Recurrence coefficients of generalized Meixner polynomials and Painlevé equations

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1 Recurrence coefficients of generalized Meixner polynomials and Painlevé equations Lies Boelen, Galina Filipuk, Walter Van Assche October 0, 010 Abstract We consider a semi-classical version of the Meixner weight depending on two parameters and the associated set of orthogonal polynomials. These polynomials satisfy a three-term recurrence relation. We show that the coefficients appearing in this relation satisfy a discrete Painlevé equation, which is a limiting case of an asymmetric dp IV equation. Moreover, when viewed as functions of one of the parameters, they satisfy one of Chazy s second-degree Painlevé equations, which can be reduced to the fifth Painlevé equation P V. 1 Introduction In this paper we are concerned with the recurrence coefficients of orthogonal polynomials, namely the generalized Meixner polynomials, and show that they satisfy a discrete and a continuous Painlevé equation. The paper is organized as follows. In the introduction we shall first review orthogonal polynomials and their main properties. Next we shall recall the Toda system and Painlevé equations and state our main results. Sections and 3 are devoted to proofs of the main theorems and more technical computations. 1.1 Orthogonal polynomials One of the most important properties of orthogonal polynomials is a three-term recurrence relation. For a sequence (p n ) n N of orthonormal polynomials with respect to a positive measure µ on the real line p n (x)p k (x)dµ(x) = δ n,k, (1) where δ n,k is the Kronecker delta, this relation takes the following form: xp n (x) = a n+1 p n+1 (x) + b n p n (x) + a n p n 1 (x) () 1

2 with the recurrence coefficients given by the following integrals a n = xp n (x)p n 1 (x)dµ(x), b n = xp n(x)dµ(x). Here the integration is over the support S R of the measure µ and it is assumed that p 1 = 0. One can define discrete orthogonal polynomials on an equidistant lattice. The measure is now supported on a discrete set {an + n 0 n A Z} and the integral is a (possibly infinite) sum. Examples are the Charlier and the Meixner polynomials (on the lattice N 0 = {0, 1,,...}) or the Krawtchouk polynomials (a sequence of N +1 polynomials orthogonal on {0, 1,..., N}). The orthogonality condition in case of a discrete measure on N 0 is given by p n (k)p m (k)w(k) = δ m,n. k=0 The recurrence coefficients can also be expressed in terms of determinants containing the moments of the orthogonality measure [9]. For classical orthogonal polynomials (e.g., Hermite, Laguerre, Jacobi) one knows these recurrence coefficients explicitly in contrast to non-classical weights. It is known that the classical orthogonal polynomials have an orthogonal sequence of derivatives []. Another characterization of classical polynomials is a Pearson equation [σ(x)w(x)] = τ(x)w(x), where σ and τ are polynomials satisfying deg σ and deg τ = 1. Semi-classical orthogonal polynomials are defined as orthogonal polynomials for which the weight function satisfies a Pearson equation for which deg σ > or deg τ 1. See Hendriksen and van Rossum [1] and Maroni [7]. The orthonormal polynomials p n (x) = γ n x n + δ n x n are determined by the orthonormality relation (1) and the fact that the leading coefficient γ n is positive. Comparing leading coefficients on both sides of identity () we can express a n as the ratio of the leading coefficients of the polynomials a 0 = 0, a n = γ n 1 γ n, n > 0. (3) The Gram-Schmidt process allows one to construct a sequence of orthonormal polynomials for a positive measure µ for which all the moments x k dµ(x), k N 0,

3 are finite. We will restrict ourselves to such positive measures, thus avoiding all existence problems. In addition, by using the Jacobi matrix, the spectral theorem for orthogonal polynomials (Favard s theorem) answers the inverse problem: a sequence of polynomials satisfying a three-term recurrence relation () with a n > 0 for n > 0 and b n R for n 0 is an orthonormal sequence, for some positive measure on the real line. 1. The Toda system In this section we derive the Toda system for the recurrence coefficients which we shall use later on to derive a differential equation. The Toda system and its relation to orthogonal polynomials are known in the literature [9] and [3,.8, p. 41], but we give a proof to be self-contained. We take the positive measure given by exp(tx)dµ(x) on the real line, where t is a real parameter, and assume that the moments exist for all t R. The coefficients of the orthogonal polynomials now depend on t. Let Q n (x, t) be the monic polynomial of degree n in the variable x, orthogonal to all polynomials of degree less than n with respect to the measure above. We have Q n (x, t)q m (x, t)exp(tx)dµ(x) = δ m,n γ n(t), (4) where integration is over the support of the measure µ on the real line and γ n (t) are the leading coefficients of the corresponding orthonormal polynomials. The monic orthogonal polynomials satisfy the following three-term recurrence relation: xq n (x, t) = Q n+1 (x, t) + b n (t)q n (x, t) + a n(t)q n 1 (x, t), (5) where a n(t) = xq n (x, t)q n 1 (x, t)exp(tx)dµ(x) Q n 1(x, t)exp(tx)dµ(x) = γn 1 (t) xq n (x, t)q n 1 (x, t)exp(tx)dµ(x) and b n (t) = xq n(x, t)exp(tx)dµ(x) Q n(x, t)exp(tx)dµ(x) = γn(t) xq n(x, t)exp(tx)dµ(x). (6) Differentiating (4) for m = n with respect to t gives (γ n) (γn = xq n(x, t)exp(tx)dµ(x) = b n ) γn, 3

4 where is differentiation d/dt with respect to t and Q n (x, t) dq n(x, t) exp(tx)dµ(x) = 0 dt due to the fact that the derivative dq n (x, t)/dt is a polynomial in x of degree n 1 and hence we can use the orthogonality condition. This yields b n = (γ n) γn. Squaring a n in (3) and differentiating with respect to t gives the first equation of the Toda system: d(a n ) = a n dt (b n b n 1 ). The second equation of the Toda system db n dt = a n+1 a n follows from differentiating (6) with respect to t and applying the identities (γ n ) xq n (x, t)exp(tx)dµ(x) = b n, γ n γ n x Q n exp(tx)dµ(x) = γ n xq n (x, t) dq n(x, t) dt ( 1 γ n+1 exp(tx) dµ(x) = γn a n + b n γn + (a n) ) γn 1 = a n+1 + b n + a n, Q n 1 (x, t) dq n(x, t) dt The last two identities follow from (5), and Q n 1 (x, t) dq n(x, t) exp(tx)dµ(x) = a n dt γn 1. exp(tx) dµ(x). Hence, the calculations above can be summarized in the following statement. Proposition 1.1. The recurrence coefficients a n (t) and b n (t) of monic polynomials which are orthogonal with respect to exp(tx) dµ(x) on the real line satisfy the Toda system (a n) = a n(b n b n 1 ) (7) b n = a n+1 a n. (8) The initial conditions a n (0) and b n (0) correspond to the recurrence coefficients of the orthogonal polynomials for the measure µ. 4

5 1.3 Painlevé equations The continuous Painlevé equations were discovered around 1900 by Painlevé and his student Gambier. They were interested in classifying all second order ordinary differential equations of the form w = R(z, w, w ), (9) where R is a rational function in w and w and meromorphic in z, which possess the so-called Painlevé property: the solutions have no movable critical points (or, alternatively, the only movable singularities of the solutions are poles). Painlevé and Gambier proved that up to Möbius transformations, only fifty equations of the form (9) exist which satisfy the Painlevé property ([14, 3, 33]). Forty-four of these equations can either be linearized, transformed to a Riccati equation or solved in terms of elliptic functions. The six remaining equations are now known as the Painlevé equations. For instance, the fifth Painlevé equation (P V ) is given by ( 1 w + 1 w 1 )(w ) w z ( Aw + B ) + Cw w z w (w 1) Dw(w + 1) = + z +, w 1 (10) where w = w(z) and A, B, C, D are arbitrary complex parameters. The six Painlevé equations are often referred to as nonlinear special functions [10], and have numerous applications in mathematics and mathematical physics. A classification problem of second-order second degree Painlevé equations was initiated by Painlevé and Chazy and later on continued by Bureau and Cosgrove (see [11] for a historical overview and the main references). It is known (see e.g., [0, 30, 8, 31]) that the tau-functions associated to the Painlevé equations satisfy second-order second degree equations. In the following we shall need the fourth Chazy equation of system (II) (in the classification of Cosgrove) given by ( d ) v dz 6v α 1 v β 1 ( ( v ) (dv ) ) = z z 4v 3 α 1 v β 1 v γ 1 dz (11) which can be reduced to the fifth Painlevé equation (10) (see [11]). It is known that recurrence coefficients of semi-classical orthogonal polynomials are solutions of nonlinear differential equations with respect to a well-chosen parameter [6]. For instance, the recurrence coefficients in xp n (x) = a n+1 p n+1 (x) + a n p n 1 (x) of the orthogonal polynomials related to the weight exp( x 4 /4 tx ) on R satisfy 4a 3 na n = (3a 4 n + ta n n)(a 4 n + ta n + n) 5

6 and a n(t) satisfies the fourth Painlevé equation with a particular choice of the parameters. It is shown in [6] that other Painlevé equations can be obtained by choosing other weights. For instance, for the generalized Jacobi weight on [ 1, 1] with three factors, w(x) = (1 x) α x β (t x) γ, one can get the sixth Painlevé equation in the variable t. The case exp(x 3 /3 + tx) on {x : x 3 < 0} C is related to the second Painlevé equation. The weight (x t) ρ exp( x ) is shown to be related to the fourth Painlevé equation. For the Hermite weight multiplied by an isolated zero exp( x ) x t K, x, t R, one gets the fourth Painlevé equation [5]. In [6] the weight x α exp( x)exp( s/x), x > 0 for α, s > 0 was used to get the third Painlevé equation. The fifth Painlevé equation is shown to be related to the weights in [3] and to (1 x) α (1 + x) β exp( tx), x ( 1, 1), t R in [8]. The discontinuous weights x α (1 x) β exp t/x, x (0, 1), t 0 x α (1 x) β (A + Bθ(x t)), x [0, 1], where θ is the Heaviside step function, give the sixth Painlevé equation [7]. See also [] for the case which leads to the fifth Painlevé equation. Discrete Painlevé equations (dp) are second-order, nonlinear difference equations which have a continuous Painlevé equation as a continuous limit. They pass an integrability test called singularity confinement [17]. This integrability detector is the discrete analogue of the Painlevé property for differential equations. The discrete Painlevé equations share many features of their continuous counterparts (degeneration cascades, Lax pairs, hierarchies, special solutions, Miura and Bäcklund transformations). However, there are a lot more discrete Painlevé equations than the six continuous equations (e.g., there are various nonequivalent dp I equations). There is a standard list [15] consisting of the earliest derived discrete Painlevé equations. For a comprehensive overview of discrete Painlevé equations, see [16]. For semi-classical weights, the recurrence coefficients obey nonlinear recurrence relations, which in many cases can be identified as discrete Painlevé equations [1, 13, 5, 34]. In this paper we are interested in discrete and continuous Painlevé equations associated with the recurrence coefficients of generalized Meixner polynomials. 1.4 Meixner polynomials and their generalization The Meixner polynomials in the Askey scheme are given by M n (x) = F 1 ( n, x, β; 1 1 ), β > 0, c (0, 1), c 6

7 which are orthogonal on N 0 with respect to the weights w(k) = (β) kc k, k N 0, k! where (β) k = β(β+1)...(β+k 1) is the Pochhammer symbol. The recurrence coefficients for the orthonormal Meixner polynomials are given by a nc(n + β 1) n = (1 c), b n = n + (n + β)c. 1 c We study the sequence (p n ) n N0 of polynomials orthonormal with respect to a semi-classical variation w of the Meixner weight w(k) = (β) kc k (k!), k N 0, β > 0, c > 0. (1) This semi-classical discrete weight is a special case of weights introduced by Ronveaux [18] who considers weights of the form w(k) = q i=1 (β i) k (k!) q µ k. Our case corresponds to q = and µ = c/β, with β. See also the open problem described in [19]. When β = 1, we get classical Charlier polynomials, for which the recurrence coefficients are 1.5 Main results a n = nc, b n = n + c. In this paper we prove the following two results. Theorem 1.1. Consider the discrete orthonormal polynomials with respect to the discrete measure on N 0 with weights (1). The recurrence coefficients a n, b n in the three-term recurrence relation () are given by a n = nc (β 1)x n, b n = n + c + β 1 (c y n ), c where x n and y n satisfy the discrete system (y n c) (x n + y n )(x n+1 + y n ) = (β 1)y n, c (x n + y n )(x n + y n 1 ) = x n (x n + c) x n nc/(β 1), (13) 7

8 with initial values x 0 = 0 and y 0 = c 1 F 1 (β 1; 1; c), 1F 1 (β; 1; c) where 1 F 1 is the confluent hypergeometric function. This result is given in [4] and the proof will be presented in Section, where it is shown that system (13) is a limiting case of an asymmetric dp IV equation. We can also consider the recurrence coefficients a n and b n as functions of the parameter c. In this case they satisfy the Toda system given by ( ) a d ( ) n := a a dc n = n c (b n b n 1 ), b n := d dc b n = 1 (14) c (a n+1 a n ). Here we have used c = e t so that we can use the equations of the Toda system (7) and (8) in Proposition 1.1. Theorem 1.. Let y n = z (v(z) 4β n + 3) 4(1 β) with c = z, then v satisfies Chazy s second degree Painlevé equation (11) with β 1 = (n + 1)(8β + 6n 5), α 1 = 4(1 6n 4β), (15) γ 1 = 4(1 + n) (3 n 4β). The solution of the discrete system in Theorem 1.1 or the Painlevé equation in Theorem 1. can also be given in terms of ratio s of determinants containing confluent hypergeometric functions (see, e.g., Okamoto [8, 31]). This follows because the recurrence coefficients a n and b n can always be written as ratio s of Hankel determinants containing the moments of the orthogonality measure. For instance, one always has (Chihara [9, Thm. 4. on p. 19]) where a n = n n, n 1 µ 0 µ 1 µ µ n µ 1 µ µ 3 µ n+1 n =.... µ n µ n+1 µ n+ µ n with µ n the nth moment of the discrete weight µ n = k=0 k n (β) k (k!) ck, 8

9 which can be expressed in terms of confluent hypergeometric functions. For the b n one has that (b 0 + b b n 1 ) is the coefficient of x n 1 of the monic orthogonal polynomial P n, which by [9, Exer. 3.1 on p. 17] is n/ n 1, where n is obtained by deleting in n the last row and the second last column. These formulas are not so convenient for computing a n and b n when n is large since they require the computation of determinants of high order matrices. Towards a discrete Painlevé equation.1 Ladder operators for discrete orthogonal polynomials on an equidistant lattice We follow the paper by Ismail et al. [4]. We will consider the lattice N 0 = {0, 1,...}, which is the lattice supporting the Charlier and the Meixner polynomials. The forward difference operator is given by f(x) = f(x + 1) f(x). Let us first consider a measure with weights w on N 0 (w( 1) = 0) and the orthonormal polynomials (p n ) n N0 with respect to this measure. The orthonormality condition reads k N 0 p m (k)p n (k)w(k) = δ n,m. Define the potential u as follows: u(x) = w(x 1) w(x), x N 0 (16) w(x) or, using the backward difference operator f(x) = f(x) f(x 1), u(x) = w(x) w(x), x N 0. We can express the action of the difference operator on p n by p n (x) = A n (x)p n 1 (x) B n (x)p n (x) (17) with and A n (x) = a n B n (x) = a n l N 0 p n (l)p n (l 1) l N 0 p n (l)p n 1 (l 1) u(x + 1) u(l) w(l) (18) x + 1 l u(x + 1) u(l) w(l). (19) x + 1 l 9

10 The following relations between the functions A n, B n hold: B n (x) + B n+1 (x) = x b n a n A n (x) u(x + 1) + n A j (x) a j, (0) a n+1 A n+1 (x) a A n 1 (x) n = (x b n )B n+1 (x) (x + 1 b n )B n (x) + 1. (1) a n 1. Proof of Theorem 1.1 We will prove Theorem 1.1 using the technique of ladder operators. The potential u from (16) is for the generalized Meixner weight given by Then u(x) = 1 + u(x + 1) u(l) x + 1 l = w(x 1) w(x) = 1 + The function A n is rational function given by with and x c(β + x 1). l (β 1)(x + 1) + c(β + l 1) c(β + x)(β + l 1). A n (x) = a n c R n + a n x + 1 c β + x T n R n = l p n (l)p n (l 1) β + l 1 w(l) l N 0 T n = (β 1) w(l) p n (l)p n (l 1) β + l 1. l N 0 For B n we have the rational function with and B n (x) = 1 c r n + 1 c x + 1 β + x t n l r n = a n p n (l)p n 1 (l 1) β + l 1 w(l) l N 0 t n = a n (β 1) w(l) p n (l)p n 1 (l 1) β + l 1. l N 0 We then elaborate the first compatibility relation (0) B n+1 + B n = x b n a n A n u(x + 1) + n A j a j. 10

11 With the expressions found above for A n, B n this relation is, after having multiplied by c(β + x), (β + x)(r n + r n+1 ) + (x + 1)(t n + t n+1 ) = (x b n )(R n (β + x) + (x + 1)T n ) +c(β + x) (x + 1) + (β + x) n R j + (x + 1) n T j. When comparing coefficients of powers of x in this polynomial equation, we get three equations βr n + βr n+1 + t n + t n+1 = βb n R n b n T n + cβ 1 + β n R j + n T j, () n n r n +r n+1 +t n +t n+1 = βr n b n R n +T n b n T n +c + R j + T j, (3) The second compatibility relation is (1), 0 = R n + T n 1. (4) a n+1 A n+1 (x) a A n 1 (x) n = (x b n )B n+1 (x) (x + 1 b n )B n (x) + 1. a n 1 This relation again gives three equations βa n+1r n+1 βa nr n 1 + a n+1t n+1 a nt n 1 = βb n r n+1 b n t n+1 β(1 b n )r n (1 b n )t n + cβ (5) a n+1r n+1 a nr n 1 + a n+1t n+1 a nt n 1 = βr n+1 b n r n+1 + t n+1 b n t n+1 βr n (1 b n )r n t n (1 b n )t n + c (6) 0 = r n+1 + t n+1 r n t n. (7) Equation (4) gives R n = 1 T n. From (7) we find, after taking a telescopic sum and bearing in mind that r 0 = t 0 = 0, that r n = t n. We now rewrite the other equations using these new substitutions, which eliminates R n and r n from the problem. For () this gives (β 1)(t n +t n+1 ) = βb n (β 1)b n T n cβ +1 β(n+1)+(β 1) For (3) we have n T j. (8) 0 = β (β 1)T n b n + c + n 1 (9) 11

12 which allows us to express b n in terms of T n and otherwise. We can use this expression for b n in (8) and we get n 1 t n + t n+1 = (β 1)Tn T n (n + c + (β 1)) + β 1 + T j. (30) For (5) we have β(a n+1 a n) (β 1)a n+1t n+1 + (β 1)a nt n 1 = (β 1)b n t n+1 + (β 1)(1 b n )t n + cβ. (31) For (6) we have a n+1 a n = (β 1)t n+1 + (β 1)t n + c. (3) Summing this equation telescopically gives a n = nc (β 1)t n. (33) Note that, combining (33) and (9) we now have explicit expressions for the recurrence coefficients when β = 1, which as noted earlier, corresponds with the Charlier polynomials. Inserting (3) in (31) we obtain a n+1t n+1 + a nt n 1 = (b n + β)t n+1 + (1 b n β)t n. (34) We rewrite this equation in terms of t n and T n only: c(n + 1)T n+1 + nct n 1 = t n+1 (n + c + β 1) t n (n + c + β ) t n+1 (β 1)(T n + T n+1 ) + t n (β 1)(T n 1 + T n ), (35) which can be summed telescopically, resulting in n 1 (nc (β 1)t n )(T n 1 + T n ) = c T j t n (n + c + (β 1)). (36) Another way of dealing with equation (34) is by using T n as an integrating factor: a n+1t n+1 T n a nt n T n 1 = t n T n (t n+1 t n )T n (β + b n ). We can write b n in terms of T n and get a n+1t n+1 T n a nt n T n 1 = t n T n (t n+1 t n )T n (β (β 1)T n + c +n 1). In this equation we replace T n using (30) and we get n n 1 a n+1t n+1 T n a nt n T n 1 = t n+1 t n (β 1)(t n+1 t n ) t n+1 T j +t n T j. 1

13 Taking a telescopic sum we get n 1 a n T nt n 1 = t n t n β + 1 T j. (37) We multiply (36) by T n and use (37): n 1 (nc (β 1)t n )Tn+t n t n β + 1 T j n 1 = ct n T j t n T n (n+c+(β 1)). Using (30) in the right hand side of this equation, we get n 1 nctn = ct n T j + t n t n+1. (38) The sum on the right hand side can be replaced using (30): c(β 1)T n (T n 1) = (t n + ct n )(t n+1 + ct n ). (39) Multiplying (36) by ct n and using (37) and (33) we obtain (nc (β 1)t n ) ( c T n T n 1 + ct n [T n + T n 1 ] ) = c t n (β 1) ct n(n+(β 1)). Rewriting this equation we get (ct n + t n )(ct n 1 + t n ) = (β 1)t n(t n + c). (40) (β 1)t n nc After the substitutions x n = t n en y n = ct n, the equations (39) and (40) reveal the system (y n c) (x n + y n )(x n+1 + y n ) = (β 1)y n c, (x n + y n )(x n + y n 1 ) = which describes the recurrence coefficients through x n (x n + c) x n nc/(β 1), (41) a n = nc (β 1)x n b n = n + c + β 1 (c y n ). c The initial values are given by x 0 = 0 (since a 0 = 0) and y 0 = c ( c + β 1 µ ) 1, β 1 µ 0 (4) 13

14 where µ k = l N 0 l k w(l) is the k-th moment of the generalized Meixner weight. We can express these moments in terms of confluent hypergeometric functions µ 0 = w(l) = 1 F 1 (β; 1; c) l=0 and It follows that µ 1 = lw(l) = βc 1 F 1 (β + 1; ; c). l=1 y 0 = c β 1 (c + β 1) 1 F 1 (β; 1; c) βc 1 F 1 (β + 1; ; c). 1F 1 (β; 1; c) This can be simplified using well-known relations between confluent hypergeometric functions ([1, 13.4, p. 506]). We evaluate the relation (1 + a b) 1 F 1 (a; b; z) a 1 F 1 (a + 1; b; z) + (b 1) 1 F 1 (a; b 1; z) = 0 at a = β, b = and z = c and combine it with the relation b 1 F 1 (a; b; z) b 1 F 1 (a 1; b; z) z 1 F 1 (a; b + 1; z) = 0 evaluated at a = β, b = 1 and z = c, obtaining y 0 = c 1 F 1 (β 1; 1; c) 1F 1 (β; 1; c) as a ratio of a transcendental functions evaluated in contiguous parameters. This proves Theorem 1.1. The system (41) is very similar to a known discrete Painlevé equation, αdp IV, (X n + Y n )(X n+1 + Y n ) = (Y n A)(Y n B)(Y n C)(Y n D) (Y n + Γ Z n )(Y n Γ Z n ) (X n + Y n )(X n + Y n 1 ) = (X n + A)(X n + B)(X n + C)(X n + D) (X n + Z n+1/ )(X n Z n+1/ ) with A + B + C + D = 0. We can actually obtain (41) from αdp IV using the following limiting procedure: set X n = x n 1/ε, Y n = y n +1/ε, A = 1/ε, B = 3/ε c, C = D = c + 1/ε, Z n = z n + 1/ε, Γ = 4c /((β 1)ε), = /ε in (41) and let ε tend to zero. It then suffices to specify the form of z n through z n = c (n 1/). β 1 14

15 3 Towards a continuous Painlevé equation The proof of Theorem 1. is by direct computations. We can express x n+1 and y n 1 from equations (41) in terms of x n and y n : and x n+1 = (β 1)y n(y n c) /c (x n + y n ) y n 1 = y n x n (x n + c) (x n + y n )(x n nc/(β 1)) x n. If we combine this with the Toda system (14), then we find two equations y n (c) = R 1(c, x n, y n ) and x n (c) = R (c, x n, y n ), where R i, i = 1,, are rational functions in their variables. Explicitly, ( x n(c) = cny n + x n c(n + β 1) αyn ) + x n (αc + c(1 + n)y n αyn) c (x n + y n ) (43) and y n (c) = (1 c)cx ny n c x n + y n(αc c(β + c 3)y n + αyn ). (44) c (x n + y n ) We use α = β 1 to make the formulas shorter. One can differentiate the second equation (44), eliminate x n by using the first equation (43) and get an equation for y n, y n, y n, x n. To eliminate x n, one can compute the resultant of this equation and equation (44). As a result, one gets an equation for y = y n given by c 6 (y ) 3c 5 y y + F 1 (c, y)y + F (c, y)y + F 3 (c, y)y + F 4 (c)y 5 + F 5 (c)y 4 + F 6 (c)y 3 + F 7 (c)y + F 8 (c)y + F 9 (c) = 0, (45) 15

16 where F 1 (c, y) = c 4 ( 4αc + y (3 16αc + 1αy) ), F (c, y) = c (c (c n β) + αy) (c(3 + c n β) + αy), F 3 (c, y) = c(y(c (c + (β + n 3)(n + β) + c(8β n 9)) F 4 (c) = 16α 3, +αy(c(6 5c 4n 8β) + 4αy)) 3αc 4 ), F 5 (c) = 4α (c (5c + 4n + 16β 14) 1), ( F 6 (c) = αc 6 4n 8β + c ( 59 44n 168β + 4(c 13c cn + n + 1(n + c)β + 4β ) )), F 7 (c) = c ( (β + n 3)(n + β) 8αc 3 + c (16αn + 8(11 5β)β 47) ) + c(1 8αn + n(8(11 6β)β 4) β(63 + 4β(8β 1))), ( ( )) F 8 (c) = αc 4 3 (1 4n 8β) + 4 c + (n + β) + c (4β n 5), F 9 (c) = 4c 6 α. Remarkably, when applying the linear transformation y n (c) = z (v(z) 4β n + 3), c = z, (46) 4(1 β) we get the Chazy equation (11) for v(z) and, hence, the statement of the theorem. This means that the functions x n (expressed in terms of y n and y n in (44)) and y n are related to the fifth Painlevé transcendent and its derivative after a change of variables. Next we recall, following [11], how the function v (and, hence, y n ) can be expressed in terms of the fifth Painlevé transcendent and its derivative. We use the formula in [11] which gives the connection of the Chazy equation (11) with the fifth Painlevé equation (10). One needs to consider the roots of the cubic equation 4q 3 + α 1 q + β 1 q + γ 1 = 0. In our case we have either q = n + 1 (a double root) or q = 4β + n 3. The root q = n + 1 gives the following parameters for the fifth Painlevé equation (10) A = (β 1), B = (β + n), C = n, D = (47) and the relation of the solution v(z) of (11) and the solution w(t) of (10) with z = t is given by v(z) = tw + (β + n 1)w + (1 4n 4β + 4t)w + (n + β). (48) w(w 1) 16

17 Remark. According to Theorem 40.3 in [0] (see also [10] for a discussion and further references), the fifth Painlevé equation with D = 1/ has a rational solution if and only if for some branch λ 0 = ±1 the parameters satisfy one of the following conditions with k, m Z: (1) A = (λ 0 C + k), B = m, m > 0, k + m is odd and A 0 when k < m; () B = (λ 0 C + k), A = m, m > 0, k + m is odd and B 0 when k < m; (3) B = (A 1 + m), λ 0 C = k, A 1 = A, m 0, k + m is even; (4) 8A = k, 8B = m, λ 0 C Z, k, m > 0 and k and m are both odd. We can scale the independent variable z of the fifth Painlevé equation so that the parameter D is 1/ and the parameter C then becomes C/. We see that either condition (1) or condition () above is satisfied when β N and β. Hence, rational solutions exist when β N and β and these rational solutions are precisely the ones we need for the recurrence coefficients, since for β N and β the initial condition becomes y 0 = c 1 F 1 (β 1; 1; c) 1F 1 (β; 1; c) = c L β ( c) L β 1 ( c), where L n is the Laguerre polynomial of degree n (and parameter α = 0). Here we have used 1 F 1 (a; b; z) = M(a, b, z) = e z M(b a, b, z) and M( n, 1, z) = L n (z) ([1, Chapter 13]). This is a rational function and hence all y n and x n will be rational. The second simple root q = 4β + n 3 gives the following parameters for the fifth Painlevé equation (10) A = 0, B = (n + 1), C = (β + n ), D = (49) and the relation of the solution v(z) of (11) and the solution w(t) of (10) with z = t is given by v(z) = tw + (1 + n)w + (4t 4n 3)w + (n + 1). w(w 1) We can also see that condition (1) of Theorem 40.3 in [0] is fulfilled when β N and β > 1. We can also differentiate equation (43) and eliminate y n, to get a seconddegree, second order equation for x n, which is of similar complexity as (45), but with a cubic polynomial in x n in the coefficient of x n. This equation is c 6 (x ) c 5 x x + G 1 (c, x)x + G (c, x)x + G 3 (c, x)x + G 4 (c)x 5 + G 5 (c)x 4 + G 6 (c)x 3 + G 7 (c)x + G 8 (c)x + G 9 (c) = 0, (50) 17

18 where x = x n (c), α = β 1 and G 1 (c, x) = 16α c x 3 1(n α)αc 3 x + (1 + 4α(α n))c 4 x 4αnc 5, G (c, x) = G 3 (c, x) = G 4 (c) = 3α 3, G 5 (c) = G 6 (c) = c (c(c 1 + α n) + 4αx)(c(1 + c n + α) + 4αx), ( c nαc 3 + x ( c ((c n) 1 + 4(c + n)α) + αx(c(4c n + α) + 4αx) )), 4α c(c 10n + 0α), 4αc((n α)c + α + n(n 1 + nα) (1 + n 1nα + 8α )c), G 7 (c) = c (1 (c n) 4α(c + 3n nc + 4n c n 3 ) G 8 (c) = G 9 (c) = 4n α c α (1 c + 18cn + 5n ) 16α 3 c), 4nαc 3 ((c n) 1 + 4α(c + n)), Let us consider system (13) and denote the first equation by E 1 (n) and the second one by E (n). In order to find a nonlinear relation between y n+1, y n and y n 1, we eliminate x n+1 between E 1 (n) and E (n + 1) by computing a resultant. Next we eliminate x n between this equation and E (n). The resulting expression is very large and we do not write it here. Let us denote it by E 3 for future reference. Next we show that equation E 3 can, in fact be obtained from the Bäcklund transformation of the fifth Painlevé equation. It is known [0, Th. 39.1] that if w = w(z) is the solution the fifth Painlevé equation (10) with parameters A, B, C, D, then we can compute another solution w 1 = w 1 (z) with new values of the parameters A 1, B 1, C 1, D 1, where dzw w 1 = 1 zw aw + (a b + dz)w + b, A 1 = 1 16D (C + d(1 a b)), B 1 = 1 16D (C d(1 a b)), C 1 = d(b a), D 1 = D, a = ε 1 A, b = ε B, d = ε3 D, ε j = 1, j {1,, 3}. We can denote such a transformation by T ε1,ε,ε 3. See also [30] for further description of the Bäcklund transformations and the isomorphism of the group of Bäcklund transformations to the affine Weyl group of A (1) 3 type. Next we consider parameters (47) as parameters (49) can be obtained from (47) by T 1, 1, 1 T 1,1,1 or T 1, 1,1 T 1,1, 1. Indeed, if w = w(z) is the solution of (10) with (47), then w 1 = w (β 1)(w 1) w zw + (1 n + z β + (β 1)w)w + n + β 18

19 is the solution of (10) with (49). Considering the composition of Bäcklund transformations T 1, 1, 1 T 1, 1,1 T 1, 1,1, we obtain the following transformation: if w = w n = w(z) is the solution of (10) with (47), then w n+1 = w n+1 (z) is the solution of (10) with and A = (β 1) (β + n + 1), B =, C = (n + 1), D = 4(n + 1)zw w n+1 = 1 + (β 1)(zw + (z + β 1 + n (β 1)w)w n β) 4z(n + β)w (β 1)(zw + (1 + n + z + (β 1)w)w n β). Similarly, considering the composition of Bäcklund transformations T 1, 1,1 T 1,1,1 T 1,1,1, we get a new solution w n 1 = w n 1 (z) of (10) with and A = (β 1) (β + n 1), B =, C = (n 1), D = 4z(n 1 + β)w w n 1 = 1 + (β 1)(zw (1 + n + z + (β 1)w) + n + β) 4nzw (β 1)(zw (n 1 + z + β (β 1)w)w + n + β). Now, using (46) with (48) for the functions w n, w n±1 and substituting into E 3 found above, we get identically zero. This shows that the discrete system (13) is obtained from the Bäcklund transformation of the fifth Painlevé equation. It is known that the fifth Painlevé equation admits a one-parameter family of solutions expressed in terms of the confluent hypergeometric functions [0]. Let w = w(t) be a solution of (10) with parameters (47) for n = 0 satisfying the following Riccati equation: Then, substituting tw = (β 1)w + (t + β 1)w β. w(t) = t y (t) β 1 y(t), y(t) = tβ Y (t), we get that the function Y = Y (z) satisfies the Kummer equation given by zy + ( z)y βy = 0. So, a particular solution to this equation can be written in terms of the confluent hypergeometric function Y (z) = 1 F 1 (β; ; z). From the well-known formula d 1 F 1 (a; b; z) dz = a b 1 F 1 (a + 1; b + 1; z) 19

20 we get that Y (z) = β 1 F 1 (β + 1; 3; z)/. Hence, using the formulas expressing v(z) and y(c) in terms of the function w(t), we can get that y 0 (c) = c v 0 (t) = w(t) + 8t 1, w(t) 1 c 1F 1 (β; ; c) 1 F 1 (β; ; c) + cβ 1 F 1 (β + 1; 3; c). Finally, using the recurrence relations for the confluent hypergeometric function and (b a) 1 F 1 (a 1; b; z) + (a b + z) 1 F 1 (a; b; z) a 1 F 1 (a + 1; b; z) = 0 we can obtain that b 1 F 1 (a; b; z) b 1 F 1 (a 1; b; z) z 1 F 1 (a; b + 1; z) = 0 y 0 (c) = c 1 F 1 (β 1; 1; c) 1F 1 (β; 1; c) which coincides with the initial value given in Theorem 1.1. It is known [8, 31] that the fifth Painlevé equation with special values of the parameters admits a classical solution written as a ratio of Wronskian determinants of (Kummer s) confluent hypergeometric functions. From the above expression of y 0 and the fact that the discrete system comes from the Bäcklund transformation of the fifth Painlevé equation, the general values (x n, y n ) can also be written explicitly in terms of the confluent hypergeometric functions. 4 Discussion As mentioned above, the generalized Meixner polynomials reduce to the Charlier polynomials when β = 1 and c = a. These are discrete orthogonal polynomials given in hypergeometric form by ( C n (x; a) = F 0 n; x; ; 1 ). a The orthogonality is with respect to the Poisson weight w(k) = ak, a > 0, k = 0, 1,,... k! The recurrence coefficients are given by a n = an, b n = n + a. In [35] Van Assche and Foupouagnigni studied the monic polynomials orthogonal with respect to a semi-classical variation on the Charlier weight, given by w(k) = ak (k!), a > 0, k N 0. (51) 0

21 These polynomials are generalized Charlier polynomials and satisfy a three-term recurrence relation (5). The authors show that the recurrence coefficients can be found as a (transformation of a) solution of dp II, with initial values in terms of the modified Bessel function Namely, I ν (z) = 1 (z/) k+ν k! Γ(k + ν + 1). k=0 { a n = a ( ) 1 x n, b n = n + ax n x n+1, where (x n ) n satisfies the discrete Painlevé II equation x n 1 + x n+1 = nx n a (1 x n ) (5) with initial values x 0 = 1, x 1 = I 1 ( a)/i 0 ( a). Moreover, the coefficients a n(a) and b n (a) satisfy the Toda system (14) (with c replaced now by a). Using the discrete equation (5) we can express x n+1 in terms of x n and x n 1. Substituting into the Toda system, we can express x n 1 in terms of x n and x n x n 1 (a) = nx n + ax n a(1 x n) and substitute into the second equation of the Toda system. This results in the following equation for the function x = x n (a): x = xx x 1 x a + 4a n 8ax + 4ax 4 x, (53) 4a (x 1) where = d/da. Applying a change of variables x n (a) = w(z) + 1 w(z) 1, a = z gives the fifth Painlevé equation (10) for the function w(z) with parameters A = B = n, C = 0, D = 8. 8 It is interesting to note that in [3, 8.3, p. 38] the fifth Painlevé equation with parameters A = B = n, C = 0, D = 8 appeared in relation to orthogonal polynomials on the unit circle. The initial conditions for (5) in [3] are x 0 = 1 and x 1 = I 1 (λ)/i 0 (λ), so that λ = a, which explains the difference in the coefficient D. Note that there is a misprint on [3, p. 38]: the minus sign in B is missing. 1

22 It can be easily seen that applying c = a/β, x n = x n /β in equation (50) and letting β tend to infinity, we get a differential equation for the function x n. Taking the change of variables x n = a(x 1) we see that the function x(a) indeed satisfies equation (53). This is due to the fact that the weights of the generalized Meixner polynomials tends to the weights of the generalized Charlier polynomials (51) in this limit. Acknowledgments We are grateful to the referees for their helpful comments and suggestions which substantially improved the paper. LB and WVA are supported by Belgian Interuniversity Attraction Pole P6/0, FWO grant G and K.U.Leuven Research Grant OT/08/033. Part of this work was carried out while GF was visiting K.U.Leuven for one month. The financial support of K.U.Leuven, MIMUW at the University of Warsaw and the hospitality of the Analysis section at K.U.Leuven is gratefully acknowledged. GF is also partially supported by Polish MNiSzW Grant N N Calculations were partially obtained in the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM), Warsaw University, within grant G We would like to thank Arno Kuijlaars and Lun Zhang for their helpful comments and illuminating discussions. References [1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, [] E. Basor and Y. Chen, Painlevé V and the distribution function of a discontinuous linear statistic in the Laguerre unitary ensembles, J. Phys. A 4 (009), 03503, 18 pp. [3] E. Basor, Estelle, Y. Chen and T. Ehrhardt, Painlevé V and time-dependent Jacobi polynomials, J. Phys. A 43 (010), 01504, 5 pp. [4] L. Boelen, Discrete Painlevé Equations and Orthogonal Polynomials, Ph.D. thesis, K.U.Leuven, 010. [5] Y. Chen and M. V. Feigin, Painlevé IV and degenerate Gaussian unitary ensembles, J. Phys. A: Math. Gen. 39 (006), [6] Y. Chen and A. Its, Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I, J. Approx. Theory 16 (010), [7] Y. Chen and L. Zhang, Painlevé VI and the unitary Jacobi ensembles, Preprint arxiv:

23 [8] Y. Chen and D. Dai, Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials, Preprint arxiv: [9] T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, [10] P. A. Clarkson, Painlevé equations nonlinear special functions, Lecture Notes in Mathematics 1883, Springer, Berlin, 006, pp [11] C. M. Cosgrove, Chazy s second-degree Painlevé equations, J. Phys. A: Math. Gen. 39 (006), [1] A.S. Fokas, A.R. Its and A.V. Kitaev, Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phys. 14 (1991), [13] G. Freud, On the coefficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A 76 (1976), 1 6. [14] B. Gambier, Sur les équations différentielles du second ordre et du premier degré dont l intégrale générale est à points critiques fixés, Acta Math. 33 (1910), [15] B. Grammaticos and A. Ramani, Discrete Painlevé equations: coalescences, limits and degeneracies, Physica A 8 (1996), [16] B. Grammaticos and A. Ramani, Discrete Painlevé equations: a review, Lect. Notes Phys., 644, Springer, 004, pp [17] B. Grammaticos, A. Ramani and V. Papageorgiou, Do integrable mappings have the Painlevé property?, Phys. Rev. Lett. 67 (1991), [18] A. Ronveaux, Discrete semi-classical orthogonal polynomials: Generalized Meixner, J. Approx. Theory 46 (1986), [19] A. Ronveaux, Asymptotics for recurrence coefficients of the generalized Meixner case, J. Comput. Appl. Math. 133 (001), [0] V.I. Gromak, I. Laine and S. Shimomura, Painlevé Differential Equations in the Complex Plane, Vol.8, Studies in Mathematics, de Gruyter, Berlin, NewYork, 00. [1] E. Hendriksen and H. van Rossum, Semi-classical orthogonal polynomials, in Polynômes Orthogonaux et Applications, Lecture Notes in Mathematics 1171, Springer-Verlag, Berlin, 1985, pp [] W. Hahn, Über die Jacobischen Polynome und zwei verwante Polynomklassen, Math. Zeit. 39 (1935), [3] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications 98, Cambridge University Press,

24 [4] M. E. H. Ismail, I. Nikonova and P. Simeonov, Difference equations and discriminants for discrete orthogonal polynomials, Ramanujan J. 8 (004), [5] A.P. Magnus, Freud s equations for orthogonal polynomials as discrete Painlevé equations, in Symmetries and Integrability of Difference Equations (Canterbury, 1996), London Math. Soc. Lecture Note Ser., 55, Cambridge University Press, 1999, pp [6] A.P. Magnus, Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, J. Comp. Appl. Math. 57 (1995), [7] P. Maroni, Prolégomènes à l étude des polynômes orthogonaux semiclassiques, Ann. Mat. Pura Appl. (4) 149 (1987), [8] T. Masuda, Classical transcendental solutions of the Painlevé equations and their degeneration, Tohoku Math. J. 56 (004), [9] J. Moser, Finitely many mass points on the line under the influence of an exponential potential an integrable system, Lecture Notes in Physics 38, Springer, Berlin, 1975, pp [30] M. Noumi, Painlevé equations through symmetry, Translations of Mathematical Monographs, 3, American Mathematical Society, Providence, RI, 004. [31] K. Okamoto, Studies on the Painlevé equations. II. Fifth Painlevé equation P V, Japan. J. Math. (N.S.) 13 (1987), [3] P. Painlevé, Mémoire sur les équations différentielles dont l intégrale générale est uniforme, Bull. Soc. Math. Phys. France 8 (1900), [33] P. Painlevé, Sur les équations différentielles du second ordre et d ordre supérieure dont l intégrale générale est uniforme, Acta Math. 1 (190), [34] W. Van Assche, Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials, in Difference Equations, Special Functions and Orthogonal Polynomials (S. Elaydi et al., eds.), World Scientific, 007, pp [35] W. Van Assche and M. Foupouagnigni, Analysis of non-linear recurrence relations for the recurrence coefficients of generalized Charlier polynomials, J. Nonlinear Math. Phys. 10 Supplement (003), Lies Boelen Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 00B box 400 4

25 BE-3001 Leuven Belgium Galina Filipuk Faculty of Mathematics, Informatics and Mechanics University of Warsaw Banacha Warsaw Poland Walter Van Assche Department of Mathematics Katholieke Universiteit Leuven Celestijnenlaan 00B box 400 BE-3001 Leuven Belgium 5