A STATIONARY PROCESS ASSOCIATED WITH THE DIRICHLET DISTRIBUTION ARISING FROM THE COMPLEX PROJECTIVE SPACE N. DEMNI

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1 A STATIONARY PROCESS ASSOCIATED WITH THE DIRICHLET DISTRIBUTION ARISING FROM THE COMPLEX PROJECTIVE SPACE N. DEMNI arxiv:143.37v1 [math.pr] 13 Mar 14 Abstract. Let (U t) t be a Brownian motion valued in the complex projective space CP N 1. Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of U 1 t and of ( U 1 t, U t ), and express them through Jacobi polynomials in the simplices of R and R respectively. More generally, the distribution of ( U 1 t,..., U k t ), k N 1 may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group U(N k + 1) yet computations become tedious. We also revisit the approach initiated in [11] and based on a partial differential equation (hereafter pde) satisfied by the Laplace transform of the density. When k = 1, we invert the Laplace transform and retrieve the expression derived using spherical harmonics. For general 1 k N, the integrations by parts performed on the pde lead to a heat equation in the simplex of R k. The complex unit sphere 1. Motivation S N 1 := {(z 1,...,z N ), z z N = 1}, N 1, is a compact manifold without boundary therefore carries a Brownian motion (U t ) t defined by means of its Laplace-Beltrami operator. This process is stationary and the random variable U t converges weakly as t to a uniformly-distributed random vector U. For the latter, it is already known that ( U 1,..., U k ), 1 k N 1, follows the Dirichlet distribution ([7]) k k (1) s k (u) du i (1 u 1 u u k ) N k 1 1 k (u) du i where k = {u i >,1 i k,u 1 + +u k < 1}is thestandardsimplex. Motivatedbyquantum information theory, the investigations of the distribution of U (k) t ( U 1 t,..., U k t ), 1 k N started in [11] yet have not been completed. There, a linear pde for the Laplace transform of this distribution was obtained and partially solved only when k = 1. Recall that for the Brownian motion on the Euclidian sphere S N 1, the density of a single coordinate is given by a series involving products of ultraspherical polynomials of index (N )/ ([8]). The main ingredients leading to this series are the expansion of the heat kernel on S N 1 in the basis of O(N)-spherical harmonics and on Gegenbauer addition Theorem ([1], p.369). In the complex setting, it is very likely known that ( Ut 1 ) t is a real Jacobi process (see [3] and references therein). Nonetheless, one wonders how does the proof written in [8] carry to the Brownian motion on S N 1 and how does it extend in order to derive the density of U (k) t. In the first part of this paper, we answer these questions by considering the heat kernel on the complex projective space CP N 1 = S 1 /S N 1 rather than S N 1. This is by no means a loss of generality since we are interested in the joint distribution of the moduli of k coordinates of U t. Besides, the space of continuous functions on CP N 1 decomposes as the direct sum of subspaces of U(N)-spherical harmonics that are homogeneous of degree zero, while the decomposition of continuous functions on S N 1 involves all spherical harmonics ([6]). Accordingly, the heat kernel on CP N 1 is expressed as a series of normalized Jacobi polynomials (Pn N, /Pn N, (1)) n which Key words and phrases. Brownian motion; complex projective space; Dirichlet distribution; Jacobi polynomials in the simplex. 1

2 for each n, gives the n-th reproducing kernel on CP N 1 ([9]). Hence, the integration over the sphere S N 3 together with an application of Koornwinder s addition Theorem ([9]) lead to the density of Ut 1 : [ ] f t (c,u) e n(n+n 1)tPN, n (c 1)Pn N, (u 1) () Pn N, s 1 (u) n= where we set c U 1 [,1] and P n is the squared L -norm of u Pn N, (u 1) with respect to s 1 (u)du. Up to an additional ingredient, the derivation of the density of U () t is quite similar. Loosely speaking, we would like to integrate the heat kernel over the sphere S N 5 (we assume N large enough) and as such, we need to decompose degree zero homogenous spherical harmonics in S N 1 under the action of the unitary group U(N 1). This decomposition is stated in [1], Theorem 5.1, and the n-th reproducing kernel in turn decomposes as a wighted sum of reproducing kernels on S N 3. Consequently, Koornwinder s addition Theorem again leads to the sought density which may be expressed through Jacobi polynomials in the simplex ([4], Proposition.3.8 p.47). More precisely, if we denote these polynomials by (Q (N) j, ) n, j n then the density reads (3) n e n(n+n 1)t n= j=,j (u 1,u ) Q (N),j Q (N),j (c 1,c )Q (N) s (u), where we set (c 1,c ) ( U 1, U ) and Q (N),j is the squared L -norm of Q (N),j with respect to s (u)du 1 du. More generally, the derivation of the density of U (k) t, k N 1 relies on the decomposition of the spherical harmonics under the action of U(N k + 1) and is expressed through orthonormal Jacobi polynomials in k as n e n(n+n 1)t τ N k, τ =n Q (N) τ (c 1,c,...,c k )Q (N) τ (u)s k (u) where (c 1,...,c k ) ( U 1,..., U k ). Yet computations become tedious and we are not willing to exhibit them here. Rather, we shall revisit and complete the investigations started in [11]. Actually, an expression for the Laplace transform of the density of Ut 1 was obtained there and involves the following sequence (a n = a n (c,n)) n of real numbers determined recursively by ([11], eq. 4.3) p ( ) p 1 (4) a n = cp n (N +n) p n p!, p, a = 1. n= In particular, the following was proved ([11] eq and eq. 4.5): which we can rewrite as a n (,N) = ( 1) n (N 1) n, a n (1,N) =, (N +n 1) n n!(n +n 1) n 1 Pn N, ( 1), (N +n 1) n 1 (N +n 1) n P N, n (1) respectively. Using a Neumann series for Bessel functions ([1]), we shall prove that for all c [,1] 1 (5) a n = a n (c,n) = Pn N, (c 1). (N +n 1) n Having these coefficients in hands, we can then invert the Laplace transform and retrieve (). At this level, we point out that the pde satisfied by the Laplace transform of the density of U 1 t leads after integrations by parts to the heat equation associated with the Jacobi operator (6) u(1 u) u +[1 Nu] u.

3 More generally, the pde satisfied by the Laplace transform of the joint distribution of U (k) t,1 k N gives rise to the heat equation on the standard simplex associated with the generalized Jacobi operator ([1], see also [4] p.46 but consult the list of errata available on the webpage of Y. Xu): (7) [1 Nu i ] i + i u (u i ) ii u i u j ij. This is an elliptic operator admitting different orthogonal basis of eigen-polynomials corresponding to the sequence of eigenvalues { n(n +n 1),n }. Among them figure the Jacobi polynomials in the simplex which agrees with our previous computations. The paper is organized as follows. The two following sections are concerned with the derivations of the densities of U (k) t,k {1,}. In section 4, we solve the system (4) and invert the Laplace transform of the density of U 1 t. In section 5, we perform integrations by parts on the pde satisfied by the Laplace transform of the density of U (k) t, omitting for a while the boundary terms. In the last section, we write down the latters and show that all of them vanish unless k = N 1.. The distribution of U 1 t Let m,n be non negative integers and recall from [9] that (m,n)-complex spherical harmonics are the restriction S N 1 of harmonic polynomials in the variables (z 1,z,...,z N,z 1,z,...,z N ) which are m-homogenousin the variables (z i ) N and n-homogeneousin the variables (z i) N. Taking m = n, we obtain the (n, n)-complex spherical harmonics that are homogenous of degree zero with respect to the action of S 1. Their restrictions to CP N 1 form a dense algebra in the space of continuous functions on CP N 1 endowed with to the uniform norm (see [6], p.189). Moreover, the spectrum of the Laplace-Beltrami operator on CP N 1 is given by the sequence { n(n + N 1),n } 1. Hence, the corresponding heat kernel is expanded in any orthonormal (with respect to the volume measure vol CP N 1) basis of homogenous of degree zero spherical harmonics (Y j ) j 1 as: R t (w,z) n= d(n,n) e n(n+n 1)t j=1 Y j (w)y j (z), w,z CP N 1. Here d(n, N) is the dimension of the eigenspace of (n, n)-complex spherical harmonics given by (Theorem 3.6 in [9]) ( ) n+n 1 (N 1)n d(n,n) =. N 1 n! Besides, the reproducing kernel formula (Theorem 3.8 in [9] ) shows that the kernel R t is real and does not depend on the choice of the basis (this is the analogue of () in [8]): (8) R t (w,z) = π e n(n+n 1)t d(n,n) Pn N, ( w,z 1) vol(s N 1 ) Pn N,, (1) where ([], p.95) and n= w,z N w i z i, P N, n vol(s N 1 ) = πn (N 1)!. (1) = (N 1) n, n! 1 We normalize the Laplacian on CP N 1 by a factor 1/4. The additional factor π comes from the fact that vol(s N 1 ) = πvol(cp N 1 ). 3

4 is the volume of S N 1. Note that R t (w,z) = (N )! π N 1 n= e n(n+n 1)tPN, n (1)Pn N, ( w,z 1) Pn N, where Pn N, 1 = n+n 1 is the squared L -norm of u Pn N, (u 1) with respect to s 1 (u)du ([], p.99). Thus Gasper s Theorem entails the positivity of R t ([5]). Now, we proceed to the derivation of the density of Ut 1 and start with the decompositions w = cosθ 1 e 1 +sinθ 1 ξ 1, z = cosθ e 1 +sinθ ξ where e 1 is the first vector of the canonical basis of C N, θ 1,θ (,π/),φ 1,φ (,π),ξ 1,ξ S N 3. The volume measure of CP N 1 in turn splits as (see [9], eq..18) vol CP N 1(dz) = cosθ (sinθ ) N 3 dθ vol S N 3(dξ ). and the next step is to integrate (8) over ξ. But U(N 1) acts transitively on S N 3 therefore we can take ξ 1 = e to be the second vector of the canonical basis. As such, we are left with the volume of S N 5 (if N is large enough) and with the integration over the distribution of the first coordinate of ξ. If this coordinate is parametrized by (r, ψ) then its distribution reads r(1 r ) N 3 1 [,1] (r)1 [,π] (ψ)drdψ. Consequently, the density of U 1 t displayed in () follows from the product formula (4.1) in [9] together with the variables change u = cos θ (c = cos θ 1 ). Remark 1. The eigenvalue of a (n, n)-spherical harmonic equals the eigenvalue of a O(N)-spherical harmonic of degree n in S N 1 viewed as a real Euclidian sphere. This coincidence is due to the fact that both polynomials are homogenous with the same total degree n and since the correspond eigenvalue comes from the action of the Euler operator N N z i zi + z i zi = N i xi + y i yi, x where z i C is identified with (x i,y i ) R. 3. The distribution of ( U 1 t, U t ) Up to an additional ingredient, the lines of the previous proof enable to derive the density of U () t. More precisely, we start with the decompositions w = cosθ 1 e 1 +sinθ 1 ξ 1 = cosθ 1 e 1 +sinθ 1 cosβ 1 e iφ1 e +sinθ 1 sinβ 1 η 1, z = cosθ e 1 +sinθ ξ = cosθ e 1 +sinθ cosβ e iφ e +sinθ sinβ 1 η, where β 1,β (,π/),φ 1,φ (,π),η 1,η S N 3 and e is the second vector of the canonical basis of C N. We also split the volume measure on CP N 1 as vol CP N 1(dz) = ( cosθ sin N 3 θ cosβ sin N 5 β dθ dβ dφ ) vols N 5(dη ). Now comes the needed additional ingredient, which is the special instance m = n,φ 1 = φ = in the formula stated in the bottom of p.5 in [1]. In order to recall it, let j (x) Pa,b j (x) P a,b, a,b > 1 j (1) p a,b 4

5 be the j-th normalized Jacobi polynomial and define the complex-valued polynomial ([9] eq.3.15) as well as ([1] p.6) R α j,q (y) y j q e i(j q)arg(y) p α, j q j q ( y 1), y C, α > 1. c j,q (n,n) N N +q +j ( n q )( n j ) (N +n 1)q (N +n 1) j (N +j) q (N +q) j. Then the n-th reproducing kernel on CP N 1 admits the following expansion Pn N, ( w,z 1) n Pn N, = c j,q (n,n)[sin(θ 1 )sinθ ] j+q [cosθ 1 cosθ ] j q p N +j+q, j q n q (cosθ 1 ) (1) j,q= p N +j+q, j q n q (cosθ 1 )R N 3 j,q ( ξ 1,ξ ). Substituting in (8), we see that the next step towardsthe joint distribution of (Ut 1,U t ) consists in integrating R N 3 j,q ( ξ 1,ξ ) = R N 3 j,q (cosβ 1 cosβ +sinβ 1 sinβ η 1,η ) over η S N 3. To this end, we can assume without loss of generality that η 1 = e 3 (the third vector of the canonical basis) and use formula (4.11) in [9]. Altogether, we get π vol(s N 5 ) (N 3)vol(S N 1 ) p N +j+q, j q n q = (N ) π p N +j+q, j q n q with respect to e n(n+n 1)t d(n,n) n= (cosθ 1 )p N +j+q, j q n q n= n c j,q (n,n)[sin(θ 1 )sinθ ] j+q [cosθ 1 cosθ ] j q j,q= (cosθ )R N 3 j,q (cosβ 1 e iφ1 )R N 3 n e n(n+n 1)t (n+n 1)[P N, n (1)] (cosθ 1 )p N +j+q, j q n q j,q= j,q (cosβ e iφ ) c j,q (n,n)[sin(θ 1 )sinθ ] j+q [cosθ 1 cosθ ] j q (cosθ )p N 3, j q j q (cosβ 1 )p N 3, j q j q (cosβ )e i(j q)(φ1+φ). cosθ sin N 3 θ cosβ sin N 5 β dθ dβ dφ. Integrating over φ (,π), then the sum over (j,q) reduces to a sum over q = j. Thus, the density of (θ,β ) given (θ 1,β 1 ) reads 4(N ) p N +j, with respect to n= e n(n+n 1)t (n+n 1)[P N, n (1)] (cosθ )p N 3, j (cosβ 1 )p N 3, j (cosβ ) Performing the variables change n j= cosθ sin N 3 θ cosβ sin N 5 β dθ dβ. u = cosθ,v = sinθ cosβ, we deduce that the density of ( Ut 1, U t ) given ( U1, U ) is: 4(N ) e n(n+n 1)t (n+n 1)[Pn N, (1)] with respect to p N +j, n= c j,j (n,n)[sin(θ 1 )sinθ ] j p N +j, (cosθ 1 ) n c j,j (n,n)[(1 U 1 )(1 u )] j j= ( U ( U 1 1)p N +j, (u 1)p N 3, j 1 U 1 1 uv(1 u v ) N 3 1 {u>,v>,u +v <1}dudv. 5 ) ( ) v p N 3, j 1 u 1

6 Finally, if ( U 1, U ) (c 1,c ) then the density of ( Ut 1, Ut ) reads n (N ) e n(n+n 1)t (n+n 1)[Pn N, (1)] c j,j (n,n)[(1 c 1 )(1 u 1 )] j p N +j, n= (c 1 1)p N +j, (u 1 1)p N 3, j j= ( c 1 c 1 1 ) ( ) p N 3, u j 1 1 u 1 with respect to s (u 1,u )du 1 du. The last expression may be put in a compact form as follows. For n, set ( ) Q (N),j (u,v) = (1 u 1) j P N +j, (u 1 1)P N 3, u j 1, j =,1,...,n. 1 u 1 These are Jacobi polynomials in the simplex and are orthogonal with respect to the Dirichlet distribution whose density is s (u 1,u ) (specialize Proposition.3.8 in [4] to α = (n j,j),κ = (1/,1/,N 5/) 3 ). After some computations, the density of U () t may be written as n e n(n+n 1)t Q (N),j (c 1,c )Q (N),j (u,v) n= j= Q (N),j where Q (N),j = 1 (n+n 1)(j +N ) = is the squared L -norm of Q (N),j with respect to s (u 1,u )du 1 du. [P N +j, (1)P N 3, j (1)] (N )(n+n 1)[Pn N, (1)] c j,j (n,n) Remark. For general k 3, the density of U (k) t may be derived in a similar way by decomposing the variable z CP N 1 and the spherical harmonics on S N 1 over the sphere S N k+1. From the point of view of representation theory, this is equivalent to the decompositions of the representation of U(N) in the space of U(N)-spherical harmonics under the action of the subgroup U(N k +1). 4. The distribution of U 1 t : another proof In this section, we shall solve the system (4) and prove (5). To this end, we rewrite (4) as, p 1 1 (9) a n = cp n!(p n)! (N +n) p n (p!) n= multiply both sides of (9) by ( 1) p (x/) p+n 1 for x lying in some neighborhood of zero then sum over p. Interchanging the order of summation, the system (4) is equivalent to a n n! Γ(N +n)( 1)n J n+n 1 (x) = J ( x ) N 1 cx)( n where J α is the Bessel function of index α R defined by ([1]): Note in passing that the estimate ([1]) shows that (9) converges provided (1) J α (x) p ( 1) p p!γ(p+α+1) ( x Γ(N +n)j n+n 1 (x) n a n n! ( ) n x ) p+α. ( ) n+n 1 x 3 Beware of the different normalization of the Jacobi polynomials used in Proposition.3.8. The reader is also invited to consult the list errata of available on the webpage of Y. Xu. 6

7 does. Now recall the Neumann series ([1], p.138) ( x ν (ν +n)γ(ν +n) = J ν+n (x), ν N. ) n! Specializing it to ν = N, we get (x/) N 1 J ( cx) = p = p n ( 1) p c p (p!) ( 1) p c p (p!) n n p = n (n+n 1) (p+n+n 1)Γ(p+n+N 1) J p+n+n 1 (x) n! (n+n 1)Γ(p+n+N 1) J n+n 1 (x) (n p)! n ( 1) p c p Γ(p+n+N 1) (p!) J n+n 1 (x). (n p)! Substituting in (9), then the uniqueness of the solution of (4) yields a n n n! Γ(N c p Γ(p+n+N 1) +n)( 1)n = (n+n 1) (p!) (n p)! or equivalently a n = p= ( 1) n n! Γ(N +n 1) ( 1) n = (N +n 1) n = n p= n p= p= ( 1) p c p Γ(p+n+N 1) (p!) (n p)! ( 1) p n! (n p)! (n+n 1) p (p!) c p ( 1) n (N +n 1) n F 1 ( n,n+n 1,1,c) where F 1 is the Gauss hypergeometric function ([], p.). But from the very definition of Jacobi polynomials ([], p.99) Pn α,β (x) = (α+1) n F 1 ( n,n+α+β +1,α+1,(1 x)/), α,β > 1, n! and the relation Pn α,β (x) = ( 1) n Pn β,α ( u) ([], p.35), we obtain (5) as required. Remark 3. The estimate Pn,N (1 c) Pn,N (1) which follows for instance from the integral representation of the Gauss hypergeometric function shows that the series (1) indeed converges absolutely everywhere. With (5) in hands, we can invert the Laplace transform ([11], eq.4.11): 1 ϕ t/n (c,λ) e λu f t/n (c,u)du = a n e Λnt λ n 1F 1 (n+1,n +n,λ), λ R, n= where 1 F 1 is the confluent hypergeometric function ([], [1]) and Λ n = n(n+n 1)/N. To proceed, recall the integral representation ([]) It follows that 1F 1 (a,b,λ) = λ n 1F 1 (n+1,n +n,λ) = Γ(b) Γ(b a)γ(a) 1 Γ(N +n) Γ(N +n 1)n! = ( 1)n Γ(N +n) Γ(N +n 1)n! e λu u a 1 (1 u) b a 1 du, b > a > λ n e λu u n (1 u) N+n du ( ) n d e λu [u n (1 u) N+n ]du du

8 after n integration by parts. But Rodriguez formula ([], p.99) ( ) n (1 x) α (1+x) β Pn α,β ( 1)n d (x) = n [(1 x) n+α (1+x) n+β ], x ( 1,1) n! du together with the variable change x = 1 u yields ( ) n d [u n (1 u) N+n ] = n!(1 u) N Pn,N (1 u). du As a result a n λ n 1F 1 (n+1,n +n,λ) = Pn,N (1 c) and Tonelli-Fubini Theorem yields () at time t/n. 1 e λu Pn,N (1 u)(1 u) N du 5. From the Laplace transform to the generalized Jacobi operator Another way to come from ϕ t (c,λ) to f t (c,λ) is as follows. For sake of simplicity, we shall drop the dependence on the parameter c. So, recall from [11] Proposition 4. that ϕ satisfies t ϕ = λϕ+ ( λ Nλ ) λ ϕ λ λ ϕ with the initial conditions ϕ (c,λ) = e λc,ϕ t (c,) = 1. Assume the density f is unknown and is smooth in both variables (t, u), then integration by parts yield where If N = then 1 e λu t f t (u)du = [e λu (1 Nu)f t (u)] e λu u [(Nu 1)f t (u)]du +[λe λu u(1 u)f t (u)] 1 [eλu u (u(1 u)f t (u))] e λu u [u(1 u)f t(u)]du = e λ ( N)f t (1)+ 1 e λu L(f t )(u)du L u u(1 u) u +[1+(N 4)u] u +(N ). L = u(1 u) u +[1 u] u is nothing else but (6) with N =. Otherwise, write f t (u) g t (u)s 1 (u) for a smooth function g and note that L(s 1 ) =. As a result, L(f t )(u) = s 1 (u) { u(1 u) u +[1 Nu] u } (gt )(u) where the RHS is the operator displayed in (6). Since f t (1) = when N 3 then we always have 1 e λu t g t (u)s 1 (u)du = 1 e λu{ u(1 u) u +[1 Nu] u } (gt )(u)s 1 (u)du. But the set of monomials (u n ) n is total in L ([,1],(1 u) N du) ([4], Theorem 3.17) then g solves the heat equation t g t (u) = { u(1 u) u +[1 Nu] u} (gt )(u). More generally, the Laplace transform of the density of U (k) t,1 k N 1 satisfies the linear pde (11) t ϕ = λ j ϕ+ (λ j Nλ j ) j ϕ λ i λ j ij ϕ. j=1 j=1 Hoping there will be no confusion, set again ϕ t (c,λ) 8 e λ,u f t (c,u)du j,

9 where du is the Lebesgue measure in the simplex. Then, Integration by parts ) λ i ϕ t (λ) e λ,u ( i f t (u) du (λ i Nλ i ) i ϕ t (λ) e λ,u [ ii +N i ](u i f t (u))du, i {1,...,k}, λ i iiϕ t (λ) e λ,u ii (u i f t(u))du, i {1,...,k}, λ i λ j ij ϕ t (λ) e λ,u ij (u i u j f t (u))du, 1 i j k, transform the pde (11) into e λ,u t f t (u)du = boundary terms+ where this time L denotes the operator k(n k 1)+ e λ,u L(f t )(u)du, ] [1+[N 4 (k 1)]u i i + i u (u i ) ii u i u j ij. If k = N 1 then L reduces to the operator displayed in (7). Otherwise, set f t (u) g t (u)s k (u) where this time g is a smooth function in both variables (t,u),u k and note that the relations i s k = 1 s k,1 i k together with the identity ) ( ( ) u i u j = u i (1 u i = 1 u i ) 1 u i u i (1 u i ) imply that L u (s k ) =. Hence L(f t )(u) gives rise to ] [1+[N 4 (k 1)]u i i + i u (u i) ii u i u j ij + 1s s (u) i (1 u i ) i u u j u i i ] = [1+[N 4 (k 1)]u i i + i u (u i) ii u i u j ij + 1s s (u) u i i 1 u i u j = [1 Nu i ] i + i u (u i ) ii u i u j ij acting on g t. Consequently, if the boundary terms vanish then Theorem 3.17 in [4] implies that g t solves the heat equation [1 Nu i ] i + i u (u i) ii u i u j ij g t = t g t. We shall see below that this is the case provided that 1 k N. 6. Analysis of the boundary terms Recall from the previous section that the integration by parts performed in the one-variable setting gave rise to the boundary term e λu [(1 Nu)f t (u) u (u(1 u)f t (u))] 9

10 which vanish at u = 1 since f t (1) = when N 3 (note that there is no such condition when N = ). For higher values k, the situation is similar provided that 1 k N and is different when k = N 1 due to the interactions between u i and u j for i j. Indeed, the boundary terms are given by 1 k 1 k [e λiui (1 Nu i )f t (u)] 1 uj [λ i e λiui u i (1 u i )f t (u)] 1 uj e λjuj 1 [,1] (u j )du j + [e λiui i {u i (1 u i )f t (u)}] 1 uj (λ j u j )[e λiui u i f t (u)] 1 m i um [e λjuj u j i {u i f t (u)}] 1 m j um By Leibniz rule, the third and the last terms split into [e λiui u i (1 u i ) i f t (u)] 1 uj [e λiui (1 u i )f t (u)] 1 uj e λjuj 1 [,1] (u j )du j e λjuj 1 [,1] (u j )du j m ie λmum 1 [,1] (u m )du m + m je λmum 1 [,1] (u m )du m. e λjuj 1 [,1] (u j )du j + e λjuj 1 [,1] (u j )du j and 1 k (k 1) j=1 u i [e λjuj u j i f t (u)] 1 m j um [e λjuj u j f t (u)] 1 m j um m je λmum 1 [,1] (u m )du m + m je λmum 1 [,1] (u m )du m respectively. Thus there are no boundary terms at u i =,1 i l, while the remaining ones are given by [e λiui (k +1 N)u i f t (u)] u uj e λjuj 1 [,1] (u j )du j + e λiui u i λ i(1 u i ) u uj λ j u j f t(u) e λjuj 1 [,1] (u j )du j e λiui u i (1 u i) i f t (u) u uj u j j f t (u) e λjuj 1 [,1] (u j )du j. 1

11 If N k + and f t = g t s k vanishes on the hyperplane {u 1 + +u k = 1} and the boundary terms reduce to e λi(1 uj) 1 u uj u j u j [ i f t (u) j f t (u)] e λjuj 1 [,1] (u j )du j. But since i s k = j s k and since s k vanishes on {u 1 + +u k = 1}, then for any 1 i j k i f t (u) = j f t (u), u 1 + +u k = 1 so that all boundary terms vanish. When k = N 1 the boundary terms read e λi(1 uj) 1 u j (λ i λ j )u j [f t(u)] u uj e λjuj 1 [,1] (u j )du j e λi(1 uj) 1 u uj u j u j [ i f t (u) j f t (u)] e λjuj 1 [,1] (u j )du j. Acknowledgments: we would like to thank D. Bakry, C.F. Dunkl, T. Hmidi and Y. Xu for stimulating discussions and for their help with appropriate references. References [1] R. Aktas, Y. Xu. Sobolev orthogonal polynomials on a simplex. Int. Math. Res. Notice, (13), no. 13, [] G. E. Andrews, R. Askey, R. Roy. Special functions. Cambridge University Press [3] D. Bakry. Remarques sur les semi-groupes de Jacobi. Hommage à P. André Meyer et J. Neveu. Astérisque. 36, 1996, [4] C. F. Dunkl, Y. Xu. Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and Its Applications. Cambridge University Press. 1. [5] G. Gasper. Banach algebras for Jacobi series and positivity of a Kernel. Ann. Math. 95, 197, [6] E. L. Grinberg. Spherical harmonics and integral geometry on projective spaces. Trans. Amer. Math. Soc., 79, no [7] F. Hiai, D. Petz. The Semicircle Law, Free Random Variables and Entropy. Mathematical Surveys and Monographs. Vol 77. A. M. S. [8] S. P. Karlin, G. McGregor. Classical diffusion processes and total positivity. J. Math. Anal. Appl. 1, 196, [9] T. Koornwinder. The addition formula for Jacobi polynomials II. The Laplace type integral and the product formula. Report TW 133/7. Mathematisch Centrum, Amsterdam [1] T. Koornwinder. The addition formula for Jacobi polynomials III. Completion of the proof. Report TW 135/7. Mathematisch Centrum, Amsterdam. 197 [11] I. Nechita, C. Pellegrini. Random pure quantum states via unitary Brownian motion. Electron. Commun. Probab. 18, 13, no. 7, [1] G. N. Watson. A treatise on the theory of Bessel functions. Cambridge Mathematical Library edition, (1995). Institut de Recherche en Mathématiques de Rennes, Université Rennes 1, France address: nizar.demni@univ-rennes1.fr 11