Reciprocal of the First hitting time of the boundary of dihedral wedges by a radial Dunkl process
aa r X i v : . [ m a t h . P R ] O c t RECIPROCAL OF THE FIRST HITTING TIME OF THE BOUNDARY OFDIHEDRAL WEDGES BY A RADIAL DUNKL PROCESS
NIZAR DEMNI
Abstract.
In this paper, we establish an integral representation for the density of the reciprocal of the firsthitting time of the boundary of even dihedral wedges by a radial Dunkl process having equal multiplicityvalues. Doing so provides another proof and extends to all even dihedral groups the main result proved in[11]. We also express the weighted Laplace transform of this density through the fourth Lauricella functionand establish similar results for odd dihedral wedges. Introduction
The radial Dunkl process associated with a finite reflection group is a diffusion valued in a Weyl chamberand depends on a set of non negative parameters called multiplicity values ([5], Chapters II and III). Itsinfinitesimal generator is the differential part of the so-called Dunkl-Laplace operator subject to normalreflections at the boundary of the chamber. In the rank-one case, the radial Dunkl process is nothing elsebut a Bessel process and if all the multiplicities values vanish, then it reduces to the reflected Brownian motionin the Weyl chamber. Another quite interesting instance of the radial Dunkl process is the Brownian motionconditioned to stay in the interior of the Weyl chamber, due to its close connections to the representationtheory of semi-simple finite-dimensional Lie algebras ([2], [3]).For small multiplicity values, the radial Dunkl process hits almost surely the boundary of the Weylchamber. In this case, the tail distribution of the first hitting time of the boundary can be computed fromthe absolute-continuity relations existing between the process distributions corresponding to different setsof multiplicity values (see [5], p.208). In particular, this probability is expressed, for the infinite familiesof irreducible root systems
A, B, C, D , through multivariate confluent hypergeometric function ([8]). Fordihedral root systems, the tail distribution is given by a series of Jacobi polynomials and one-variableconfluent hypergeometric functions, which generalize the expansions derived in [6] for the planar Brownianmotion in dihedral wedges (in [10], the formulas were only given for even dihedral groups and similar onesmay be derived for odd dihedral groups). Note in passing that combinatorial techniques were used in [12]where the tail distribution of the first exit time from Weyl chambers by a Brownian motion was representedthrough pfaffians.Back to the radial Dunkl process in dihedral wedges, an integral representation of the density of thereciprocal of the first hitting time of the boundary was obtained in [11] for the special angle π/ π/
4, thedomain of integration is the multidimensional standard simplex and its integrand still involves a one-variableconfluent hypergeometric function yet with a more complicated argument. When the radial Dunkl processstarts at a point lying on the bisector of the dihedral wedge, we express the weighted Laplace transform of
Date : October 12, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Generalized Bessel function; Dihedral groups; Gegenbauer polynomials; Modified Bessel functions;Radial Dunkl process; First hitting time of the boundary of a dihedral wedge. he density through the fourth Lauricella function, thereby extending to all even dihedral groups Corollary3.5 proved in [11]. As to the extension Corollary 3.3 in [11], it is straightforward and we will only allude toit without details in a aside remark. Using an analytic analogy between the infinitesimal generators of radialDunkl processes corresponding on one side to odd dihedral groups and on the other side to even dihedralgroups with equal multiplicity values, we derive an analogous integral representation for odd dihedral wedgeswhere this time, the integrand involves rather a Fox-Wright confluent hypergeometric function. The weightedLaplace transform is then computed but is not expressed through a multivariable hypergeometric series. Letus finally stress that the choice of equal multiplicity values is not a restriction and is rather made for sakeof simplicity since the formulas are already cumbersome. Nonetheless, we hope dealing with the case ofdifferent multiplicity values in a future paper.The paper is organized as follows. The next section recalls some facts on reflection groups and systems, aswell as on the radial Dunkl process in dihedral wedges. In the third section, we recall the definitions and someproperties of various special functions occurring in the paper. The fourth section is devoted to the derivationof the integral representation of the density for even dihedral wedges and equal multiplicity values, and ofits weighted Laplace transform. In the last section, we derive analogous results for odd dihedral wedges.2. Reminder: Dunkl processes in dihedral wedges
Dihedral systems and groups.
For general facts on root systems and reflection groups, we referthe reader to the monographs [13] and [16]. The dihedral group, D ( n ) , n ≥
3, consists of orthogonaltransformations leaving invariant a regular n -sided polygon centered at the origin. Actually, it contains n rotations: r j : z ze iπj/n , j = 1 , . . . , n, and n reflections: s j : z ze iπj/n , j = 1 , . . . , n, where z ∈ C and i = √− D ( n ) corresponds tothe dihedral root system I ( n ), which may be defined as the set of planar vectors: I ( n ) := {± ie iπl/n , ≤ l ≤ n } . To the following choice of positive root system {− ie iπl/n , ≤ l ≤ n } , corresponds a cone, named the positive Weyl chamber C , and is the wedge of dihedral angle π/n parametrizedin polar coordinates as: C := { ( r, θ ) , r > , < θ < π/n } . On the other hand, the dihedral group acts on its root systems in a natural way and this action gives riseto two orbits when n := 2 p, p ≥ , (the roots forming the diagonals and those forming the lines joining themidpoints of the polygon) while it is transitive otherwise. Accordingly, we shall assign to the two orbitsnon negative real parameters k , k , respectively when n is even and only one non negative real parameter k otherwise (these parameters are known in the realm of Lie groups as multiplicity values). With these data,we are now ready to recall the definition of the radial Dunkl process associated with dihedral groups (see[10] for further details).2.2. Radial Dunkl processes in dihedral wedges.
The radial Dunkl process X associated with a givenreduced root system is a C -valued diffusion reflected at the boundary ∂C (see [5] for further details). In par-ticular, for odd dihedral groups D ( n ) , n ≥
3, its infinitesimal generator acts on smooth functions supportedin C as ([10]):(1) 12 (cid:20) ∂ r + 2 nk + 1 r ∂ r (cid:21) + 1 r (cid:20) ∂ θ nk cot( nθ ) ∂ θ (cid:21) , subject to Neumann boundary conditions. While for even dihedral groups D (2 p ), the action of the infini-tesimal generator of X is given by:(2) 12 (cid:20) ∂ r + 2 p ( k + k ) + 1 r ∂ r (cid:21) + 1 r (cid:20) ∂ θ p ( k cot( pθ ) − k tan( pθ )) ∂ θ (cid:21) . hese polar decompositions show that the Euclidean norm of X is a Bessel process and that the cosine ofits angular part is, up to a time change, a real Jacobi process on the interval [0 , π/n ] ([20], [10]). This lastobservation was the key ingredient in the study of the first hitting time T of ∂C by X : T := inf { t ≥ , X t ∈ ∂C } . In this respect, recall from [4], Proposition 6 and from [9], Proposition 1, that T is almost surely finite ifand only if (at least) one of the multiplicity values is strictly less than 1 /
2. In particular, let n = 2 p, p ≥ , x = ρe iφ ∈ C and assume ( k , k ) ∈ (1 / , X starting at X = x andassociated with the multiplicity values (1 − k , − k ) hits almost surely the boundary of the dihedral wedgeof angle π/ (2 p ). As such, the corresponding reciprocal of T V := ρ T is a almost surely positive random variable. A similar statement holds true for odd dihedral wedges.3. Special functions
In this section, we record the definitions of various special functions occurring in the remainder of thepaper as well as some of their properties we will need in our subsequent computations. The reader is referredfor instance to [1], [14], [15], [18]. We start with the Gamma integral:Γ( z ) = Z ∞ e − u u z − du, ℜ ( z ) > , and the Legendre duplication formula:(3) √ π Γ(2 z + 1) = 2 z − Γ (cid:18) z + 12 (cid:19) Γ( z + 1) . Then, we recall the Pochhammer symbol:( a ) k = ( a + k − . . . ( a + 1) a, a ∈ C , k ∈ N , which may be written when a > a ) k = Γ( a + k )Γ( a ) . Next comes the generalized hypergeometric function defined by the series: r F q (( a i , ≤ i ≤ r ) , ( c j , ≤ j ≤ q ); z ) = X m ≥ Q ri =1 ( a i ) m Q qj =1 ( c j ) m z m m ! , provided it converges absolutely. Here, an empty product equals one and the parameters ( a i , ≤ i ≤ r )are complex numbers while ( c j , ≤ j ≤ q ) ∈ C \ − N . If a i = − n ∈ − N for some 1 ≤ i ≤ r , then thehypergeometric series terminates and as such, reduces to a polynomial of degree n . For instance, the j -thGegenbauer polynomial of parameter λ is defined through the Gauss hypergeometric function F :(4) C ( λ ) j ( z ) := (2 λ ) j j ! F (cid:18) − j, j + 2 λ, λ + 12 , − z (cid:19) , λ = 0 . For this polynomial and this hypergeometric function, the following argument transformations hold:(5) C ( λ ) j ( − z ) = ( − j C ( λ ) j ( z ) , (6) F ( a, c , c ; z ) = (1 − z ) − a F (cid:18) a, c − c , c ; zz − (cid:19) , | arg(1 − z ) | < π. Now, the confluent hypergeometric series F converges in the whole complex plane:(7) F ( a, c, z ) = X j ≥ ( a ) j ( c ) j z j j ! , nd admits the following integral representation valid for ℜ ( c ) > ℜ ( a ) > F ( a, c, z ) = Γ( c )Γ( a )Γ( c − a ) Z e uz u a − (1 − u ) c − a − du. (8)We shall also need the fourth Lauricella function F ( p − D in p − , p ≥ F ( p − D ( a, d , . . . , d p − , d p ; z , . . . , z p − ) := X m ,...,m p − ≥ ( a ) m + ... + m p − ( d p ) m + ... + m p − p − Y s =1 ( d s ) m s z m s s , which converges for | z s | < , ≤ s ≤ p −
1, where a, d , . . . , d p − ∈ C and d p ∈ C \ N . When ℜ ( d s ) > , ≤ s ≤ p, it admits the following Euler-type integral representation (see e.g. [17], eq. (2.1)):(10) F ( p − D ( a, d , . . . , d p − , d + . . . + d p − + d p ; z , . . . , z p − ) = Γ( a + d + . . . + d p − )Γ( a ) Q p − s =1 Γ( d s ) Z Σ p du . . . du p − (1 − u z − . . . − u p − z p − ) − a p Y s =1 u d s − s , where Σ p := { ( u , . . . , u p ) , u , . . . , u p ≥ , u + . . . + u p = 1 } is the standard simplex in R p . Finally, we recall the Dirichlet integral: for any β , . . . , β p > Z Σ p du . . . du p − p Y s =1 u β s − s = Γ( β ) . . . Γ( β p )Γ( β + . . . + β p ) , and we denote µ s ( du ) = Γ( s + 1 / √ π Γ( s ) (1 − u ) s − [ − , ( u ) du, s > , the symmetric Beta distribution.4. Integral representation of the density of V : even dihedral wedges and equalmultiplicity values Let k = k ≡ k ∈ (1 / ,
1] and consider the radial Dunkl process in an even dihedral wedge of angle π/ (2 p ) , p ≥ , and associated with (1 − k, − k ). In [11], it was shown that the density of V is, up to anormalizing constant, the even part of the following series (see eq. (3.2), p.146):(12) sin ν (2 pφ ) e − v v pν − X j ≥ Γ( p ( j + 1))Γ(2 p ( j + k )) v pj F ( p ( j + 1) , p ( j + k ) + 1 , v ) C ( k ) j (cos(2 pφ )) , where v > , ν := k − . For the particular value p = 2, the following integral representation of (12) was obtained in [11], Lemma 3.2:(13) X j ≥ Γ(2( j + 1))Γ(4( j + k )) v j F (2( j + 1) , j + k ) + 1 , v ) C ( k ) j (cos(4 φ )) = 1Γ(4 k ) Z F (cid:18) , ν + 32 ; v (1 − cos(2 φ ) u )2 (cid:19) µ k ( du ) . The extension of (13) to all values p ≥ The result is also valid for p = 1 roposition 1. For any p ≥ , (14) X j ≥ Γ( p ( j + 1))Γ(2 p ( j + k )) v pj F ( p ( j + 1) , p ( j + k ) + 1 , v ) C ( k ) j (cos(2 pφ )) = Γ( p )Γ( pk )[Γ( k )] p Γ(2 kp ) Z Σ p du . . . du p − p Y s =1 u k − s F p, pk + 12 ; v p X s =1 u s cos (cid:18) φ + sπp (cid:19)! . Proof.
From (7), we readily deriveΓ( p ( j + 1))Γ(2 p ( j + k )) F ( p ( j + 1) , p ( j + k ) + 1 , v ) = 2 p ( j + k ) X m ≥ Γ( a j + m )Γ( b j + m + 1) v m m ! , whence X j ≥ Γ( p ( j + 1))Γ(2 p ( j + k )) v pj F ( p ( j + 1) , p ( j + k ) + 1 , v ) C ( k ) j (cos(2 pφ )) = (2 p ) X j,m ≥ ( j + k )Γ( pj + m + p )Γ(2 pj + m + 2 pk + 1) m ! v pj + m C ( k ) j (cos(2 pφ ))= X N ≥ v N Γ( N + p ) X m,j ≥ N = pj + m p ( j + k )Γ(2 pj + m + 2 pk + 1) m ! C ( k ) j (cos(2 pφ )) . Now, recall from [7], Proposition 1, the following identity: for any integers M ≥ , q ≥
1, any real numbers k > , ξ ∈ [0 , π ], we have:(15) X m,j ≥ M =2 m + qj q ( j + k ) m !Γ( q ( j + k ) + m + 1) C ( k ) j (cos ξ ) = 2 M Γ( M + qk ) X j ,...,j q ≥ j + ··· j q = M ( k ) j . . . ( k ) j q [ b ,q ( ξ )] j j ! · · · [ b q,q ( ξ )] j q j q ! , where b s,q ( ξ ) := cos (cid:18) ξ + 2 sπq (cid:19) , s = 1 , . . . , q. In particular, if M = 2 N, q = 2 p, ξ = 2 pφ , then (15) specializes to: X m,j ≥ N = m + pj p ( j + k ) m !Γ(2 p ( j + k ) + m + 1) C ( k ) j (cos(2 pφ )) = 2 N Γ(2 N + 2 pk ) X j ,...,j p ≥ j + ··· j p =2 N p Y s =1 ( k ) j s j s ! cos j s (cid:18) φ + sπp (cid:19) . As a result, X j ≥ Γ( p ( j + 1))Γ(2 p ( j + k )) v pj F ( p ( j + 1) , p ( j + k ) + 1 , v ) C ( k ) j (cos(2 pφ )) = X N ≥ v N N Γ( N + p )Γ(2 N + 2 pk ) X j ,...,j p ≥ j + ··· j p =2 N p Y s =1 ( k ) j s j s ! cos j s (cid:18) φ + sπp (cid:19) = Γ( p )Γ(2 pk ) X N ≥ v N ( p ) N ( pk ) N ( pk + (1 / N X j ,...,j p ≥ j + ··· j p =2 N p Y s =1 ( k ) j s j s ! cos j s (cid:18) φ + sπp (cid:19) . Now, since cos( a + π ) = − cos( a ), then the finite sum in last equality may be written as:(16) X j ,...,j p ≥ j + ··· j p =2 N ( − j + ··· + j p p Y s =1 ( k ) j s j s ! p Y s =1 cos j s + j s + p (cid:18) φ + sπp (cid:19) , hich may be further simplified as follows. Consider a 2 p -tuple of integers in the sum (16) arranged in p pairs: ( j , j p +1 ) , . . . , ( j p , j p ) , p X s =1 j s = 2 N, and assume that one pair consists of an even and an odd integers. Then, the 2 p -tuple obtained from theprevious one by permuting the even and the odd integers in the given pair (while keeping the other pairsunchanged) cancels the latter since the expression p Y s =1 ( k ) j s j s ! p Y s =1 cos j s + j s + p (cid:18) φ + sπp (cid:19) is the same for both tuples. As a matter of fact, the only tuples that remains in (16) after cancellations arethose for which j s + j s + p is even, therefore(17) X j ,...,j p ≥ j + ··· j p =2 N ( − j + ··· + j p p Y s =1 ( k ) j s j s ! p Y s =1 cos j s + j s + p (cid:18) φ + sπp (cid:19) = X j ,...,j p ≥ j + ··· j p =2 Nj s + j s + p is even ( − j + ··· + j p p Y s =1 ( k ) j s j s ! p Y s =1 cos j s + j s + p (cid:18) φ + sπp (cid:19) . Setting m s = j s + j s + p and using the identity m s X j s =0 ( − j s ( k ) j s ( k ) m s − j s j s !(2 m s − j s )! = ( k ) m s m s ! , which is readily checked by equating generating functions of both sides, then the right-hand side of (17) maybe written as X m ,...,m p ≥ m + ··· m p = N p Y s =1 cos m s (cid:18) φ + sπp (cid:19) m s X j s =0 ( − j s ( k ) j s ( k ) m s − j s j s !(2 m s − j s )! = X m ,...,m p ≥ m + ··· m p = N p Y s =1 cos m s (cid:18) φ + sπp (cid:19) ( k ) m s m s ! . Consequently, X j ≥ Γ( p ( j + 1))Γ(2 p ( j + k )) v pj F ( p ( j + 1) , p ( j + k ) + 1 , v ) C ( k ) j (cos(2 pφ )) = Γ( p )Γ(2 pk ) X N ≥ v N ( p ) N ( pk ) N ( pk + (1 / N X m ,...,m p ≥ m + ··· m p = N p Y s =1 cos m s (cid:18) φ + sπp (cid:19) ( k ) m s m s ! = Γ( p )Γ(2 pk ) X m ,...,m p ≥ ( p ) m + ··· + m p ( pk ) m + ··· + m p ( pk + (1 / m + ··· + m p p Y s =1 ( k ) m s m s ! (cid:26) v cos (cid:18) φ + sπp (cid:19)(cid:27) m s . Finally, substituting β s = k + m s in (11) and using the multinomial theorem:( a + . . . + a p ) N = X m + ... + m p = N N ! m ! . . . m p ! p Y s =1 a m s s , ogether with (7), we end up with :Γ( p )Γ(2 pk ) X m ,...,m p ≥ ( p ) m + ··· + m p ( pk ) m + ··· + m p ( pk + (1 / m + ··· + m p p Y s =1 ( k ) m s m s ! (cid:26) v cos (cid:18) φ + sπp (cid:19)(cid:27) m s = Γ( p )Γ( pk )[Γ( k )] p Γ(2 pk ) Z Σ p du . . . du p − X m ,...,m p ≥ ( p ) m + ··· + m p ( pk + (1 / m + ··· + m p p Y s =1 u k + m s − s m s ! (cid:26) v cos (cid:18) φ + sπp (cid:19)(cid:27) m s = Γ( p )Γ( pk )[Γ( k )] p Γ(2 pk ) Z Σ p du . . . du p − p Y s =1 u k − s X N ≥ ( p ) N ( pk + (1 / N X m + ... + m p = N p Y s =1 u m s s m s ! (cid:26) v cos (cid:18) φ + sπp (cid:19)(cid:27) m s = Γ( p )Γ( pk )[Γ( k )] p Γ(2 pk ) Z Σ p du . . . du p − p Y s =1 u k − s X N ≥ ( p ) N ( pk + (1 / N N ! ( v p X s =1 u s cos (cid:18) φ + sπp (cid:19)) N . Using (7), the proposition is proved. (cid:3)
Remark 1. If p = 2 , then (14) simplifies to Γ(2 k )[Γ( k )] Γ(4 k ) Z u k − (1 − u ) k − F (cid:18) , k + 12 ; v (cid:8) u sin ( φ ) + (1 − u ) cos ( φ ) (cid:9)(cid:19) du = Γ(2 k )[Γ( k )] Γ(4 k ) Z u k − (1 − u ) k − F (cid:18) , k + 12 ; v (cid:26) − u ) cos(2 φ )2 (cid:27)(cid:19) du. Performing the variable change u (1 − u ) / in this integral and using Legendre duplication formula (3) ,one recovers (13) (recall k = ν + (1 / . Keeping in mind (12), we get:
Corollary 1.
Let k ∈ (1 / , and p ≥ . Then, the density of V admits the following integral representation(up to a normalizing constant depending only on p, k ): sin ν (2 pφ ) e − v v pν − Z Σ p du . . . du p − p Y s =1 u k − s ( F p, pν + ( p + 1)2 ; v p X s =1 u s cos (cid:18) φ + sπp (cid:19)! + F p, pν + ( p + 1)2 ; v p X s =1 u s cos (cid:18) φ − (2 s − π p (cid:19)!) . Proof.
As claimed in the beginning of this section, the density of V is half of the sum of both series: X j ≥ ( ± j Γ( p ( j + 1))Γ(2 p ( j + k )) v pj F ( p ( j + 1) , p ( j + k ) + 1 , v ) C ( k ) j (cos(2 pφ )) . Using the symmetry relation (5), we get:( − j C ( k ) j (cos(2 pφ )) = C ( k ) j (cid:18) cos (cid:18) p (cid:18) π p − φ (cid:19)(cid:19)(cid:19) , whence(18) X j ≥ Γ( p ( j + 1))Γ(2 p ( j + k )) v pj F ( p ( j + 1) , p ( j + k ) , v ) ( − j C ( k ) j (cos(2 pφ )) = Γ( p )Γ( pk )[Γ( k )] p Γ(2 kp ) Z Σ p du . . . du p − p Y s =1 u k − s F p, pk + 12 ; v p X s =1 u s cos (cid:18) φ − π (2 s + 1)2 p (cid:19)! . ince cos (cid:18) φ − π (2 p + 1)2 p (cid:19) = cos (cid:18) φ − π p (cid:19) , then equality (18) is rewritten:(19) X j ≥ Γ( p ( j + 1))Γ(2 p ( j + k )) v pj F ( p ( j + 1) , p ( j + k ) , v ) ( − j C ( k ) j (cos(2 pφ )) = Γ( p )Γ( pk )[Γ( k )] p Γ(2 kp ) Z Σ p du . . . du p − p Y s =1 u k − s F p, pk + 12 ; v p X s =1 u s cos (cid:18) φ − π (2 s − p (cid:19)! , and the corollary is proved. (cid:3) Remark 2.
With (13) in hands, one readily extends Corollary 3.3 in [11] to all p ≥ provided p ( k −
1) + 12 > , ensuring the validity of the Euler-type integral representation (8) . As to the extension of Corollary 3.5 in [11] , it goes as follows: Corollary 2.
Assume the radial Dunkl process starts at a point lying on the bisector of the Weyl chamber: x = ρe iπ/ (4 p ) , ρ > . Denote E ( − ν ) ρ,π/ (4 p ) the distribution of this process associated with the common multiplicity value (1 − k ) . Then,for any y > , E ( − ν ) ρ,π/ (4 p ) (cid:16) V ( p +1) / − pν e − yV (cid:17) ∝ sin ν (2 pφ )(1 + y ) p ( ν − p +1) / (cid:20) y + sin (cid:18) π p (cid:19)(cid:21) − p F ( p − D p, k, . . . , k | {z } p − , pk ; − sin[3 π/ (2 p )] sin[( π/p )] y + sin ( π/ (4 p )) , . . . , − sin[(2 p + 1) π/ (2 p )] sin[(( p − π/p )] y + sin ( π/ (4 p )) , where the coefficient of proportionality only depends on k, p .Proof. The choice φ = π/ (4 p ) ensures the equality: p X s =1 u s cos (cid:18) π p + sπp (cid:19) = p X s =1 u s cos (cid:18) (4 s + 1) π p (cid:19) = p X s =1 u s cos (cid:18) π p − π (2 s − p (cid:19) , so that both confluent hypergeometric functions displayed in Corollary 1 coincide. Now, set M p ( u , . . . , u p ) := p X s =1 u s cos (cid:18) (4 s + 1) π p (cid:19) . Then, the Gamma integral together with the generalized binomial Theorem entail: Z ∞ v pν +( p − / e − v (1+ y )1 F (cid:18) p, pν + ( p + 1)2 ; vM p ( u , . . . , u p ) (cid:19) = Γ( pν + ( p + 1) / y ) pν +( p +1) / ∞ X N =0 ( p ) N N ! (cid:20) M p ( u , . . . , u p )(1 + y ) (cid:21) N = 1(1 + y ) p ( ν − p − / y − M p ( u , . . . , u p )] p . ext, we compute1 + y − M p ( u , . . . , u p ) = 1 + y − p − X s =1 u s cos (cid:18) (4 s + 1) π p (cid:19) − (1 − u − . . . − u p − ) cos (cid:18) π p (cid:19) = y + sin (cid:18) π p (cid:19) − p − X s =1 u s (cid:26) cos (cid:18) (4 s + 1) π p (cid:19) − cos (cid:18) π p (cid:19)(cid:27) = y + sin (cid:18) π p (cid:19) − p − X s =1 u s (cid:26) cos (cid:18) (4 s + 1) π p (cid:19) − cos (cid:18) π p (cid:19)(cid:27) = y + sin (cid:18) π p (cid:19) − p − X s =1 u s sin (cid:18) (2 s + 1) π p (cid:19) sin (cid:18) − sπp (cid:19) = (cid:20) y + sin (cid:18) π p (cid:19)(cid:21) − ( − p − X s =1 u s sin[(2 s + 1) π/ (2 p )] sin[ − ( sπ/p )] y + sin ( π/ (4 p )) ) . Using (10), we are done. (cid:3)
Remark 3. If p = 2 , then sin (cid:16) π (cid:17) = √ − √ and the Lauricella function F (1) D coincides with the Gauss hypergeometric function: F (cid:18) , k, k ; 21 − √ y ) (cid:19) . Using the argument transformation (6) , this hypergeometric function is transformed into: F (cid:18) , k, k ; 21 − √ y ) (cid:19) = (1 − √ y )) (1 + √ y )) F (cid:18) , k, k ; 21 + √ y ) (cid:19) . As a result, E ( − ν ) ρ,π/ (8) (cid:16) v (3) / − ν e − yV (cid:17) ∝ sin ν (4 φ )(1 + y ) ν − (1 / √ y )) F (cid:18) , k, k ; 21 + √ y ) (cid:19) which is the expression derived in the proof of Corollary 3.5 in [11] . Integral representation of the density of V : odd dihedral wedges Our previous reasoning applies to odd dihedral groups. Indeed, notice that if k = k , then the infinitesimalgenerator (2) reduces to12 (cid:20) ∂ r + 4 pk + 1 r ∂ r (cid:21) + 1 r (cid:20) ∂ θ pk cot(2 pθ ) (cid:21) , θ ∈ [0 , π/ (2 p )] , which is exactly the infinitesimal generator (1) after the identifications k ↔ k, n ↔ p . Since the taildistribution of T starting at x is the unique solution of the heat equation with appropriate boundaryconditions (see e.g. [6]), then this probability has the same expression in both cases (i.e., corresponding toodd dihedral wedges on the one side and to even ones with equal multiplicity values on the other side) underthe previous identifications. Of course, one can also mimic the derivation of the tail distribution writtenin [10], paragraph 7.1 and retrieve the same expression. Hence, if n ≥ k ∈ (1 / , D ( n ) and corresponding to the multiplicity value1 − k ∈ [0 , /
2) hits ∂C almost surely and the density of V is (up to a normalizing constant) the even partof the following series:(20) sin ν ( nφ ) e − v v nν − X j ≥ Γ( n ( j + 1) / n ( j + k )) ( √ v ) j F ( n ( j + 1) / , n ( j + k ) + 1 , v ) C ( k ) j (cos( nφ )) , v > . sing again (7) followed by (15), we may rewrite (20) as: X j ≥ Γ( n ( j + 1) / n ( j + k )) ( √ v ) j F ( n ( j + 1) / , n ( j + k ) + 1 , v ) C ( k ) j (cos( nφ )) = X j,m ≥ n ( j + k )Γ( n ( j + 1) / m ) m !Γ( nj + m + nk + 1) ( √ v ) nj +2 m C ( k ) j (cos( nφ )) = X N ≥ ( p v ) N Γ(( N + n ) / X N = nj +2 m n ( j + k ) m !Γ( nj + m + nk + 1) C ( k ) j (cos( nφ )) = X N ≥ (2 √ v ) N Γ(( N + n ) / N + nk ) X j ,...,j n ≥ j + ··· j n = N ( k ) j . . . ( k ) j n [ b ,n ( nφ )] j j ! · · · [ b n,n ( nφ )] j n j n ! . Finally, the Dirichlet integral (11) and the multinomial Theorem lead again to:
Proposition 2.
Let n ≥ be an integer and k > . Then, for any v > , X j ≥ Γ( n ( j + 1) / n ( j + k )) v j/ F ( n ( j + 1) / , n ( j + k ) + 1 , v ) C ( k ) j (cos( nφ )) = 1[Γ( k )] n Z Σ n du . . . du n − n Y s =1 u k − s X N ≥ Γ(( N + n ) / nk ) N (2 √ v ) N N ! " n X s =1 u s cos (cid:18) φ + 2 sπn (cid:19) N . The series occurring in the integrand is a Fox-Wright confluent hypergeometric function Ψ ([18]): X N ≥ Γ(( N + n ) / nk ) N z n N ! = Γ( nk ) Ψ [(( n/ , (1 / , ( nk, z ] , z ∈ C . Moreover, one obtains a similar integral representation for the density of V along the same lines written inthe proof of Corollary 1: Corollary 3.
Let n be an odd integer and k ∈ (1 / , . Then, the density of V admits the following integralrepresentation (up to a constant depending only on n, k ): sin ν ( nφ ) e − v v nν − Z Σ n du . . . du n − n Y s =1 u k − s ( Ψ " (( n/ , (1 / , ( nk, √ v n X s =1 u s cos (cid:18) φ + 2 sπn (cid:19) + Ψ " (( n/ , (1 / , ( nk, √ v n X s =1 u s cos (cid:18) φ − (2 s − πn (cid:19) . As to the analogue of Corollary 2, we similarly notice that n X s =1 u s cos (cid:18) π n + 2 sπn (cid:19) = n X s =1 u s cos (cid:18) π n − (2 s − πn (cid:19) and integrate the density specialized to φ = π/ (2 n ) with respect to the Gamma weight v ( n +1) / − nν e − yv fora given y >
0. Using the Gauss duplication formula, the resulting weighted Laplace transform is then givenby:
Corollary 4.
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Bernoulli . , (2013). 2000-2009. IRMAR, Universit´e de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France
E-mail address : [email protected]@univ-rennes1.fr