Asymptotic zero distribution of Jacobi-Piñeiro and multiple Laguerre polynomials
aa r X i v : . [ m a t h . C A ] J a n Asymptotic zero distribution of Jacobi-Pi˜neiro andmultiple Laguerre polynomials
Thorsten Neuschel, Walter Van AsscheKatholieke Universiteit Leuven, BelgiumAugust 15, 2018
Abstract
We give the asymptotic distribution of the zeros of Jacobi-Pi˜neiro polynomialsand multiple Laguerre polynomials of the first kind. We use the nearest neighborrecurrence relations for these polynomials and a recent result on the ratio asymp-totics of multiple orthogonal polynomials. We show how these asymptotic zerodistributions are related to the Fuss-Catalan distribution.
In this paper we obtain the asymptotic distribution of the zeros of two families of multipleorthogonal polynomials: the Jacobi-Pi˜neiro polynomials and the multiple Laguerre poly-nomials of the first kind [9, Ch. 23], [2], [19]. These are two families of multiple orthogonalpolynomials for which explicit formulas are known and which are useful for a number ofapplications. For instance, the zeros of Jacobi-Pi˜neiro polynomials (and Wronskian-typedeterminants of Jacobi-Pi˜neiro polynomials) form the unique solution of certain BetheAnsatz equations [14] and multiple orthogonal polynomials are also useful for investigat-ing determinantal point processes [10]. Recently the Jacobi-Pi˜neiro ensemble and themultiple Laguerre ensemble were introduced for random matrix minor processes relatedto percolation theory [1] which are based on the Jacobi-Pi˜neiro and multiple Laguerrepolynomials of the first kind.Let ~n = ( n , n , . . . , n r ) ∈ N r be a multi-index of size | ~n | = n + n + · · · + n r .The Jacobi-Pi˜neiro polynomials P ~n , with parameters ~α = ( α , . . . , α r ) and β , are typeII multiple orthogonal polynomials on [0 ,
1] for r Jacobi weights, i.e., P ~n is a monicpolynomial of degree | ~n | satisfying Z P ~n ( x ) x k x α j (1 − x ) β dx = 0 , k = 0 , , . . . , n j − , for j = 1 , , . . . , r , where β > − α j > − ≤ j ≤ r . They were introducedby Pi˜neiro for β = 0 [17]. A multi-index ~n is normal if the monic multiple orthogonalpolynomial P ~n of degree | ~n | exists and is unique. All multi-indices for Jacobi-Pi˜neiropolynomials are normal when α i − α j / ∈ Z because then the measures form an AT-system19, § − | ~n | r Y j =1 ( | ~n | + α j + β + 1) n j (1 − x ) β P ~n ( x )= r Y j =1 (cid:18) x − α j d n j dx n j x n j + α j (cid:19) (1 − x ) | ~n | + β , (1.1)where the product of the differential operators can be taken in any order, since theseoperators are commuting [9, § L ~n are given by the Rodrigues formula( − | ~n | e − x L ~n ( x ) = r Y j =1 (cid:18) x − α j d n j dx n j x n j + α j (cid:19) e − x , (1.2)where the product of the differential operators can be taken in any order [9, § ~α = ( α , α , . . . , α r ) are such that α i > − i and α i − α j / ∈ Z (1 ≤ i, j ≤ r ), then all multi-indices are normal and the polynomials satisfy the followingorthogonality properties Z ∞ L ~n ( x ) x k x α j e − x dx = 0 , k = 0 , , . . . , n j − , for j = 1 , , . . . , r . An explicit expression is given by L ~n ( x ) = n X k =0 · · · n r X k r =0 ( − | ~k | n !( n − k )! · · · n r !( n r − k r )! × (cid:18) n r + α r k r (cid:19)(cid:18) n r + n r − + α r − − k r k r − (cid:19) · · · (cid:18) | ~n | − | ~k | + k + α k (cid:19) x | ~n |−| ~k | . (1.3)We will obtain the asymptotic distribution of the zeros of these multiple orthogonalpolynomials by using a result on the asymptotic behavior of the ratio of two neighboringpolynomials [21]. This result uses the nearest neighbor recurrence relations for multipleorthogonal polynomials xP ~n ( x ) = P ~n + ~e k ( x ) + b ~n,k P ~n ( x ) + r X j =1 a ~n,j P ~n − ~e j ( x ) , ≤ k ≤ r, where ~e j = (0 , . . . , , , , . . . ,
0) with 1 in the j th entry, and some knowledge about theasymptotic behavior for the recurrence coefficients a ~n,j , b ~n,j (1 ≤ j ≤ r ). The ratio asymp-totic behavior for Jacobi-Pi˜neiro polynomials will be obtained in Section 2 and for multipleLaguerre polynomials of the first kind in Section 5. The asymptotic distribution of thezeros of Jacobi-Pi˜neiro polynomials will be obtained in Section 4, where the followingresult will be proved. We will use the multi-index ~ , , . . . ,
1) so that the diagonalindex is ( n, n, . . . , n ) = n~
1. 2 heorem 1.1.
Let < x ,rn < x ,rn < · · · < x rn,rn < be the zeros of the Jacobi-Pi˜neiropolynomial P n~ with multi-index n~ n, n, . . . , n ) . Then for every continuous function f on [0 , one has lim n →∞ rn rn X k =1 f ( x k,rn ) = Z f ( t ) v r ( t ) dt, where the density v r on [0 , is given by means of a density w r on [0 , c r ] as v r ( x ) = c r w r ( c r x ) , c r = ( r + 1) r +1 r r , and with the change of variables ˆ x = c r x = (cid:0) sin( r + 1) ϕ (cid:1) r +1 sin ϕ (cid:0) sin rϕ (cid:1) r , < ϕ < πr + 1 , the density w r is w r (ˆ x ) = r + 1 π | ˆ x ′ ( ϕ ) | = r + 1 π ˆ x sin ϕ sin rϕ sin( r + 1) ϕ ( r + 1) sin rϕ − r ( r + 1) sin( r + 1) ϕ sin rϕ cos ϕ + r sin ( r + 1) ϕ . (1.4)The density w r is in fact the uniform density on [0 , πr +1 ] in the variable ϕ since Z c r f (ˆ x ) w r (ˆ x ) d ˆ x = Z πr +1 f (ˆ x ( ϕ )) w r (ˆ x ( ϕ )) | ˆ x ′ ( ϕ ) | dϕ = r + 1 π Z πr +1 f (ˆ x ( ϕ )) dϕ. In this sense Theorem 1.1 is the extension to multiple orthogonal polynomials of theequidistribution result for zeros of orthogonal polynomials [18, Thm. 12.7.2] for the case ofJacobi-Pi˜neiro polynomials. In fact, the same asymptotic distribution of zeros will hold forall multiple orthogonal polynomials for which the nearest neighbor recurrence coefficientsbehave as in (2.3)–(2.4), provided the zeros of neighboring polynomials interlace. We haveplotted the density v r on [0 ,
1] for 1 ≤ r ≤ x = c r − (cid:18) r + 12 (cid:19) c r ϕ + O ( ϕ ) , ϕ → , and ˆ x = r + 1sin πr +1 ! r +1 (cid:18) πr + 1 − ϕ (cid:19) r +1 + O (cid:18) πr + 1 − ϕ (cid:19) r +2 ! , ϕ → πr + 1 , so that the density v r behaves as (ˆ x = c r x ) v r ( x ) = O ( ϕ − ) = O (cid:0) (1 − x ) − / (cid:1) , x → , and v r ( x ) = O (cid:18) πr + 1 − ϕ (cid:19) − r ! = O (cid:16) x − rr +1 (cid:17) , x → . v r for Jacobi-Pi˜neiro polynomials: r = 1 (solid), r = 2 (dash), r = 3 (dash dot), r = 4 (long dash), and r = 5 (dots).Hence the densities v r have a square root singularity at 1 but a higher order singularityat 0 when r >
1, which means that the zeros are more dense near the endpoints 0 and1, and even more so near 0 than near 1 when r >
1. For r = 1 the density v is thewell-known arcsin density on [0 , v ( x ) = 1 π p x (1 − x ) , < x < , which is the equilibrium measure for [0 ,
1] in logarithmic potential theory. For r = 2 thedensity can explicitly be written as v ( x ) = √ π (1 + √ − x ) / + (1 − √ − x ) / x / √ − x , x ∈ (0 , , and this asymptotic zero distribution was already found in [3, Thm. 2.5]. The momentsof w r are integers given by Z c r x n w r ( x ) dx = r + 1 π Z πr +1 x ( ϕ ) n dϕ = (cid:18) ( r + 1) nn (cid:19) , n ∈ N = { , , , . . . } , which follows from [16, Remark 3.4].For multiple Laguerre polynomials we need to use a scaling to prevent the zeros fromgoing to infinity. The appropriate scaling is to divide all the zeros of L ~n by | ~n | , so thatwe are in fact investigating the zeros of L n~ ( rnx ) for the multi-index n~ n, n, . . . , n )on the diagonal. In Section 6 we obtain the asymptotic distribution of the scaled zeros,where we prove the following result. 4 heorem 1.2. Let < x ,rn < x ,rn < · · · < x rn,rn be the zeros of the multiple Laguerrepolynomials L n~ with multi-index n~ n, n, . . . , n ) . Then for every continuous function f on [0 , c r /r ] one has lim n →∞ rn rn X k =1 f (cid:16) x k,rn rn (cid:17) = Z c r f ( t/r ) u r ( t ) dt, c r = ( r + 1) r +1 r r , where the density u r on [0 , c r ] is given by u r (ˆ x ) = 1 rπ (sin rϕ ) r +1 (cid:0) sin( r + 1) ϕ (cid:1) r , < ϕ < πr + 1 , (1.5) where ˆ x = (cid:0) sin( r + 1) ϕ (cid:1) r +1 sin ϕ (cid:0) sin rϕ (cid:1) r , < ϕ < πr + 1 . The densities u r for 1 ≤ r ≤ u r for multiple Laguerre polynomials of the firstkind: r = 1 (solid), r = 2 (dash), r = 3 (dash dot), r = 4 (long dash), and r = 5 (dots).The density of the scaled zeros { x k,rn rn , ≤ k ≤ rn } is therefore given by ru r ( rx ) for0 < x < c r /r . Note that the densities u r behave as u r (ˆ x ) = O ( ϕ ) = O (cid:0) ( c r − ˆ x ) / (cid:1) , ˆ x → c r , and u r (ˆ x ) = O (cid:18) πr + 1 − ϕ (cid:19) − r ! = O (cid:16) ˆ x − rr +1 (cid:17) , ˆ x → . Hence the densities u r tend to zero as a square root near the endpoint c r and have thesame singularity near 0 as in the Jacobi-Pi˜neiro case. For r = 1 the density is the5archenko-Pastur density [12] u (ˆ x ) = 12 π r − ˆ x ˆ x , < ˆ x < , (1.6)which is also the known asymptotic distribution of the (scaled) zeros of Laguerre polyno-mials (see, e.g., [6]). For r = 2 we have u (ˆ x ) = g ( x ), where g ( y ) = 3 √ π (1 + 3 √ − y )(1 − √ − y ) / − (1 − √ − y )(1 + √ − y ) / y / , and the asymptotic zero distribution of the zeros of multiple Laguerre polynomials forthat case was already obtained in [3, Thm. 2.6]. An interesting observation is that themoments of u r are given by Z c r x n u r ( x ) dx = 1 n + 1 (cid:18) ( r + 1) nn (cid:19) , n ∈ N . The simple expressions for the moments of w r and u r on [0 , c r ] is the main reason why weprefer to express the asymptotic zero densities in terms of densities on [0 , c r ], rather thanon [0 ,
1] and [0 , c r /r ] respectively. In Section 3 we will show that these densities and theasymptotic behavior of the ratio of Jacobi-Pi˜neiro and multiple Laguerre polynomials ofthe first kind are related to the Fuss-Catalan distribution with density g r ( x ) = 1 π sin ϕ (cid:0) sin rϕ (cid:1) r − (cid:0) sin( r + 1) ϕ (cid:1) r , < x < c r , where x = (cid:0) sin( r + 1) ϕ (cid:1) r +1 sin ϕ (cid:0) sin rϕ (cid:1) r , < ϕ < πr + 1 , for which the moments are the Fuss-Catalan numbers [7, p. 347] Z c r x n g r ( x ) dx = 1 rn + 1 (cid:18) ( r + 1) nn (cid:19) , n ∈ N . The nearest neighbor recurrence relations are xP ~n ( x ) = P ~n + ~e k ( x ) + b ~n,k P ~n ( x ) + r X j =1 a ~n,j P ~n − ~e j ( x ) , ≤ k ≤ r, where the recurrence coefficients are given by a ~n,j = n j ( n j + α j )( | ~n | + β )( | ~n | + n j + α j + β + 1)( | ~n | + n j + α j + β )( | ~n | + n j + α j + β − × r Y i =1 | ~n | + α i + β | ~n | + n i + α i + β r Y i =1 ,i = j n j + α j − α i n j − n i + α j − α i , ≤ j ≤ r, (2.1)6nd b ~n,k = ( | ~n | + β + 1) Q rj =1 ( | ~n | + β + α j + 1)( | ~n | + n k + β + α k + 2) Q j = k ( | ~n | + n j + β + α j + 1) − ( | ~n | + β ) Q rj =1 ( | ~n | + β + α j ) Q rj =1 ( | ~n | + n j + β + α j ) , ≤ k ≤ r. (2.2)(see, e.g., [20]).If we take the multi-index ~n = ( ⌊ q n ⌋ , . . . , ⌊ q r n ⌋ ), where q j > P rj =1 q j = 1, and ⌊·⌋ is the floor function (i.e., ⌊ a ⌋ = k whenever k ≤ a < k + 1), then the asymptotic behaviorof the recurrence coefficients islim n →∞ a ~n,j = q r +1 j (1 + q j ) r Y k =1
11 + q k Y i = j q j − q i =: a j , ≤ j ≤ r, (2.3)and with a bit of elementary calculuslim n →∞ b ~n,j = r Y k =1
11 + q k r + 1 − r X k =1
11 + q k −
11 + q j ! =: b j , ≤ j ≤ r. (2.4)In order to have finite values of a j , we assume for the moment that q i = q j whenever i = j , but later on we will take the limit q j → /r for every j . This passage to the limitis allowed since the asymptotic distribution of the zeros is continuous in the parameters( q , . . . , q r ), which can be shown as in [4, Thm. 2]. We will use the notation p ( ~q ) = r Y k =1
11 + q k , s = r + 1 − r X k =1
11 + q k , so that a j = p ( ~q ) q r +1 j (1 + q j ) Y i = j q j − q i , b j = p ( ~q ) (cid:18) s −
11 + q j (cid:19) . (2.5)According to [21, Thm. 1.1], the ratio asymptotics for the Jacobi-Pi˜neiro polynomialswith multi-index ~n = ( ⌊ q n ⌋ , . . . , ⌊ q r n ⌋ ) will then be given bylim n →∞ P ~n + ~e k ( x ) P ~n ( x ) = z ( x ) − b k , ≤ k ≤ r, (2.6)uniformly on compact subsets of C \ [0 , z is the solution of the algebraic equation( z − x ) B r ( z ) + A r − ( z ) = 0 (2.7)for which z ( x ) − x → x → ∞ . In [21] the convergence was given uniformly oncompact subsets of C \ R , but since all the zeros of Jacobi-Pi˜neiro polynomials are in [0 , C \ [0 , B r ( z ) = Q rj =1 ( z − b j ) and A r − is the polynomial of degree r − A r − ( z ) B r ( z ) = r X j =1 a j z − b j . A r − /B r at b j is given by a j : a j = A r − ( b j ) B ′ r ( b j ) = A r − ( b j ) Q i = j ( b j − b i ) . (2.8)Observe that Y i = j ( b j − b i ) = p ( ~q ) r (1 + q j ) r − Y i = j ( q j − q i ) , so that the condition on the residues (2.8) becomes A r − ( b j ) = (cid:18) p ( ~q ) q j q j (cid:19) r +1 , ≤ j ≤ r. (2.9)This is a Lagrange interpolation problem. If we use (2.5) to write q j in terms of b j , then q j = 1 s − b j /p ( ~q ) − , so that p ( ~q ) q j q j = p ( ~q )(1 − s ) + b j , ≤ j ≤ r. The interpolation problem (2.9) then becomes A r − ( b j ) = (cid:0) b j + p ( ~q )(1 − s ) (cid:1) r +1 , ≤ j ≤ r, hence A r − ( z ) is a polynomial of degree r − (cid:0) z + p ( ~q )(1 − s ) (cid:1) r +1 at the points b j (1 ≤ j ≤ r ). If we take the limit where q j → /r for every j , then p ( ~q ) → (cid:18) rr + 1 (cid:19) r =: p, s → r + 1 r + 1 , b j → p (cid:18) s − rr + 1 (cid:19) = p, hence all the interpolation points coincide. It is well known that the Lagrange inter-polating polynomial for which all the interpolation points coincide corresponds to theTaylor polynomial of degree r − f ( z ) = (cid:0) z + p (1 − s ) (cid:1) r +1 around thecommon interpolation point p . This Taylor polynomial of degree r − (cid:0) z + p (1 − s ) (cid:1) r +1 of degree r + 1 from which we subtract the last two terms of the Taylorexpansion around p : A r − ( z ) = (cid:0) z + p (1 − s ) (cid:1) r +1 − ( z − p ) r +1 f ( r +1) ( p )( r + 1)! − ( z − p ) r f ( r ) ( p ) r != (cid:0) z + p (1 − s ) (cid:1) r +1 − ( z − p ) r +1 − ( r + 1) p (2 − s )( z − p ) r . (2.10)The algebraic equation (2.7) for multi-indices on the diagonal then becomes( z − x )( z − p ) r + (cid:0) z + p (1 − s ) (cid:1) r +1 − ( z − p ) r +1 − ( r + 1) p (2 − s )( z − p ) r = 0 , which simplifies to x ( z − p ) r = (cid:16) z − prr + 1 (cid:17) r +1 . (2.11)8 Relation with the Fuss-Catalan numbers
Recently the Fuss-Catalan distribution and other related distributions (Raney distribu-tions) appeared as limiting distributions of eigenvalues and singular values of certainrandom matrices [5], [15], [16]. In this section we will show how the ratio asymptotics in(2.6) is related to the Stieltjes transform of the Fuss-Catalan distribution. The weights w r and u r in Theorem 1.1 and 1.2 cannot be identified with the Fuss-Catalan distributionor any of the Raney distributions (except u , which is the Catalan distribution) becausetheir behavior near the endpoints of the interval differs from the behavior of the Raneydistributions given in [13].If we scale the variables ˆ x = c r x and ˆ z = c r z , where c r = ( r + 1) r +1 r r = r + 1 p , then the algebraic equation (2.11) becomesˆ x (ˆ z − r − r = (ˆ z − r ) r +1 . (3.1)If we define ω = ˆ z − r ˆ z − r − , ˆ z = ( r + 1) ω − rω − , (3.2)then the algebraic equation becomes ω r +1 + ˆ x − ˆ xω = 0 . (3.3)This is the algebraic equation for the generating function G (1 / ˆ x ) of the Fuss-Catalannumbers [7, p. 347] [16, Eq. (3.12)]. As in [16, § ω = ρe iϕ , where ρ > ϕ is real. Then inserting this in (3.3) gives ρ r +1 e i ( r +1) ϕ + ˆ x − ˆ xρe iϕ = 0 . This gives for the real and the imaginary part ρ r +1 cos( r + 1) ϕ + ˆ x − ˆ xρ cos ϕ = 0 , (3.4) ρ r +1 sin( r + 1) ϕ − ˆ xρ sin ϕ = 0 . (3.5)From (3.5) we find ˆ x = ρ r sin( r + 1) ϕ sin ϕ , (3.6)and inserting this in (3.4) gives ρ (ˆ x ) = sin( r + 1) ϕ sin rϕ , (3.7)from which ˆ x = (cid:0) sin( r + 1) ϕ (cid:1) r +1 sin ϕ (cid:0) sin rϕ (cid:1) r . (3.8)Observe that ρ ( x ) > < ϕ < πr +1 , and ˆ x is a monotonically decreasing functionmapping [0 , πr +1 ] into [0 , c r ]. So for ˆ x ∈ [0 , c r ] there is a solution ρe iϕ of the algebraic9quation (3.3). The conjugate function ρe − iϕ is also a solution for ˆ x ∈ [0 , c r ]. In fact bothsolutions are the boundary value of a function ω which is analytic on C \ [0 , c r ] and ω + = lim ǫ → ω (ˆ x + iǫ ) = ρe − iφ , ω − = lim ǫ → ω (ˆ x − iǫ ) = ρe iφ , (3.9)because this ω is G (1 / ˆ x ) = ˆ xF (ˆ x ), where F is the Stieltjes transform of the Fuss-Catalandistribution F ( z ) = Z c r g r ( y ) z − y dy, z ∈ C \ [0 , c r ] , with g r the Fuss-Catalan density, and a Stieltjes transform has the property thatIm F ( z ) ( < , Im z > ,> , Im z < . . Observe that 1 z − p = lim n →∞ P ~n ( x ) P ~n + ~e k ( x )is the Stieltjes transform of a probability measure on [0 , P ~n ( x ) P ~n + ~e k ( x ) = | ~n | +1 X j =1 c j,~n x − x j,~n + ~e k , and c j,~n > P ~n and P ~n + ~e k interlace [8, Thm.2.1], and P c j,~n = 1since we are dealing with monic polynomials. With the change of variables ˆ x = c r x andˆ z = c r z it follows that 1 / (ˆ z − r −
1) = Z c r dµ ( y )ˆ x − y is the Stieltjes transform of a probability distribution µ on [0 , c r ]. Note that (3.2) implies1ˆ z − r − ω − xF (ˆ x ) − , where F is the Stieltjes transform of the Fuss-Catalan distribution, F (ˆ x ) = 1ˆ x ∞ X n =0 rn + 1 (cid:18) ( r + 1) nn (cid:19) x n , so that 1 / (ˆ z − r −
1) is the Stieltjes transform of the probability measure for which themoments are the Fuss-Catalan numbers shifted by one Z c r y n dµ ( y ) = 1 r ( n + 1) + 1 (cid:18) ( r + 1)( n + 1) n + 1 (cid:19) , and hence this probability distribution has a density ˆ xg r (ˆ x ), where g r is the Fuss-Catalandensity g r (ˆ x ) = 1 π sin ϕ (sin rϕ ) r − (cid:0) sin( r + 1) ϕ (cid:1) r , < ϕ < πr + 1 , x is given in (3.8). In particular this gives1ˆ z − r − Z c r yg r ( y )ˆ x − y dy. The weight is explicitly given byˆ xg r (ˆ x ) = 1 π sin ϕ sin( r + 1) ϕ sin rϕ , ≤ ϕ < πr + 1 , with ˆ x as in (3.8). So far we found that for ~n near the diagonal (i.e., n j /n → /r for every j ) one haslim n →∞ P ~n + ~e k ( x ) P ~n ( x ) = z ( x ) − p = 1 c r (ˆ z − r − , (4.1)uniformly for x on compact subsets of C \ [0 , x on compact subsets of C \ [0 , c r ].However we are interested in the asymptotic behavior of1 | ~n | P ′ ~n ( x ) P ~n ( x ) , where the prime ′ denotes the derivative with respect to x , because the limit will give theStieltjes transform of the asymptotic distribution of the zeros of P ~n . By taking derivativeswith respect to x in (4.1) we findlim n →∞ P ~n + ~e k ( x ) P ~n ( x ) (cid:18) P ′ ~n + ~e k ( x ) P ~n + ~e k ( x ) − P ′ ~n ( x ) P ~n ( x ) (cid:19) = z ′ = ˆ z ′ c r , uniformly for x on compact subsets of C \ [0 , n →∞ (cid:18) P ′ ~n + ~e k ( x ) P ~n + ~e k ( x ) − P ′ ~n ( x ) P ~n ( x ) (cid:19) = ˆ z ′ ˆ z − r − . If we use this result successively for each k , 1 ≤ k ≤ r , then we find for multi-indices onthe diagonal n~ n, n, . . . , n ) and ( n + 1) ~ n + 1 , n + 1 , . . . , n + 1)lim n →∞ P ′ ( n +1) ~ ( x ) P ( n +1) ~ ( x ) − P ′ n~ ( x ) P n~ ( x ) ! = r ˆ z ′ ˆ z − r − . Then by taking averages (Ces`aro’s lemma) we getlim n →∞ n n − X k =0 P ′ ( k +1) ~ ( x ) P ( k +1) ~ ( x ) − P ′ k~ ( x ) P k~ ( x ) ! = r ˆ z ′ ˆ z − r − , and since this contains a telescoping sum, this becomeslim n →∞ rn P ′ n~ ( x ) P n~ ( x ) = ˆ z ′ ˆ z − r − , x on compact subsets of C \ [0 , P n~ for n~ n, n, . . . , n ).From the Stieltjes transform we can find the density by using Stieltjes’ inversion formula2 πiv r ( x ) = (cid:18) ˆ z ′ ˆ z − r − (cid:19) − − (cid:18) ˆ z ′ ˆ z − r − (cid:19) + . Taking derivatives in (3.1) (and recalling that ˆ x = c r x ) gives c r (ˆ z − r − r + ˆ xr (ˆ z − r − r − ˆ z ′ = ( r + 1) (ˆ z − r ) r ˆ z ′ , so that ˆ z ′ ˆ z − r − c r ˆ x ˆ z − r ˆ z − r − . Writing this in terms of ω using (3.2) givesˆ z ′ ˆ z − r − c r ˆ x ω − rω + r + 1 . Now use ω + = ρe − iϕ and ω − = ρe iϕ to find the density v r ( x ) = r + 1 πx ρ sin ϕ | rρe iϕ − r − | , and clearly v r ( x ) = v r (ˆ x/c r ) = c r w r (ˆ x ), with the weight in (1.4). Observe that ˆ x :[0 , πr +1 ] → [0 , c r ] is a monotonically decreasing function withˆ x ′ ( ϕ ) = − ˆ x sin ϕ sin rϕ sin( r + 1) ϕ | ( r + 1) sin rϕ − e iϕ r sin( r + 1) ϕ | so that w r (ˆ x ) = r + 1 π | ˆ x ′ ( ϕ ) | , < ϕ < πr + 1 . The nearest neighbor recurrence relations for multiple orthogonal polynomials of the firstkind are given by xL ~n ( x ) = L ~n + ~e k ( x ) + b ~n,k L ~n ( x ) + r X j =1 a ~n,j L ~n − ~e j ( x ) , ≤ k ≤ r, where the recurrence coefficients are given by a ~n,j = n j ( n j + α j ) r Y i =1 ,i = j n j + α j − α i n j − n i + α j − α i , ≤ j ≤ r, (5.1)and b ~n,k = | ~n | + n k + α k + 1 , ≤ k ≤ r. (5.2)12see, e.g., [20]). We can now proceed as in the case of Jacobi-Pi˜neiro polynomials. Therecurrence coefficients are somewhat easier but they are unbounded so that we need touse a scaling. Suppose again that ~n = ( ⌊ q n ⌋ , . . . , ⌊ q r n ⌋ ), where q i = q j whenever i = j .It then follows that lim n →∞ a ~n,j n = q r +1 j Y i = j q j − q i =: a j , ≤ j ≤ r, and lim n →∞ b ~n,j n = 1 + q j =: b j , ≤ j ≤ r. According to [21, Thm. 1.2] we then havelim n →∞ L ~n + ~e k ( nx ) nL ~n ( nx ) = z ( x ) − b k , ≤ k ≤ r, (5.3)uniformly on compact subsets of C \ [0 , ∞ ), where z is the solution of the algebraic equation( z − x ) B r ( z ) + A r − ( z ) = 0 , where B r ( z ) = Q rj =1 ( z − b j ) and A r − is obtained from A r − ( z ) B r ( z ) = r X j =1 a j z − b j . The uniform convergence on compact subsets of C \ R in [21] can be extended to C \ [0 , ∞ )because the zeros of multiple Laguerre polynomials of the first kind are on [0 , ∞ ). Onecan even extend this further to C \ [0 , c r /r ] since all the scaled zeros are dense on [0 , c r /r ],but we will not need this here. Observe that Y i = j ( q j − q i ) = Y i = j ( b j − b i ) , so that we get the interpolation condition A r − ( b j ) = q r +1 j = ( b j − r +1 , ≤ j ≤ r. Hence A r − ( z ) is the Lagrange interpolating polynomial of degree r − f ( z ) = ( z − r +1 for the interpolation points b , . . . , b r . Now let q j → /r for every j ,then b j → r + 1 r , and A r − ( z ) will be the Taylor polynomial of degree r − r +1 r for the function f ( z ) = ( z − r +1 . This gives A r − ( z ) = ( z − r +1 − (cid:18) z − r + 1 r (cid:19) r +1 − r + 1 r (cid:18) z − r + 1 r (cid:19) r . The algebraic equation for multi-indices near the diagonal then becomes x (cid:18) z − r + 1 r (cid:19) r = ( z − r +1 . (5.4)The change of variables rz = ˆ z and rx = ˆ x gives the same algebraic equation as in (3.1).13 Proof of Theorem 1.2
As in Section 4 we use L ′ n~ ( x ) L n~ ( x ) = n − X k =0 L ′ ( k +1) ~ ( x ) L ( k +1) ~ ( x ) − L ′ k~ ( x ) L k~ ( x ) ! , where k~ k, k, . . . , k ) and ( k + 1) ~ k + 1 , k + 1 , . . . , k + 1). However, because of thescaling, we need to consider (observe that | n~ | = rn ) L ′ n~ ( rnx ) rnL n~ ( rnx ) = n − X k =0 L ′ ( k +1) ~ ( rnx ) rnL ( k +1) ~ ( rnx ) − L ′ k~ ( rnx ) rnL k~ ( rnx ) ! , so that we can not use Ces`aro’s lemma to get the asymptotic behavior. We modify theproof as follows. For kn ≤ t < k +1 n one has ⌊ nt ⌋ = k , hence the sum can be written as anintegral L ′ n~ ( rnx ) rnL n~ ( rnx ) = n Z L ′ ( ⌊ nt ⌋ +1) ~ ( rnx ) rnL ( ⌊ nt ⌋ +1) ~ ( rnx ) − L ′⌊ nt ⌋ ~ ( rnx ) rnL ⌊ nt ⌋ ~ ( rnx ) ! dt, and the integrand can be written as L ( ⌊ nt ⌋ +1) ~ ( rnx ) rnL ⌊ nt ⌋ ~ ( rnx ) ! ′ / L ( ⌊ nt ⌋ +1) ~ ( rnx ) rnL ⌊ nt ⌋ ~ ( rnx ) ! . So we need to know the asymptotic behavior of the ratiolim n →∞ L ( ⌊ nt ⌋ +1) ~ ( rnx ) rnL ⌊ nt ⌋ ~ ( rnx ) . If we change n to rn in Section 5 then for q j → r (1 ≤ j ≤ r ) we get the multi-index n~ n, n, . . . , n ) and (5.3) becomeslim n →∞ L n~ ~e k ( rnx ) rnL n~ ( rnx ) = z ( x ) − r + 1 r , but we need to extend this for multi-indices containing the parameter 0 < t ≤
1. Forthis we need to use the following asymptotic behavior of the recurrence coefficients: if ~n = ( ⌊ nq ⌋ , ⌊ nq ⌋ , . . . , ⌊ nq r ⌋ ) and ~m = ( ⌊ ntq ⌋ , ⌊ ntq ⌋ , . . . , ⌊ ntq r ⌋ ), thenlim n →∞ a ~m,j n = t q r +1 j Y i = j q j − q i = t a j , ≤ j ≤ r, and lim n →∞ b ~m,j n = t (1 + q j ) = tb j , ≤ j ≤ r. The required asymptotic behavior is then for 0 < t ≤ n →∞ L ~m + ~e k ( nx ) nL ~m ( nx ) = z ( x, t ) − tb k , (6.1)14niformly for x on compact subsets of C \ [0 , ∞ ), where z ( x, t ) satisfies the algebraicequation (cid:0) z ( x, t ) − x (cid:1) B r ( z, t ) + A r − ( z, t ) = 0 , with B r ( z, t ) = Q rj =1 ( z − tb j ) = t r B r ( z/t ) and A r − ( z, t ) B r ( z, t ) = r X j =1 t a j z − tb j , so that A r − ( z, t ) = t r +1 A r − ( z/t ). Here we used A r − ( z ) = A r − ( z,
1) and B r ( z ) = B r ( z, q j → r (1 ≤ j ≤ r ) then b j → r +1 r (1 ≤ j ≤ r ) and the algebraic equation for z ( x, t ) becomes x (cid:18) z ( x, t ) − t r + 1 r (cid:19) r = (cid:0) z ( x, t ) − t (cid:1) r +1 . (6.2)Now change n to rn so that we can deal with the multi-index n~ n, n, . . . , n ). By goingfrom the multi-index n~ n, n, . . . , n ) to ( n + 1) ~ n + 1 , n + 1 , . . . , n + 1) in r steps(each time increasing one coefficient) we then getlim n →∞ L ( ⌊ nt ⌋ +1) ~ ( rnx )( rn ) r L ⌊ nt ⌋ ~ ( rnx ) = (cid:18) z ( x, t ) − t r + 1 r (cid:19) r , so that lim n →∞ rn L ′ n~ ( rnx ) rnL n~ ( rnx ) = 1 r Z ddx (cid:0) z ( x, t ) − t r +1 r (cid:1) r (cid:0) z ( x, t ) − t r +1 r (cid:1) r dt = Z z ′ ( x, t ) z ( x, t ) − t r +1 r dt, uniformly on compact subsets of C \ [0 , ∞ ), where the prime ′ means the derivative withrespect to x . This limit is the Stieltjes transform of the asymptotic zero distribution Z c r /r r u r ( rs ) x − s ds = Z c r u r ( y ) x − y/r dy, and hence we have Z z ′ ( x, t ) z ( x, t ) − t r +1 r dt = Z c r u r ( y ) x − y/r dy. Observe that the change of variables rz = t ˆ z and rx = t ˆ x transforms the algebraicequation (6.2) to (3.1), so that z ( t ˆ x/r, t ) = t ˆ z (ˆ x ) /r . From our analysis in Sections 2–4 wefound that ˆ z ′ ˆ z − r − Z c r w r ( s )ˆ x − s ds, hence z ′ ( x, t ) z ( x, t ) − t r +1 r = rt Z c r w r ( s ) rxt − s ds = Z c r w r ( s ) x − tsr ds. Therefore Z z ′ ( x, t ) z ( x, t ) − t r +1 r dt = Z Z c r w r ( s ) x − tsr ds dt = Z Z tc r w r ( y/t ) x − yr dyt dt = Z c r x − yr Z y/c r w r ( y/t ) dtt dy, ts = y in the second equality and Fubini’s theoremfor the third equality. This means that u r ( y ) = Z y/c r w r ( y/t ) dtt = Z c r y w r ( x ) dxx , (6.3)and hence the asymptotic density of the scaled zeros u r is the Mellin convolution of thedensity w r given in (1.4) and the uniform distribution on [0 , Z c r y n u r ( y ) dy = Z c r x n w r ( x ) dx Z t n dt = 1 n + 1 (cid:18) ( r + 1) nn (cid:19) . We still need to show that the density u r is given by the expression in (1.5). Observe thatthe derivative of (1.5) with respect to ϕ is r + 1 π sin ϕ (sin rϕ ) r (cid:0) sin( r + 1) ϕ (cid:1) r +1 = r + 1 π ˆ x . On the other hand, taking the derivative in (6.3) with respect to ϕ gives du r (ˆ x ) dϕ = − w r (ˆ x )ˆ x ˆ x ′ = r + 1 π ˆ x , where we used (1.4) for the last equality. Thus, using u r ( c r ) = 0, we find that u r (ˆ x ) = 1 rπ (sin rϕ ) r +1 (cid:0) sin( r + 1) ϕ (cid:1) r . There is yet another family of multiple orthogonal polynomials for which the asymptoticdistribution of the zeros is of the same flavor. These are multiple orthogonal polynomialsassociated with Meijer G-functions, which appear in the study of products of Ginibrerandom matrices [11]. The polynomials on the stepline for | ~n | = n are given by P n ( x ) = ( − n r Y j =1 ( n + ν j )! n X k =0 (cid:18) nk (cid:19) ( − x ) k ( k + ν )! · · · ( k + ν r )! , and the asymptotic distribution of the scaled zeros { x k,n /n r , ≤ k ≤ n } is given in [16,Thm. 3.2]. The density is g r ( x ) = 1 π sin ϕ (sin rϕ ) r − (cid:0) sin( r + 1) ϕ (cid:1) r , where again x = (cid:0) sin( r + 1) ϕ (cid:1) r +1 sin ϕ (sin rϕ ) r , < ϕ < πr + 1 . (7.1)This is the density of the Fuss-Catalan distribution, for which the moments are the Fuss-Catalan numbers Z c r x n g r ( x ) dx = 1 rn + 1 (cid:18) ( r + 1) nn (cid:19) , n ∈ N . g r is a Mellin convolution of the density w r in (1.4) and thebeta( r ,
1) density: g r ( y ) = 1 r Z y/c r w r ( y/t ) t /r − dtt = 1 r Z c r y w r ( x ) (cid:16) yx (cid:17) /r − dxx = y /r − r + 1 rπ Z θ dϕx /r , (7.2)where y ( θ ) = (cid:0) sin( r + 1) θ (cid:1) r +1 sin θ (sin rθ ) r . This can most easily be seen from ddϕ (sin ϕ ) /r +1 (cid:0) sin( r + 1) ϕ (cid:1) /r ! = r + 1 r x /r , with x given in (7.1), which enables a straightforward computation of the last integralin (7.2). The case r = 1 corresponds to the asymptotic zero distribution of Laguerrepolynomials (the Marchenko-Pastur distribution (1.6)). The case r = 2 was obtainedearlier in [3] and corresponds to multiple orthogonal polynomials for modified Besselfunctions K ν and K ν +1 . The weight is then explicitly given by g ( x ) = h ( x ), where h ( y ) = 3 √ π (1 + √ − y ) / − (1 − √ − y ) / y / , < y < . Acknowledgements
This research was supported by KU Leuven research grant OT/12/073, FWO researchgrant G.0934.13 and the Belgian Interuniversity Attraction Poles Programme P7/18.Thorsten Neuschel is a Research Associate (charg´e de recherches) of FRS-FNRS (Bel-gian Fund for Scientific Research).
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Ratio asymptotics for multiple orthogonal polynomials , Contempo-rary Mathematics (to appear); arXiv:1408.1829 [math.CA].Walter Van AsscheDepartment of MathematicsKU LeuvenCelestijnenlaan 200 B box 2400BE-3001 LeuvenBelgium [email protected]
Thorsten Neuschelcurrent address:IRMPUniversit´e Catholique de LouvainChemin du Cyclotron 2BE-1348 Louvain-la-NeuveBelgium [email protected]@uclouvain.be