q was computed in thiscase by Trimeche [T].We next turn to the classification of all ˜ K -spherical functions ϕ ∈ C ( G ). In order totranslate it into a standard version in the Heckman-Opdam theory, we recapitulate thefollowing result: For ϕ ∈ C ( G ) the following properties are equivalent: (1) ϕ is ˜ K -spherical, i.e., ˜ K -biinvariant with ϕ ( g ) ϕ ( h ) = R ˜ K ϕ ( gkh ) dk for g, h ∈ G . (2) ϕ ( e ) = 1 , and there exists a unique l ∈ Z such that ϕ ( k gk ) = χ l ( k k ) − · ϕ ( g ) for g ∈ G, k , k ∈ K (2.10) and ϕ ( g ) ϕ ( h ) = Z K ϕ ( gkh ) χ l ( k ) dk for g, h ∈ G. (2.11)(3) ϕ is a spherical function of type χ l for some l ∈ Z in the sense of Definition 5.2.1of Heckman [HS], i.e., ϕ is an eigenfunction with respect to all members of a certainalgebra of G -invariant differential operators.Proof. For abbreviation define the diagonal matrix d z := (cid:18) I p zI q (cid:19) ∈ K for z ∈ T .For (1) = ⇒ (2) consider a ˜ K -spherical function ϕ . Then ϕ | K is ˜ K -spherical, and as K/ ˜ K ≡ T , we find a unique l ∈ Z with ϕ | K = χ − l . Moreover, as ϕ ( kg ) = ϕ ( k ) ϕ ( g ) = ϕ ( gk )for g ∈ G and k ∈ K , (2.10) is clear. For (2.11), we use the normalized Haar measure dz on T and observe that for g, h ∈ G , Z K ϕ ( gkh ) · χ l ( k ) dk = Z T Z ˜ K ϕ ( gd z kh ) χ l ( d z k ) dk dz = Z T ϕ ( gd z ) ϕ ( h ) χ l ( d z ) dz = ϕ ( g ) ϕ ( h ) Z T ϕ ( d z ) χ l ( d z ) dz = ϕ ( g ) ϕ ( h )as claimed.For (2) = ⇒ (1) consider ϕ as in (2). Then by (2.10), ϕ | K = χ − l , and ϕ is ˜ K -biinvariant.Moreover, for g, h ∈ G and z ∈ T , ˜ Kgd z ˜ K = ˜ Kd z g ˜ K , and thus ϕ ( g ) ϕ ( h ) = Z K ϕ ( gkh ) · χ l ( k ) dk = Z T Z ˜ K ϕ ( gd z kh ) χ l ( d z k ) dk dz = Z T Z ˜ K ϕ ( d z gkh ) χ l ( d z ) dk dz = Z ˜ K ϕ ( gkh ) dk as claimed.The equivalence of (2) and (3) is already mentioned in Section 5.2 of Heckman [HS] andcan be checked in the same way as for classical spherical functions as it is carried out e.g. inSection IV.2 of Helgason [Hel]. 7he elementary spherical functions ϕ on G of type χ l for l ∈ Z are classified by Heckmanin Section 5 of [HS]. For a description, we consider the root system R := 2 · BC q with thepositive roots R + := { e i , e i : i = 1 , . . . , q } ∪ { e i − e j ) : 1 ≤ i < j ≤ q } as well as the associated Heckman-Opdam hypergeometric functions according to [H], [HS],[O1], [O2] which we denote by F BC q ( λ, k ; t ) with λ ∈ C q , t ∈ C q , and with multiplicity k = ( k , k , k ) where the k i belong to the roots as in the ordering of R + above. Note thatthe Heckman-Opdam hypergeometric functions F BC q ( λ, k ; t ) exists for all λ ∈ C q , t ∈ C q whenever the multiplicity k is contained in some open regular subset K reg ⊂ C . It iswell-known (see Remark 4.4.3 of [HS]) that, in our notation, { k = ( k , k , k ) : Re k ≥ , Re k + k ≥ } ⊂ K reg . (2.12)Taking Lemma 2.3 into account, we obtain the following known classification from Theorem5.2.2 of [HS]: If ˜ K -spherical functions on G are regarded as functions on C q × T as above,then the ˜ K -spherical functions on G are given precisely by ϕ pλ,l ( t, z ) = z l · q Y j =1 cosh l t j · F BC q ( iλ, k ( p, q, l ); t ) with λ ∈ C q , l ∈ Z , and the multiplicity k ( p, q, l ) = ( p − q − l,
12 + l, ∈ K reg . For q = 1, the parameter k is irrelevant and usually suppressed. If onecompares the one-dimensional example of Heckman-Opdam functions on p. 89f of [O1] withthe classical definition of the Jacobi functions ϕ ( α,β ) λ with α = k + k − / β = k − / F BC ( iλ, k ; t ) = ϕ ( α,β ) λ ( t );see also Example 3.4 in [R3]. Therefore, by Theorem 2.4, the U ( p )-spherical functions on U ( p,
1) are given by ϕ pλ,l ( t, z ) = z l · cosh l t · ϕ ( p − ,l ) λ ( t ) ( t ≥ , z ∈ T )with l ∈ Z and λ ∈ C . This is a classical result of Flensted-Jensen [F].Let us summarize the results above. For integers q ≥ q , we obtain from Proposition 2.2that the double coset convolution of point measures on the double coset space U ( p, q ) // ( U ( p ) × SU ( q )) ≃ C q × T is given by( δ ( s,z ) ∗ p δ ( t,z ) )( f ) := (2.13)= 1 κ p Z B q Z SU ( q ) f (cid:16) d ( t, s ; v, w ) , z z · h ( t, s ; v, w ) (cid:17) · ∆( I q − w ∗ w ) p − q dv dw s, t ∈ C q , z , z ∈ T and all bounded continuous functions f ∈ C b ( C q × T ) with theabbreviations d ( t, s ; v, w ) := ( d j ( t, s ; v, w )) j =1 ,...,q := arcosh ( σ sing (sinh t w sinh s + cosh t v cosh s )) ∈ C q (2.14)and h ( t, s ; v, w ) := ∆(sinh t w sinh s + cosh t v cosh s ) . (2.15)It is well known (see e.g. [J]) that this double coset convolution can be uniquely extended ina bilinear, weakly continuous convolution on the Banach space M b ( C q × T ) of all boundedregular Borel measures on C q × T , and that ( C q × T , ∗ p ) forms a commutative hypergroup.Moreover, by Theorem 2.4, the functions ϕ pλ,l ( λ ∈ C q , l ∈ Z ) form the set of all multiplicativefunctions on these hypergroups. For integers p ≥ q , the image ω p ∈ M + ( C q × T ) of the Haar measure on U ( p, q ) under the canonical projection U ( p, q ) → U ( p, q ) // ( U ( p ) × SU ( q )) ≃ C q × T is given, as a measure on C q × T , by dω p ( t, z ) = const · q Y j =1 sinh p − q +1 t j cosh t j · Y ≤ i 1) as above. Thesefunctions are related to the spherical functions ϕ pλ,l of the preceding section by ϕ pλ,l ( t, e iθ ) = ψ λ,l ( t, θ ) ( t ∈ C q , θ ∈ R , l ∈ Z ) . (3.2)In a second step we notice that both sides of this product formula depend analytically on theparameters l and p and extend the formula to a positive product formula for all l ∈ C andall p ∈ R with p > q − .1 Theorem. Let f ( z ) be holomorphic in a neighbourhood of { z ∈ C : Re z ≥ } satisfying f ( z ) = O (cid:0) e c | z | (cid:1) on Re z ≥ for some c < π . If f ( z ) = 0 for all nonnegative integers z , then f is identically zero for Re z > . To state our product formula on C q × R , we need the fact from complex analysis that ananalytic function f : G → C on a connected, simply connected set G ⊂ C n with 0 f ( G )admits an analytic logarithm g : G → C with f = e g . In fact, this known result can be shownlike in the well-known one-dimensional case by using the fact that a closed 1-form is exacton a simply connected domain. We also recapitulate that SU ( q ) is simply connected. Theseresults and the results of Section 2 lead to the following extended product formula: Let q ≥ be an integer. For all l ∈ C and p ∈ ]2 q − , ∞ [ , the functions ψ pλ,l of Eq. (3.1) satisfy the product formula ψ pλ,l ( t, θ ) ψ pλ,l ( s, θ ) = (3.3)= 1 κ p Z B q Z SU ( q ) ψ pλ,l (cid:16) d ( s, t ; v, w ) , θ + θ + Im ln h(s , t; v , w) (cid:17) · ∆(I q − w ∗ w) p − dv dw . for all s, t ∈ C q , θ , θ ∈ R where κ p , dw , the functions d, h , and other data are defined as inSection 2, and where ln denotes the unique analytic branch of the logarithm of the function ( s, t, w, v ) h ( s, t ; v, w ) = ∆(sinh t w sinh s + cosh t v cosh s ) on the simply connected domain C q × C q × B q × SU ( q ) with ln ∆( I q ) = 0 .Proof. In a first step take a parameter l ∈ Z and an integer p ≥ q . In this case, (3.3) followsimmediately from (3.2) and the product formula (2.9).We next observe that both sides of (3.3) are analytic in the variables p, l, λ . We nowwant to employ Carleson’s theorem to extend (3.3) to p ∈ ]2 q − , ∞ [ and l ∈ C . How-ever, for this we need some exponential growth estimates for the hypergeometric functions F BC q ( iλ, k ( p, q, l ); t ) with respect to the parameters p and l in some suitable right half planes,and such suitable exponential estimates are available only for real, nonnegative multiplicities;see Proposition 6.1 of [O1], [Sch], and Section 3 of [RKV]. We thus proceed in several steps,follow the proof of Theorem 4.1 of [R3], and restrict our attention first to a discrete set ofspectral parameters λ for which F BC q is a product of the c -function and Jacobi polynomialssuch that in this case the growth condition can be checked. Carleson’s theorem then leadsto (3.3) for this discrete set of spectral parameters λ and all p ∈ ]2 q − , ∞ [ and l ∈ C . Ina further step we fix p ∈ ]2 q − , ∞ [ and l ∈ [ − / , p − q ] and extend (3.3) by Carleson’stheorem to all spectral parameters λ ∈ C q . Finally, usual analytic continuation leads to thegeneral result in the theorem for l ∈ C .Let us go into details. We need some notations and facts from [O1], [O2], and [HS]. Forour root system R := 2 · BC q with the set R + of positive roots as in Section 2, we define thehalf sum of roots ρ ( k ) := 12 X α ∈ R + k ( α ) α = ( k + 2 k ) q X j =1 e j + 2 k q X j =1 ( q − j ) e j (3.4)as well as the c -function c ( λ, k ) := Y α ∈ R + Γ( h λ, α ∨ i + k ( α ))Γ( h λ, α ∨ i + k ( α ) + k ( α )) · Y α ∈ R + Γ( h ρ ( k ) , α ∨ i + k ( α ) + k ( α ))Γ( h ρ ( k ) , α ∨ i + k ( α )) (3.5)10ith the usual inner product on C q and the conventions α ∨ := 2 α/ h α, α i and k ( α ) = 0 for α / ∈ R . Notice that c is meromorphic on C q × C . We now consider the dual root system R ∨ = { α ∨ : α ∈ R } , the coroot lattice Q ∨ = Z .R ∨ , and the weight lattice P = { λ ∈ R q : h λ, α ∨ i ∈ Z ∀ α ∈ R } of R . Further, denote by P + = { λ ∈ P : h λ, α ∨ i ≥ ∀ α ∈ R + } theset of dominant weights associated with R + . Then, by Eq. (4.4.10) of [HS] and by [O1], weobtain for all k ∈ K reg and all λ ∈ P + , F BC q ( λ + ρ ( k ) , k ; t ) = c ( λ + ρ ( k ) , k ) P λ ( k ; t ) (3.6)where c ( λ, k ) is the c -function (3.5) which is meromorphic on C q × K , and where the P λ arethe Heckman-Opdam Jacobi polynomials of type BC q . We now consider the parameters k p,l := ( p − q − l, / l, ∈ K reg (see (2.12)) as well as the associated half sum of roots ρ ( k p,l ) = ( p − q + l + 1) q X j =1 e j + 2 q X j =1 ( q − j ) e j . (3.7)Using the asymptotics of the gamma function, we now check the growth of c ( λ + ρ ( k p,l ) , k p,l )for fixed λ ∈ P + and parameters p, l → ∞ in suitable half planes. Indeed, by Stirling’sformula, Γ( z + a ) / Γ( z ) ∼ z a for z → ∞ , Re z ≥ . Moreover, for ρ = ρ ( k p,l ), c ( λ + ρ, k ) == q Y i =1 Γ( λ i + ρ i ) Γ( ρ i + k )Γ( λ i + ρ i + k ) Γ( ρ i ) · q Y i =1 Γ (cid:0) λ i + ρ i + k (cid:1) Γ( ρ i + k + k )Γ (cid:0) λ i + ρ i + k + k (cid:1) Γ( ρ i + k ) · Y i 1, a degenerate version of the product formula (3.3) is available.For this we need some notations and facts. We here follow Section 3 of [R2] and Remark 2.14of [V2] where also such limit cases of product formulas for Bessel and Laguerre functions onmatrix cones were considered.We fix the dimension q and consider the matrix ball B q := { w ∈ M q,q ( C ) : w ∗ w ≤ I q } asabove as well as the ball B := { y ∈ C q : k y k < } and the sphere S := { y ∈ C q : k y k = 1 } .By Lemma 3.6 and Corollary 3.7 of [R2], the mapping P : B q → B q from the direct product B q to B q with P ( y , . . . , y q ) := y y ( I q − y ∗ y ) / ... y q ( I q − y ∗ q − y q − ) / · · · ( I q − y ∗ y ) / (3.10)establishes a diffeomorphism such that the image of the measure∆( I q − w ∗ w ) p − q dw P − is given by Q qj =1 (1 −k y j k ) p − q − j dy . . . dy q . Therefore, for p > q − 1, the productformula (3.3) may be rewritten as ψ pλ,l ( t, θ ) ψ pλ,l ( s, θ ) = 1 κ p Z B q Z SU ( q ) ψ pλ,l (cid:16) d ( t, s ; v, P ( y )) , θ + θ + Im ln h ( t, s ; v, P ( y )) (cid:17) ·· q Y j =1 (1 − k y j k ) p − q − j dy . . . dy q dw (3.11)with y = ( y , . . . , y q ) ∈ B q , where dy , . . . , dy q means integration with respect to the Lebesguemeasure on C q . Moreover, for p ↓ q − 1, we obtain from (3.11) by continuity the followingdegenerated product formula: ψ q − λ,l ( t, θ ) ψ q − λ,l ( s, θ ) = 1 κ q − Z B q Z SU ( q ) ψ q − λ,l (cid:16) d ( t, s ; v, P ( y )) , θ + θ + Im ln h ( t, s ; v, P ( y )) (cid:17) ·· q − Y j =1 (1 − k y j k ) q − − j dy . . . dy q − dσ ( y q ) dw (3.12)where σ ∈ M ( S ) is the uniform distribution on S and κ q − := Z B q q − Y j =1 (1 − k y j k ) q − − j dy . . . dy q − dσ ( y q ) . C q × R The positive product formulas (3.3) for real parameters p > q − p = 2 q − C q × R parametrizedby p ≥ q − 1. In fact, these convolutions form commutative hypergroups, which havethe functions ψ pλ,l ( λ ∈ C q , l ∈ C ) as multiplicative functions. For q = 1, our convolutionstructures are closely related to those studied by Trimeche [T].Before going into details, we briefly recapitulate some notions from hypergroup theory.For more details we refer to [J] and the monograph [BH]. Hypergroups generalize the con-volution of bounded measures on locally compact groups such that the convolution δ x ∗ δ y oftwo point measures δ x , δ y is a probability measure with compact support, but not necessarilya point measure. A hypergroup is a locally compact Hausdorff space X with a weakly contin-uous, associative, bilinear convolution ∗ on the Banach space M b ( X ) of all bounded regularBorel measures on X such that the following properties hold:(1) For all x, y ∈ X , δ x ∗ δ y is a compactly supported probability measure on X such thatthe support supp ( δ x ∗ δ y ) depends continuously on x, y with respect to the Michaeltopology on the space of all compacta in X (see [J] for details).(2) There exists a neutral element e ∈ X with δ x ∗ δ e = δ e ∗ δ x = δ x for all x ∈ X .(3) There exits a continuous involution x ¯ x on X such that for all x, y ∈ X , e ∈ supp ( δ x ∗ δ y ) holds if and only if y = ¯ x .134) If for µ ∈ M b ( X ), µ − is the image of µ under the involution, we require that ( δ x ∗ δ y ) − = δ ¯ y ∗ δ ¯ x for all x, y ∈ X .Due to weak continuity and bilinearity, the convolution of arbitrary bounded measureson a hypergroup is determined uniquely by the convolution of point measures.A hypergroup is called commutative if so is the convolution. We recapitulate from [J]that for a Gelfand pair ( G, K ), the double coset convolution on the double coset space G//K forms a commutative hypergroup.For a commutative hypergroup we define the space χ ( X ) = { ϕ ∈ C ( X ) : ϕ , ϕ ( x ∗ y ) := ( δ x ∗ δ y )( ϕ ) = ϕ ( x ) ϕ ( y ) ∀ x, y ∈ X } of all nontrivial continuous multiplicative functions on X , as well as the dual space b X := { ϕ ∈ χ ( X ) : ϕ is bounded and ϕ ( x ) = ϕ ( x ) ∀ x ∈ X } . The elements of b X are called characters.Using the positive product formulas (3.3) and (3.12) for p > q − p = 2 q − X := C q × R dependingon p : For s, t ∈ C q and θ , θ ∈ R , we define the probability measures ( δ ( s,θ ) ∗ p δ ( t,θ ) ) withcompact supports by( δ ( s,θ ) ∗ p δ ( t,θ ) )( f ) := (4.1)= 1 κ p Z B q Z SU ( q ) f (cid:16) d ( t, s ; v, w ) , θ + θ + Im ln h ( t, s ; v, w ) (cid:17) · ∆( I q − w ∗ w ) p − q dv dw for p > q − 1, and by( δ ( s,θ ) ∗ q − δ ( t,θ ) )( f ) := (4.2)= 1 κ q − Z B q Z SU ( q ) f (cid:16) d ( t, s ; v, P ( y )) , θ + θ + Im ln h ( t, s ; v, P ( y )) (cid:17) ·· q − Y j =1 (1 − k y j k ) q − − j dy . . . dy q − dσ ( y q ) dw for p = 2 q − f ∈ C ( C q × R ), where the functions d, h and the other data are givenas in Sections 2 and 3. Let q ≥ be an integer and p ∈ [2 q − , ∞ [ . Then ∗ p can be extendeduniquely to a bilinear, weakly continuous convolution on the Banach space M b ( C q × R ) . Thisconvolution is associative, and ( C q × R , ∗ p ) is a commutative hypergroup with (0 , as identityand with the involution ( r, a ) := ( r, − a ) .Proof. It is clear by the definition of the convolution that the mapping( C q × R ) × ( C q × R ) → M b ( C q × R ) , (( s, θ ) , ( t, θ )) δ ( s,θ ) ∗ p δ ( t,θ ) is probability preserving and weakly continuous. It is now standard (see [J]) to extendthis convolution uniquely in a bilinear, weakly continuous way to a probability preservingconvolution on M b ( C q × R ). 14o prove commutativity, it suffices to consider point measures. But in this case, for p > q − 1, commutativity follows easily by using transposition of matrices and the fact thatthe integration in (3.3) over w ∈ B q and v ∈ SU ( q ) remains invariant under transposition.The commutativity for p > q − p = 2 q − p ≥ q . In this case, we first identify C q × T with the double coset space U ( p, q ) // ( U ( p ) × SU ( q )) as in Section 2 where the associated double coset convolution on C q × T is given by the product formula (2.9). After bilinear, weakly continuous extension, thisdouble coset convolution is associative by its very construction. We now compare convolutionproducts w.r.t. this convolution on C q × T with the corresponding one defined by (3.3) on C q × R for ( r, θ ) , ( s, θ ) , ( t, θ ) ∈ C q × R for small r, s, t ∈ C q . Taking (3.8) into account, wesee readily that the complex logarithm in (3.3) for the triple products( δ ( r,θ ) ∗ p δ ( s,θ ) ) ∗ p δ ( t,θ ) and δ ( r,θ ) ∗ p ( δ ( s,θ ) ∗ p δ ( t,θ ) )is just the usual main branch of the logarithm on the open right halfplane { z : Re z > } for all integration variables in B q and SU ( q ) respectively. Using this elementary logarithm,we see immediately that the associativity of the convolution on C q × T implies the asso-ciativity on C q × R for small r, s, t ∈ C q . We now extend the associativity to arbitrary( r, θ ) , ( s, θ ) , ( t, θ ) ∈ C q × R . For this we use (3.8) and find an open, relatively compact set K ⊂ C q × R withsupp (( δ (˜ r, ˜ θ ) ∗ p δ (˜ s, ˜ θ ) ) ∗ p δ (˜ t, ˜ θ ) ) ∪ supp ( δ (˜ r, ˜ θ ) ∗ p ( δ (˜ s, ˜ θ ) ∗ p δ (˜ t, ˜ θ ) )) ⊂ K for all ˜ r, ˜ s, ˜ t ∈ C q and ˜ θ , ˜ θ , ˜ θ ∈ R with k ˜ r k ∞ , k ˜ s k ∞ , k ˜ t k ∞ ≤ k r k ∞ , k s k ∞ , k t k ∞ ) and | ˜ θ | , | ˜ θ | , | ˜ θ | ≤ | θ | , | θ | , | θ | ) . Let f ∈ C c ( C q × R ) be a continuous function with compact support which is analytic on K .Then by analyticity of the product formulas (3.3) and (3.12) w.r.t. the variables in C q × R ,(( δ (˜ r, ˜ θ ) ∗ p δ (˜ s, ˜ θ ) ) ∗ p δ (˜ t, ˜ θ ) )( f ) and ( δ (˜ r, ˜ θ ) ∗ p ( δ (˜ s, ˜ θ ) ∗ p δ (˜ t, ˜ θ ) ))( f ) (4.3)are analytic in the variables ˜ r, ˜ s, ˜ t, ˜ θ , ˜ θ , ˜ θ where both expressions are equal for small ˜ r, ˜ s, ˜ t .Therefore, they are equal in general for all such functions f . As both measures have compactsupport, a Stone-Weierstrass argument leads to the general associativity for integers p ≥ q − 1. We now extend the associativity to arbitrary p > q − f as above which is analytic also in thevariable p ∈ C with Re p > q − | κ p | Z B q | ∆( I − w ∗ w ) p − q | dw is of polynomial growth for p → ∞ in the right halfplane { p ∈ C : Re p ≥ q } ; see Theorem3.6 of [R2]. This completes the proof of associativity.For the remaining hypergroup axioms we first notice that (0 , 0) is obviously the neutralelement. Moreover, for ( t, θ ) ∈ C q × R we have(0 , ∈ supp ( δ t,θ ∗ p δ t, − θ )15y taking the integration variables v ∈ SU ( q ) as the identity matrix I q and w := − I q ∈ B q inthe convolution (4.1) and y := − e , . . . , y q := − e q for the usual unit vectors in C q in (4.2).We next check the converse part of axiom (3) of a hypergroup. For this take ( s, θ ) , ( t, θ ) ∈ C q × R with (0 , ∈ supp ( δ s,θ ∗ p δ t,θ ). As the support is independent of p ∈ ]2 q − , ∞ [ withsupp ( δ s,θ ∗ q − δ t,θ ) ⊂ supp ( δ s,θ ∗ q δ t,θ )by (4.1) and (4.2), we may restrict our attention to integers p ≥ q . In this case we nowcompare the convolution (4.1) on C q × R with the convolution (2.13) on C q × T which isthe double coset convolution for U ( p, q ) // ( U ( p ) × SU ( q )). As here axiom (3) is availableautomatically, we conclude from our assumption that s = t and θ − θ ∈ π Z holds. Forthe proof of θ = − θ , we analyze (2.13) more closely: Recapitulate from Section 2 that theidentification C q × T ≃ U ( p, q ) // ( U ( p ) × SU ( q )) is done via the representatives a t,z ∈ U ( p, q )( t ∈ C q , z ∈ T ) of double cosets. It is clear that for all t ∈ C q and z ∈ T , the matrix J := (cid:18) − I p I q (cid:19) ∈ U ( p ) × SU ( q )is the only element of U ( p ) × SU ( q ) with a t,z · J · a t,z − = I p + q = a , . By the proof ofProposition 2.2 this means that for v ∈ SU ( q ) and w = 0 σ ∗ uσ ∈ B q with σ as in (2.6), wehave d ( t, t ; v, w ) = 0 and arg h ( t, t ; v, w ) = arg ∆(sinh t w sinh t + cosh t v cosh t ) = 1only for v = I q and w = − I q . In this case, d ( t, t ; I q , − I q ) = 0 and h ( t, t ; I q , − I q ) = 1, wherefor all t ∈ C q obviously the branch of the complex logarithm in the product formula (4.1)satisfies ln h ( t, t ; I q , − I q ) = 1. This proves θ = − θ above and completes the proof of axiom(3).Furthermore, axiom (4) is clear, and the continuity of the supports of convolution productscan be checked in a straightforward, but technical way. We skip the details.We next turn to subgroups of the commutative hypergroups ( C q × R , ∗ p ) for p ≥ q − H ⊂ C q × R is a subhypergroup ifsupp( δ x ∗ p δ ¯ y ) ⊂ H holds for all x, y ∈ H . Moreover, H is called a subgroup, if the convolutionrestricted to H is the convolution of a group structure on H . It is clear from (3.3) and (3.12),that { } × R is a subgroup of ( C q × R , ∗ p ) which is isomorphic to the group ( R , +).Now let H be a subgroup of a commutative hypergroup ( X, ∗ ). Then the cosets x ∗ H := [ y ∈ H supp ( δ x ∗ p δ y ) ( x ∈ X )form a disjoint decomposition of X , and the quotient X/H := { x ∗ H : x ∈ X } is again alocally compact Hausdorff space with respect to the quotient topology. Moreover,( δ x ∗ H ∗ δ y ∗ H )( f ) := Z X f ( z ∗ H ) d ( δ x ∗ δ y )( z ) ( x, y ∈ X, f ∈ C b ( X/H )) , (4.4)establishes a well-defined quotient convolution and an associated quotient hypergroup ( X/H, ∗ ).For these quotient convolutions we refer to [J], [R] and [V1]. We now apply this concept tothe subgroups { } × R and { } × Z of our hypergroups ( C q × R , ∗ p ) for p ≥ q − 1. By(4.4) and (3.3) we can identify the quotient spaces in the obvious way with C q and C q × T respectively, and we obtain immediately: 16 .3 Lemma. Let p > q − . (1) ( C q × R ) / ( { } × R ) ≃ C p is a commutative hypergroup with the convolution ( δ s ∗ p δ t )( f ) := 1 κ p Z B q Z SU ( q ) f (cid:16) d ( t, s ; v, w ) (cid:17) · ∆( I q − w ∗ w ) p − q dv dw (4.5) for s, t ∈ C q and f ∈ C b ( C q ) . This is precisely the hypergroup studied in Section 5 of[R3]. For integers p ≥ q , this is just the double coset hypergroup U ( p, q ) / ( U ( p ) × U ( q )) . (2) ( C q × R ) / ( { } × Z ) ≃ C p × Z is a commutative hypergroup with the convolution ∗ p asdefined in Eq. (2.13), but here for arbitrary real numbers p > q − . In particular, forintegers p ≥ q , this is just the double coset hypergroup U ( p, q ) / ( U ( p ) × SU ( q )) . For p = 2 q − 1, a corresponding result holds on the basis of (3.12).By using Weil’s integral formula for Haar measures on hypergroups (see [Her] and [V1]),we now can determine the Haar measures on the hypergroups ( C q × R , ∗ p ) from the knownHaar measures on the hypergroups of Lemma 4.3(1) in Theorem 5.2 of [R3]. For this reca-pitulate that each commutative hypergroup ( X, ∗ ) admits a (up to a multiplicative constantunique) Haar measure ω X , which is characterized by the condition ω X ( f ) = ω X ( f x ) for allcontinuous functions f ∈ C c ( X ) with compact support and x ∈ X , where the translate f x ∈ C c ( X ) is given by f x ( y ) := ( δ y ∗ δ x )( f ). For p ≥ q − , the Haar measure of the commutative hypergroup ( C q × R , ∗ p ) , is given by dω p ( t, θ ) = const · q Y j =1 sinh p − q +1 t j cosh t j · Y ≤ i 1. Weconjecture that in fact each continuous multiplicative function on ( C q × R , ∗ p ) has thisform. In fact, one has to prove similar to Lemma 5.3 of [R3] that continuous multi-plicative functions are eigenfunctions of a corresponding family of differential operatorsdiscussed in Section I.5 of [HS].(2) The multiplicative functions ψ pλ,l satisfy several symmetry conditions in the spectralvariables which are immediate consequences of corresponding symmetries for F BC q in[HS], [O1], [O2]. In particular, similar to [R3], we have:For λ, ˜ λ ∈ C q , l, ˜ l ∈ C , and the Weyl group W q of type B q acting on C q , ψ pλ,l = ψ p ˜ λ, ˜ l on C q ⇐⇒ ˜ λ ∈ W λ, ˜ l = l. Moreover, ψ pλ,l = ψ p ¯ λ, − l . In particular, ψ pλ,l satisfies ψ pλ,l (( t, θ ) − ) = ψ pλ,l ( t, θ ) for all ( t, θ ) ∈ C q × R (4.7)if and only if l ∈ R and ¯ λ ∈ W λ holds.(3) It is an interesting task to determine the dual space ( C q × R ) ∧ which consists of allbounded, continuous multiplicative functions satisfying (4.7). For the correspondinghypergroups on C q , we refer to [R3] and [NPP] for this problem. We fix the dimension q ≥ 1, a real parameter p > q − l ∈ R . We knowfrom Section 2 that the functions ψ λ,l with ψ pλ,l ( t, 0) = q Y j =1 cosh l t j · F BC q ( iλ, k ( p, q, l ); t )satisfy the product formula (3.3). We now apply q Y j =1 cosh l d j ( s, t ; v, w ) · e il · Im ln h(s , t;v , w) = h ( s, t ; v, w ) l = ∆(sinh t w sinh s + cosh t v cosh s ) l for s, t ∈ C q , w ∈ B q , v ∈ SU ( q ) and the analytical branch of the l -th power functionassociated with the analytical branch of the logarithm in the setting of (3.3). This branchwill be taken always from now on. This leads immediately to the following product formulafor the hypergeometric functions ϕ p,lλ ( t ) := F BC q ( iλ, k ( p, q, l ); t ): Fix an integer q ≥ , p ∈ [2 q − , ∞ [ , and all l ∈ R . Then the functions ϕ p,lλ satisfy the product formula ϕ p,lλ ( s ) · ϕ p,lλ ( t ) = 1 κ p Q qj =1 (cid:16) cosh t j · cosh s j (cid:1) l · (5.1) · Z B q Z SU ( q ) ϕ p,lλ ( d ( t, s ; v, w )) · Re( h ( t, s ; v, w ) l ) · ∆( I q − w ∗ w ) p − q dv dw or s, t ∈ C q and all λ ∈ C .Proof. Our considerations above lead to ϕ p,lλ ( s ) · ϕ p,lλ ( t ) = 1 κ p Q qj =1 (cid:16) cosh t j · cosh s j (cid:1) l ·· Z B q Z SU ( q ) ϕ p,lλ ( d ( t, s ; v, w )) · h ( t, s ; v, w ) l · ∆( I q − w ∗ w ) p − q dv dw. for s, t ∈ C q , λ ∈ C . Now take λ ∈ R q in which case ϕ p,lλ is real on C q . Therefore, takingreal parts above, we obtain the product formula of the theorem for λ ∈ R q . The general casefollows by analytic continuation.We next present a condition on l which ensures positivity of the product formula (5.1)for all s, t ∈ C q . It is based on the following: For all l ∈ R with | l | ≤ /q and all s, t ∈ C q , w ∈ B q , v ∈ SU ( q ) , Re (cid:16) (∆(sinh t w sinh s + cosh t v cosh s )) l (cid:17) ≥ . Proof. We have∆(sinh t w sinh s + cosh t v cosh s ) = ∆(cosh t · cosh s ) · ∆( ˜ w + I q ) (5.2)for the matrix ˜ w := v − · tanh s w tanh t . We now check ˜ w ∈ B q , i.e., ˜ w ∗ ˜ w ≤ I q . Infact, this is equivalent to tanh t w ∗ tanh s w tanh t ≤ I q , which is clearly a consequence of w ∗ tanh s w ≤ I q which is obviously correct.As all eigenvalues τ ∈ C of a matrix ˜ w ∈ B q satisfy | τ | ≤ 1, we obtain that all eigenvaluesof ˜ w + I q are contained in { z ∈ C : Re z ≥ } . The lemma now follows from (5.2).We notice that it can be easily seen that Lemma 5.2 is not correct for a larger range ofparameters l ∈ R . It is however unclear for which precise range of parameters l ∈ R thereis a positive product formula for the functions ϕ p,lλ . We expect that this range depends on q and p ; see also the example for q = 1 below.We also remark that the results above for p > q − p = 2 q − 1, and that for l ∈ R with | l | ≤ /q and p ≥ q − 1, our positive productformulas for the ϕ p,lλ lead to commutative hypergroup structures on C q . This can be shownin the same way as in Section 4. We here skip the details and remark only that ϕ p,liρ ≡ ρ = ρ ( k p,l ) as in (3.7) ensures that the corresponding positive measures on the right handsides are in fact probability measures.In summary: For all integers q ≥ , all p ∈ [2 q − , ∞ [ and all l ∈ R with | l | ≤ /q , theHeckman-Opdam hypergeometric functions ϕ p,lλ ( λ ∈ C ) satisfy some positive product formula(namely (5.1) for p > q − and a corresponding one for p = 2 q − ). Moreover, in thiscase the ϕ p,lλ ( λ ∈ C ) are multiplicative functions of some associated unique commutativehypergroup structures ( C q , ∗ p.l ) . It is not difficult to determine the Haar measures of these hypergroups:19 .4 Proposition. For integers q ≥ , p ∈ [2 q − , ∞ [ and l ∈ R with | l | ≤ /q , the Haarmeasure on the hypergroup ( C q , ∗ p.l ) is given by dω p,l ( t ) = const · q Y j =1 sinh p − q +1 t j cosh l +1 t j · Y ≤ i 0, the symmetry of the integral w.r.t. θ ∈ [ − π, π ] and the correct integrationconstant lead to the product formula ϕ ( α,β ) λ ( s ) · ϕ ( α,β ) λ ( t ) = 2 απ · (cosh s · cosh t ) β Z Z π ϕ ( α,β ) λ (arcosh | re iθ sinh t sinh s + cosh t cosh s | ) · Re (cid:16) ( re iθ sinh t sinh s + cosh t cosh s ) β (cid:17) · (1 − r ) α − r dr dθ (5.4)for s, t ≥ λ ∈ C . For α = 0 we obtain the degenerate formula ϕ (0 ,β ) λ ( s ) · ϕ (0 ,β ) λ ( t ) = 1 π (cosh s · cosh t ) β Z π ϕ (0 ,β ) λ (arcosh | re iθ sinh t sinh s + cosh t cosh s | ) · Re (cid:16) ( re iθ sinh t sinh s + cosh t cosh s ) β (cid:17) dθ (5.5)For β = 0, these formulas coincide with the well known product formulas for Jacobi functions;see Section 7 of [K]. However, for β = 0, (5.4) and (5.5) do not seem to be much used inliterature. In fact, to our knowledge, they are only considered in Section 6 of [RV]. It shouldbe noticed that some details in Section 6 are not correct.Let us compare our product formulas with those of Koornwinder [K]. Our Eqs. (5.4) and(5.5) are available for all α ≥ β ∈ R , and they are positive for α ≥ | β | ≤ 1. Onthe other hand, Koornwinder’s formulas in Section 7 of [K] are available with positivity for α ≥ β ≥ − / 2. It is well known (see Section 7 of [K], and [J] and [BH] for the hypergroupbackground) that for all α ≥ β ≥ − / 2, Koornwinder’s product formulas for the ϕ ( α,β ) λ areassociated with unique commutative hypergroup structures on [0 , ∞ [, the so-called Jacobihypergroups ([0 , ∞ [ , ∗ α,β ). As here injectivity of the Jacobi transform (as a special case ofthe injectivity of the Fourier transform on commutative hypergroups in [J]) ensures that for α, β, s, t there is at most one bounded signed measure µ α,βs,t ∈ M b ([0 , ∞ [) with ϕ (0 ,β ) λ ( s ) ϕ (0 ,β ) λ ( t ) = Z ϕ (0 ,β ) λ ( u ) dµ α,βs,t ( u )for all λ ∈ C , we conclude that for α ≥ max( β, 0) and β ≥ − / 2, the product formulas (5.4)and (5.5) are equivalent to those of [K] and thus positive.This in particular shows that for q = 1 the range of parameters p, l in Theorem 5.3 witha positive product formula is larger than described there. We expect that this holds also for q ≥ 2. 20aking our results and the results of Koornwinder into account, we obtain in summarythat the Jacobi functions ϕ (0 ,β ) λ ( λ ∈ C ) admit associated commutative hypergroup structuresfor the set of parameters { ( α, β ) ∈ R : α ≥ β ≥ − / α ≥ , β ∈ [ − , } . By classical results in the case of Koornwinder (see [BH] for details on the Jacobi hypergroups)and by (5.3) in our case, the Haar measures on these hypergroups are given in both cases by const · sinh α +1 t · cosh β +1 t ( t ≥ . C q We show in this section that for all p ≥ q − l ∈ R , the (not necessarily positive)product formulas of Section 5 for the Heckman-Opdam functions ϕ p,lλ on C q are related tosome so-called signed hypergroup structure ( C q , • p,l ). For this we return to the commutativehypergroups ( C q × R , ∗ p ) of Section 4 for p ≥ q − ψ pλ,l asmultiplicative functions. As in Section 4, we consider the closed subgroup G := { } × R of( C q × R , ∗ p ), and identify the quotient ( C q × R ) /G with C q .For l ∈ R consider the functions σ l ∈ C b ( C q × R ) with σ l ( t, θ ) := e ilθ . They satisfy | σ l ( t, θ ) | = 1 , σ l ( t, − θ ) = σ l ( t, θ ) , and ( δ (0 ,τ ) ∗ p δ ( t,θ ) )( σ l ) = σ l ( t, θ ) · σ l (0 , τ )for t ∈ C q and θ, τ ∈ R , i.e., the σ l are partial characters on ( C q × R , ∗ p ) w.r.rt. G in thesense of Definition 4.1 of [RV]. For l ∈ R we now consider the mapping( C q × R ) /G × ( C q × R ) /G −→ M b ( C q × R ) /G )(( s, θ ) ∗ G, ( t, θ ) ∗ G ) δ ( s,θ ) ∗ G • p,l δ ( t,θ ) ∗ G :=:= σ l ( s, θ ) · σ l ( t, θ ) · pr ( σ l · ( δ ( s,θ ∗ p δ ( t,θ ) ))with pr as the canonical projection pr : C q × R → ( C q × R ) /G as well as its extension toimages of measures. It can be easily checked (see also Section 4 of [RV] for a general theory)that this mapping is well-defined, i.e., independent of representatives of the cosets, andweakly continuous. Moreover, by Section 4 of [RV], it can be uniquely extended in a bilinear,weakly continuous way to an associative convolution ∗ l on M b (( C q × R ) /G ). Moreover,( M b (( C q × R ) /G ) , ∗ l ) is a commutative Banach- ∗ -algebra with respect to the total variationnorm with the identity δ (0 , ∗ G as neutral element where the involution on M b (( C q × R ) /G )is inherited from that on the hypergroup C q × R . We also consider the the image ˜ ω p := pr ( ω p ) ∈ M + (( C q × R ) /G ) of the Haar measure ω p on C q × R in (4.6). By Theorem 4.6of [RV], the quotient ( C q × R ) /G with the convolution • p,l and the measure ˜ ω p forms a so-called commutative signed hypergroup (( C q × R ) /G, • p,l , ˜ ω p ) with the identity mapping asinvolution, i.e., a so-called hermitian commutative signed hypergroup. For details on thisnotion we refer to [R1], [RV] and references cited there. We only remark that for a Haarmeasure ω p in this setting we require the conjugation relation Z ( C q × R ) /G T x f · g dω p = Z ( C q × R ) /G T x g · f dω p (6.1)for all f, g ∈ C c (( C q × R ) /G ) and x ∈ ( C q × R ) /G where the translates T x are defined by T x f ( y ) := ( δ x • p,l δ y )( f ). It is well-known (see [J]) that for usual hermitian commutative21ypergroups, the usual Haar measures satisfy (6.1), and that by [RV], Haar measures oncommutative signed hypergroups in the above sense are unique up to a constant.We now identify ( C q × R ) /G with C q as usual. In this case, we obtain from (4.1) that for p > q − • p,l on C q is given by( δ s • p,l δ t )( f ) = Z C q × R e ilθ d ( δ s, ∗ p δ s, )( t, θ ) (6.2)= 1 κ p Z B q Z SU ( q ) f ( d ( s, t ; v, w )) · ( arg h ( s, t ; v, w )) l · ∆( I q − w ∗ w ) p − q dv dw. Moreover, for p = 2 q − ω p ∈ M + ( C q ) of our signed hypergroups is independent of l and is given by d ˜ ω p ( t ) = const · q Y j =1 sinh p − q +1 t j cosh t j · Y ≤ i 1, we have two choices to define convolution structures for measures on C q associated with the hypergeometric functions F BC q , namely the convolutions of Section 5for the functions F BC q directly as well as the convolutions of Section 6 for the functions˜ ϕ p,lλ . These both convolutions are equal for l = 0 in which case C q carries the usual classicalquotient convolution which was studied in [R3].For l = 0 with l ∈ [ − /q, /q ], we have a positive convolution. In this case, the convo-lution of Section 5 for the functions F BC q is probability preserving and generates classicalhypergroup structures, which is not the case for the convolutions of Section 6, which are onlypositive and norm-decreasing. 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