Plancherel Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems

Transcription

1 Constr Approx : DOI.7/s Plancherel Rotach Asymptotic Expansion for Some Polynomials from Indeterminate Moment Problems Dan Dai Mourad E.H. Ismail Xiang-Sheng Wang Received: 7 February / Accepted: 8 June / Published online: October Springer Science+Business Media New York Abstract We study the Plancherel Rotach asymptotics of four families of orthogonal polynomials: the Chen Ismail polynomials, the Berg Letessier Valent polynomials, and the Conrad Flajolet polynomials I and II. All these polynomials arise in indeterminate moment problems, and three of them are birth and death process polynomials with cubic or quartic rates. We employ a difference equation asymptotic technique due to Z. Wang and R. Wong. Our analysis leads to a conjecture about large degree behavior of polynomials orthogonal with respect to solutions of indeterminate moment problems. Keywords Asymptotics The Chen Ismail polynomials The Berg Letessier Valent polynomials The Conrad Flajolet polynomials Turning points Difference equation technique Indeterminate moment problems Nevanlinna functions Asymptotics of zeros Plancherel Rotach asymptotics Mathematics Subject Classification Primary A C7 Secondary E5 Communicated by Tom H. Koornwinder. D. Dai Department of Mathematics, City University of Hong Kong, Hong Kong, Hong Kong SAR dandai@cityu.edu.hk M.E.H. Ismail Department of Mathematics, University of Central Florida, Orlando, FL 8, USA mourad.eh.ismail@gmail.com M.E.H. Ismail King Saud University, Riyadh, Saudi Arabia X.-S. Wang B Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO 7, USA xswang@semo.edu

2 Constr Approx : Introduction The Plancherel Rotach asymptotics for orthogonal polynomials refer to asymptotics of the orthogonal polynomials scaled by the largest zero. There are three types of Plancherel Rotach asymptotics depending on whether we are in the oscillatory region between the largest and smallest zeros, in the exponential region beyond the largest or smallest zero, or in the transition region near the largest or smallest zero. If the extreme zero moves as the polynomial degree increases, the last type of asymptotics is related to the soft edge asymptotics in the random matrix theory and describes the eigenvalues of large random Hermitian matrices near the tail end, see ]. The Plancherel Rotach asymptotics for Hermite and Laguerre polynomials are givenin] and ]. The corresponding asymptotics for the exponential weights were developed by using the Deift Zhou steepest descent analysis for corresponding Riemann Hilbert problems in 8] and 5]. Every birth and death process leads to a family of orthogonal polynomials. The birth rates {λ n } and the death rates {μ n } are assumed to satisfy the conditions that λ n and μ n+ are positive and μ. The recurrence relation of the birth and death process polynomials {Q n x} is xq n x = λ n Q n+ x + μ n Q n x λ n + μ n Q n x, n >,. with the initial conditions Q x =, Q x = λ + μ x/λ. It is known in the literature that some classical continuous and discrete orthogonal polynomials can be written as birth and death process polynomials with certain special rates. For example, one can find the birth and death rates for Laguerre polynomials, Jacobi polynomials, and Charlier polynomials in, Sect. 5.]. Of course, for these classical polynomials, their weight functions are well-known and unique. However, for quite a few choices of λ n and μ n, the corresponding moment problem is indeterminate. This means that the orthogonality measures of such polynomials are not unique. For example, Letessier and Valent 8] introduced polynomials associated with quartic rates λ n = n + n + n +, μ n = n n n +.. These polynomials were also studied by Berg and Valent 5], and we refer to them as Berg Letessier Valent polynomials. Van Fossen Conrad and Flajolet 5, ] considered the following two families with cubic rates: λ n = n + c + n + c +, μ n = n + c n + c +, c >,. λ n = n + c + n + c +, μ n = n + c n + c, c>;.

3 Constr Approx : seealso9]. The moment problems associated with the above rates are all indeterminate. For the Conrad Flajolet polynomials, this will be proved in Sect.. In fact, the orthogonality measures of the above polynomials are still unknown in the literature. Chen and Ismail 7] considered the orthogonal polynomials arising from the following recurrence relation: F n+ x = xf n x n n F n x,.5 with F x = and F x = x. They indicated that these polynomials are also related to the indeterminate moment problems. Moreover, they found the following family of weight functions for F n x: α w α x = cosρ x/ + coshρ x/] {cosρ x/ + coshρ x/} α sin ρ x/ sinh ρ x/], x R,. where α,, ρ is a constant defined in.5 below and is defined as i; cf. 7, Eq. 5.]. Note that the weight function. has a singularity at, which is similar to the Freud weight e x β. One may want to use the Riemann Hilbert method to obtain asymptotic formulas for the Chen Ismail polynomials, as has been done by Kriecherbauer and McLaughlin 5] for Freud polynomials. However, since the orthogonality measures are usually unknown or too complicated cf. Chen Ismail polynomials for orthogonal polynomials related to the indeterminate moment problems, it is difficult to apply the powerful Deift Zhou steepest descent analysis for Riemann Hilbert problems. In this paper, we intend to use the difference equation method developed by Wong and his colleagues in 8 ] to get the asymptotic expansions. This will be the first time to apply this technique to derive Plancherel Rotach asymptotics for orthogonal polynomials corresponding to indeterminate moment problems. It must be emphasized that we only use the recurrence relation or difference equations to achieve our asymptotic expansion in this paper. This is a significant improvement compared with the previous works, where some extra asymptotic information is needed to determine the asymptotic expansion; for example, see 7, 9, ]. Because we are dealing with polynomials orthogonal with respect to infinitely many measures, it is important to use a technique which only depends on the moments but not on the specific orthogonality measure used. Before we proceed to the next section, we would like to mention that one useful tool in estimating the largest and smallest zeros of orthogonal polynomials is the following theorem of Ismail and Li ]. We did not consider any q-orthogonal polynomials in this work. Ismail and Li ] considered the Plancherel Rotach asymptotics for symmetric q-polynomials when the recursion coefficients grow exponentially. We hope to treat these types of polynomials in a future work using difference equation techniques. Theorem. Let {P n x} be a sequence of monic polynomials satisfying xp n x = P n+ x + α n P n x + β n P n x,.7

4 Constr Approx : with β n >, for n<n, and let {c n } be a chain sequence. Set B := max{x n : <n<n}, and A := min{y n : <n<n},.8 where x n and y n, x n y n, are the roots of the equation x α n x α n c n = β n,.9 that is, x n, y n = α n + α n ± α n α n + β n /c n.. Then the zeros of P N x lie in A, B. In practice, one usually chooses the simple chain sequence c n / for all n in the above theorem. The present paper is organized as follows. In Sect., we study the Chen Ismail polynomials. After applying results in Wang and Wong ] to obtain two linearly independent solutions for the difference equation.5, we develop ideas used in Wang and Wong 8] to determine the coefficients of these two solutions. Then the Plancherel Rotach asymptotic expansion is obtained, which is given in terms of Airy functions. Moreover, we study the limiting zero distribution as well as the behavior of the largest zero. In the next three sections, we follow procedures similar to Sect. and derive asymptotic expansions for Berg Letessier Valent polynomials and Conrad Flajolet polynomials, respectively. Since Bessel-type asymptotic expansions appear in these cases, we need to apply the recent results by Cao and Li ] to get the two linearly independent solutions of our difference equations. As Sects. 5 are independent of each other, we shall use the same notation for the functions and variables but it will mean different things in different sections. We hope that this will not cause any confusion. In the last section of this paper, we list some remarks about the moment problem and formulate a conjecture about large degree Plancherel Rotach behavior of orthogonal polynomials associated with indeterminate moment problems. In this section, we also apply asymptotic results to show that the moment problem associated with the Conrad Flajolet polynomials is indeterminate. We used the term Conrad Flajolet polynomials because the proper name Van Fossen Conrad Flajolet polynomials is just too long, and we are sure that Eric Van Fossen Conrad will not mind. Chen Ismail Polynomials Before we derive the Plancherel Rotach asymptotics for F n x, one can easily get a bound for the largest and smallest zeros from Theorem.. Proposition. Let x n,k be zeros of F n x such that x n, >x n, > >x n,n. Then we have the following bounds for all n : x n, < 8n n and x n,n > 8n n.

5 Constr Approx : 5 Proof Recall the recurrence relation of F n x in.5, and choose c n = / in Theorem.. Then the result follows.. Difference Equation Method To derive the Plancherel Rotach asymptotics of F n x, we start from the recurrence relation in.5 and apply Wang and Wong s difference equation method developed in 9, ]. To use their results, we transform the recurrence relation.5 into the following standard form they need: p n+ x A n x + B n p n x + p n x =.. Let n + K n := n Γ Γ and F n x = K n p n x. Then.5 becomes n + / Γ n + / K n+ p n+ x = xk n p n x n n K n p n x.,. Since K n+ = n n K n, the above recurrence equation reduces to the standard form. with A n = K n K n+ and B n =. As n, the recurrence coefficient A n satisfies the following expansion: with θ = and A n n θ α s n s,. α =, α =.. Since A n is of order On when n is large, to balance the term A n x in., we introduce x = ν t and ν = n + τ, where τ is a constant to be determined. Then the characteristic equation for. is: λ α tλ+ =, with α given in.. The roots of this equation are λ = t t ± ], and they coincide when the quantity inside the above square root vanishes, that is, t = t ± =±8. These points t ± are called transition points for difference equations by Wang and Wong in 9, ] because in their neighborhood, the behaviors of solutions to. change dramatically.

6 Constr Approx : Since the polynomials p n x and p n+ x are even and odd functions, respectively, let us consider the asymptotics near the large transition point t + = 8 only. According to the main theorem in 9, p. 89], we have the following proposition. Proposition. When n is large, p n x in. can be expressed as p n x = C xp n x + C xq n x, where C x and C x are two n-independent functions. In the above formula, P n x and Q n x are two linearly independent solutions of. satisfying the following Airy-type asymptotic expansions in the neighborhood of t + = 8: P n ν t Ut t Ai ν Ã s U Ut + Ai ν ] B s U Ut, + and Q n ν t Ut t Bi ν Ã s U Ut + Bi ν Ut where ν = n + and Ut is defined as ] Ut t π = F, F arcsin ] 8 t, ] B s U, + log t + t, 8 t 8,.5 ] Ut = cos t t 8 8 B t, 8 <t<8.. Here Fϕ,k is the elliptic integral of the first kind, F ϕ,k ϕ dθ =,.7 k sin θ B x a, b is the incomplete Beta function B x a, b = x and the leading coefficients are given by y a y b dy,.8 Ã U =, B U =. Proof Recall the asymptotic expansion for A n in.; Eq.. falls into the case θ and t + considered in 9]. Following their approach, we choose τ = α t + θ = and ν = n + τ. Then our proposition follows from the main theorem in 9].

7 Constr Approx : 7 Remark. Note that the terms involving the elliptic integral Fϕ,k and the incomplete Beta function B x a, b in.5 and. can be written as ] π 8 t F, F arcsin t, = 8 t t 8 B t, 8 = t t/s s ds. t/s s ds, Moreover, one can verify that Ut defined in.5 and. is a monotonically increasing function in the neighborhood of 8. Also, we have the following asymptotic formula: Ut= t Ot 8 as t Determination of C x and C x In this subsection, we will determine the coefficients C x and C x in Proposition. via a matching method. To this end, we shall derive the asymptotic formulas of F n x or, p n x in the exponential region and oscillatory region, respectively. In the following lemma, we provide the asymptotic formula in the exponential region of the solution to a general class of difference equations. A similar result was obtained by Van Assche and Geronimo in ]. Here, we adopt the approach developed by Wang and Wong in 8]. Lemma. Let π n x be monic polynomials defined from the following recurrence relation: π n+ x = x a n π n x b n π n x, with initial conditions π x = and π x = x a. Here the constants a n and b n are assumed to be polynomials in n and have the following asymptotic behaviors as n : a n = an p + αn p + O n p, b n = b n p + βn p + O n p, where a and b are not identically zero and p> is a positive integer. Let I be the smallest convex and closed interval which contains, a b, and a + b, namely, I =a b,a + b] if a b< <a+ b; I =,a+ b] if a b ; I =a b,] if a + b. Note that b but a could be negative. So, it is possible although rarely that a + b. Rescale the variable x by x = x n := n + σ p y with y C \ I. Then we have the following asymptotic formula for

8 8 Constr Approx : π n x n : n p n y a + y a π n x n b ] / y { exp n log y ar p y + ar p } b r p] dr Proof Let exp exp exp a y ar b r dr + pσy y as p b s ds p We obtain w x = x a and b ] r + ay ar y ar b r ] dr α p y ar b r dr βr p y ar b r y ar+ y ar b r ] dr π n x = n w k x. k= ] ].. w k+ x = x a k b k w k x. for k. Since x = x n = n + σ p y, we have from successive approximation that for large n, where ε k = = w k x n = x n a k + x n a k b k + ε k,. a k+ a k + b k+ b k + x n a k a k+ a k x n a k b k x n a k + O n b k ] pak p yn p ak p b k + pb k p + pak p yn p ak p p yn p ak p b k p + O n ] uniformly for k =,,...,n. Here we have made use of the facts that a k and b k are polynomials in k, and k/n = O uniformly in k =,,...,n. The leading term in. is derived from solving the characteristic equation for.. The first-order term of ε k can be obtained at least formally from a standard successive approximation. A rigorous proof of the above asymptotic formula can be given via induction. On the other hand, noting that x n yn p + ξn p with ξ = pσy, we

9 Constr Approx : 9 obtain x n a k + x n a k b k = yn p + ξn p ak p αk p + yn p + ξn p ak p αk p b k p βk p + O n p. A simple calculation yields where ˆε k = x n a k + x n a k b k yn p ak p + yn p ak p b k p = +ˆε k, ξn p αk p yn p ak p b k p βk p yn p ak p b k p yn p ak p + yn p ak p b k p ] + O n uniformly for k =,,...,n. By the trapezoidal rule, we have the following three asymptotic formulas: and k= and n k= n k= log yn p ak p yn + p ak p b k p] np log n + n log y as p y + as p b s p] ds + y a + y a log b, y n pas p ε k y as p b s + pb s p + pas p y as p p y as p b s p ds ] a = y ar b r + b r + ay ar y ar b r ] dr, ˆε k ξ αs p ds y as p b sp βs p y as p b s p y as p + y as p b s p ] ds

10 7 Constr Approx : ξ = y as p b s ds α p p y ar b r dr βr p y ar b r y ar + y ar b r ] dr. Combining the above three asymptotic formulas yields the desired formula. A direct application of the above lemma yields the following result. Proposition. As n, we have F n 8n + / y n n exp{ n log y + y e + n + } yf arcsin y, y + y /. y for complex y bounded away from the interval, ], where Fϕ,k is the elliptic integral of the first kind in.7. Proof Choose a = α =, b =, β =, p =, σ = /, and replace y by 8y. We obtain from.: F n x n n exp n y + y y log ] y + y t y dt + y t dt /. Note that log y + y t dt = t log y + y t + t y t y + y t dt = log y + y + y / y = log y + y + y s s + s ds / y s ds = log y + y + yf arcsin, y

11 Constr Approx : 7 and y y t dt = / y y ds = yf arcsin,. s y Combining the above three equations yields.. Once we have the asymptotic formula for F n x outside the interval where the zeros are located, using an argument similar to that in the proof of 8, Theorem.], we obtain the following result in the oscillatory region. The main idea of this argument is to match the asymptotic formula in the outer region with that in the oscillatory region. Note that the second-order recurrence relation. has two linearly independent solutions in the oscillatory region. The unique solution with given initial conditions can be written in terms of a linear combination of these two solutions, while the coefficients can be determined via asymptotic matching. Due to the symmetry of F n x, we consider positive x only. Proposition. Let δ> be any fixed small number. As n, we have F n 8n + / cos θ cos θ n n e n en+ ρ sin θ + sin n + θ cos n + cos θ θ B sin θ, ] cos θ B sin θ,. for θ δ, π δ], where B xa, b is the incomplete Beta function in.8 and the constant ρ is π ρ = ds = F s, = πγ 5 Γ..5 Finally, we obtain our theorem. Theorem. Let ν = n +. With K n and Ut defined in.,.5, and., respectively, we have F n ν t K n Γ 5 exp πt Ut π / Γ ν t Ai ν Ã s U Ut + Ai ν Ut uniformly for t in compact subsets of,. ] B s U +.

12 7 Constr Approx : Proof Let y = n+ t. Then 8n y = ν t = x. Because 8n n y we have from. and., y u u du n + 8 t t/s s ds p n x = F n x K n nn e n 8e n+ ρ t/8 K n t cos n + cos t8 n + 8 t + sin n + cos t8 n + 8 t Recall the asymptotic expansions for Airy function, as x, then, we have when t<8, Ai x πx / cos x π Bi x πx / sin x π p n x = C xp n x + C xq n x ν C x t cos Ut π π ν C x t sin Ut π π = C x t cos ν cos t8 8 π t C x t sin ν cos t8 8 π Comparing.7 and., we have Γ 5 C x = exp π / Γ πx t t, t/s ds s t/s s ds ]..7,.8,.9 t/s ds π s t t/s ds π. s., C x =. This completes the proof of the theorem.

13 Constr Approx : 7 Remark. Recall the weight functions for F n x given in.. Let α =. Then we have the weight function wx = cosρ x/ + coshρ, x/] x R.. When x is a large number as in., that is, x = ν t. Then we have wx ρ x Γ 5 exp exp πt Γ ν. So the main formula in the above theorem can be stated as wx K n Ut Fn x = π / t Ai ν Ã s U Ut + Ai ν ] B s U Ut.. + Remark. In fact, the three-term recurrence relation contains a lot of useful information. It can also give us the asymptotic zero distribution of the rescaled polynomials F n 8n x as n. Using the method developed in Kuijlaars and Van Assche ], one can obtain the following limiting zero distribution for F n 8n x from the recurrence relation.5: π ds, x, ]. x s x Since the above theorem is uniformly valid in the neighborhood of the large transition point 8, we have the following asymptotic formula for F n x. Corollary. Let ν = n+, K n, and wx be given in. and., respectively. Uniformly for a bounded real number s, we have, for x = 8ν + 8 sν, wx Fn x = K nν Ais + O ν ] 7/ π / as ν.. Proof Let t = sν in., and recall the asymptotic formula for Ut in.9. We obtain after some computations wx K n Fn x = / + O ν ν Ais + O ν ] π /. This above formula proves our corollary. Remark. We may rewrite the polynomial on the left-hand side of. intoan orthonormal one. Note that if P n x satisfies a three-term recurrence relation in.7,

14 7 Constr Approx : then P n x := n k= β k P n x is the corresponding orthonormal polynomial and satisfies the following recurrence relation: x P n x = β n+ P n+ x + α n P n x + β n P n x. Let F n x be the orthonormal Chen Ismail polynomials. Then. can be rewritten as wx F n x = ν 5 Ais + O ν ] 7/ as ν.. From the above corollary, an asymptotic approximation of extreme zeros of F n x can be easily obtained by using the zeros of the Airy function Aix. To get it, we need the following result of Hethcote ]. Lemma. In the interval a ρ,a + ρ], suppose that ft= gt + εt, where ftis continuous, gt is differentiable, ga =, m = min g t >, and Et = max εt < min ga ρ, ga + ρ. Then there exists a zero c of ftin the interval such that c a E m. We have the following approximation. Corollary. Let x n,k be the zeros of F n x such that x n, >x n, > >x n,n, and a k be the zeros of the Airy function Aix in descending order. Then we have for fixed k and large n, where ν = n +. x n,k = 8ν + 8 a k ν / + O ν /, Proof A combination of Corollary. and Lemma. immediately gives us the result. Berg Letessier Valent Polynomials Now we are going to study some birth and death process polynomials Q n x. Before we derive the Plancherel Rotach asymptotics for Berg Letessier Valent polynomials Q n x with the rates in., we can use the chain sequence method to get the bounds for the largest and smallest zeros as we did in Proposition.. Proposition. Let x n,k be zeros of Q n x such that x n, >x n, > >x n,n. Then we have the following bounds for all n : x n, < n n + 5 n and x n,n >.9.

15 Constr Approx : 75 Proof Recall the recurrence relation of Q n x in. and the recurrence coefficients in.. We choose c n = / in Theorem.. Note that the solutions x n and y n in. are monotonically increasing and decreasing with n, respectively. Then the result follows.. Difference Equation Method Like what we have done in the previous section, we introduce Q n x = n K n p n x, with K n := Γn+ n+ Γ. Γ n+ Γ n+ 7 to arrive at the standard form. Note that K n+ n n = K n n + n +. Then the recurrence relation. with. reduces to the following standard form: p n+ x A n x + B n p n x + p n x =.. As n, the recurrence coefficients A n and B n satisfy the following expansions: A n = K n λ n K n+ n θ α s n s, B n = λ n + μ n K n λ n K n+ β s n s,. with θ =, and α = 5, α = 5,. β =, β = β =..5 Let us introduce x = n + / t. Then the characteristic equation for. is λ α t + β λ + =,. with α and β given in. and.5, respectively. The roots of this equation coincide when α t ± + β =±, which gives us two transition points, t + =, t =. Near the large transition point t +, we get the Airy-type asymptotic expansion as in the previous section. But since the small transition point t is located at the origin, we obtain the Bessel-type not Airy-type expansion in its neighborhood. This is similar to the case of Laguerre polynomials, in which case Bessel asymptotics are obtained

16 7 Constr Approx : near the origin. Although we do not have the weight functions for Berg Letessier Valent polynomials, we guess they are supported on R +. For a detailed explanation why the case t = is so special to give us the Bessel type asymptotics, one may refer to discussions in ]. We have the following two different types of expansions. Proposition. When n is large, p n x in. can be expressed as p n x = C xp n x + C xq n x, where C x and C x are two n-independent functions, and P n x and Q n x are two linearly independent solutions of. satisfying the following Airy-type asymptotic expansions in the neighborhood of t + = : P n ν t Ut 9/ tt Ai ν à s U Ut +Ai ν ] B s U Ut, +.7 and Q n ν t Ut 9/ tt Bi ν à s U Ut +Bi ν ] B s U Ut. +.8 Here ν = n +, the leading coefficients are given by à U =, B U =, and Ut is defined as ] Ut t = B 8 t, log t 9 + tt 9, t,.9 ] Ut t = cos 9 t B 8 t, <t< ;. 9 see.8 for the definition of B x a, b. Proof With τ = α t + + β β θ = and ν = n + τ, our results follow from the main Theorem in 9]. Proposition. When n is large, p n x in. can be expressed as p n x = C x n P n x + C x n Q n x,

17 Constr Approx : 77 where C x and C x are two n-independent functions, and P n x and Q n x are two linearly independent solutions of. satisfying the following Bessel-type asymptotic expansions in the neighborhood of t = : Pn ν t 9/ ν U t t J νu t à s U t + J νu t ] B s U,. and Q n ν t 9/ ν U t t W νu t à s U t + W νu t ] B s U.. Here ν = n +, W α x := Y α x ij α x,. the leading coefficients are given by and U t is defined as U t ] = U t ] = t t 8 à U =, B U =, t/s ss ds log 9 t + tt 9, t,. B t see.8 for the definition of B x a, b., cos 9 t 9 <t< ;.5 Proof With the asymptotic expansions for A n and B n in., our results follow from the main theorem in ]. Remark. Note that the integral in. can be written as a hypergeometric function as follows: t t/s t ss ds = F,, 5.,t The two functions Utand U t are both monotonically increasing functions in the neighborhood of and, respectively. In fact, from their definitions in the above

18 78 Constr Approx : two propositions, we have the following asymptotic formulas: and Ut= t + O t, as t,. U t = t 8 + O t as t.. Determination of C x and C x For convenience, we state a special case of Lemma. which will be used in determining C x and C x for all three types of birth and death polynomials considered in this paper. Lemma. Assume π n x satisfies the following recurrence relation: π n+ x = x λ n + μ n ] π n x λ n μ n π n x, with π x = and π x = x λ + μ. Here, λ n and μ n areassumedtobe polynomials in n and satisfy the following asymptotic formulas as n : λ n = b n p + un p + O n p, μ n = b n p + vn p + O n p, where b> and p>. Rescale the variable x by x = x n := n + σ p y with y C \, b]. Then we obtain the following asymptotic formula: yn p n + b/y π n x n e p b/y / { exp ] n+ u+v p n + σp bs p /y ds }..7 Proof We will show that.7 is a special case of. with a = b, α = bu + v, and β = b u + v p. Due to the fact that a = b, we have following formulas for the integrals in.: and and a y ar b r dr = a y ayr dr = a/y, b r + ay ar y ar b r ] dr = ay y ayr] dr = log y y a, α p α a/y dr =, y ar b r pa

19 Constr Approx : 79 and βr p y ar b r y ar + y ar b r ] dr ] a/y log. = β pa Therefore, we have from., + a/y n p n + a/y π n x n a/y / { exp n log y as p + Furthermore, since ] + β pa y ays p] ds + exp α pa + β ] a/y pa. y as p + y ays p = y + y as p /, we have y log as p ] + y ays p ds = log + } pσy y ays ds p log y + y as p ] ds. According to integration by parts, the right-hand side becomes log + log ] y y as p y + y a + p ds y as p Therefore, we obtain = log y + y a p y p + y as p ds. yn p n + a/y π n x n e p a/y / { n + σp exp as p /y ds + α pa + β pa ] n++ β pa a/y }. Note that α pa + β pa = u + v p + u + v p =. p The asymptotic formula.7 forπ n x n follows.

20 8 Constr Approx : Since the birth and death process polynomials Q n x defined in. are related to the monic one as follows, Q n x = n n k= λ k ] π n x,.8 we have the following results for Berg Letessier Valent polynomials from the above lemma. Proposition. Let x = x n := n + / t with t C \, ]. We have as n, p n x n 8n 5/ n + t n /t / + /t { n + / exp /t / F arcsin /t } /,. ] n+/ Here, the function F is the elliptic integral of the first kind defined in.7. Proof Recall the definition of K n in.. We have K n = Γn+ n+ Γ Γ n+ Γ n+ 7 n +, which yields Q n x = n K n p n x n p n+ n x. Moreover, by applying Stirling s formula, we obtain n k= It is readily seen that λ k = 8n Γ/ + nγ / + n Γ/ + n Γ/Γ / Γ/ 8n Γn n Γ/Γ / Γ/ 8n π n/e n Γ/Γ / Γ/ 8n n/e n. n π n x = n λ k Q n x 8n+/ n/e n n + p n x. k= Since π n x satisfies the asymptotic formula in.7 with σ = /, p =, b = 8, u =, and v =, our result follows from the above formula.

21 Constr Approx : 8 Again, using an argument similar to that in the proof of 8, Theorem.], we obtain the following result in the oscillatory region. Proposition.5 Let x = x n := n + / t with t = cos θ,θ δ, π δ]. We have as n, p n x n n + t { π cos n + /θ + exp n + ] t 5/ ρ n + t t/s / s s ds },.9 where ρ = du u = u u πγ Γ. Then we have a result in the interval containing t + =. Theorem. Let ν = n +. With K n and Ut defined in.,.9, and., respectively, we have Q n ν t n πγ K n πνt exp Ut 5/ Γ νt tt Ai ν Ã s U Ut + Ai ν ] B s U Ut +. uniformly for t in compact subsets of,. Proof When t<, recall the asymptotic expansions for the Airy function in.8 and.9 again. We have from.7 and.8, p n x = C xp n x + C xq n x 9/ ν C x cos Ut π t t] π 9/ ν C x sin Ut π t t] π 9/ t = C x cos ν cos 9 t t] π 9 t/s t s s ds π

22 8 Constr Approx : 9/ sin ν π C x t t] t/s t s s ds π. t cos 9 Comparing the above formula and.9, we have πx C x = exp x πγ 5/ Γ, C x =. 9 This completes the proof of the theorem. Remark. As in Remark., we can obtain the following limiting zero distribution for Q n n x from the recurrence relation.: ds, x, ]. π x / xs x Since the above theorem is uniformly valid in the neighborhood of the large transition point, we have the following asymptotic formula for Q n x. Corollary. Let ν = n + and K n be given in.. Uniformly for a bounded real number s, we have, for x = ν + sν, x πγ exp 5/ Γ x Q n x = n K n πν Ais+O ν ] as ν.. Proof Let t = + sν in., and recall the asymptotic formula of Ut in.. We have after some computations x πγ exp 5/ Γ x Q n x = n K n π ν Ais + O ν ]. This above formula proves our corollary. Remark. We can put the left-hand side of. into its orthonormal form as in Remark.. Let Q n x be the orthonormal Berg Letessier Valent polynomials. Then. can be rewritten as x πγ exp 5/ Γ x Q n x = ν Ais + O ν ] as ν.. Unlike., we don t have wx on the left-hand side of the above equation because the weight function wx is unknown for the Berg Letessier Valent polynomials.

23 Constr Approx : 8 But it seems reasonable to conjecture that at least one of the weight functions for the Berg Letessier Valent polynomials should behave like x πγ exp / Γ x as x. We shall revisit this issue in Sect.. We have the following approximation. Corollary. Let x n,k be the zeros of Q n x such that x n, >x n, > >x n,n, and a k be the zeros of the Airy function Aix in descending order. Then we have for fixed k and large n, where ν = n +. x n,k = ν + a k ν / + O ν 8/,. Proof A combination of Corollary. and Lemma. immediately gives us the result. We also have a result in the interval containing t =. Theorem. Let ν = n +. With K n and U t defined in.,., and.5, respectively, we have Q n ν t πγ K n πν t exp 5/ Γ νt { πγ sin 5/ Γ νt + J νu t πγ cos 5/ Γ νt + Y νu t U t t t J νu t ] B s U Y νu t ]} B s U Ã s U Ã s U. uniformly for <t M<.

24 8 Constr Approx : Proof When <t M, we have from. and., p n x = n C xp n x + n C xq n x n 9/ ν U t C t t xj νu t + C xw νu t ]. As p n x is real, we may choose C x = Ĉ x + iĉ x and C x = Ĉ x such that p n x n 9/ ν U t t t = n 5 t t cos 9 ] t Since 9 Ĉ xj π {Ĉ x sin t t/s 9 Ĉ x cos ν s s ds t cos t νu t + Ĉ xy νu t ] t ν t/s s s ds t/s / t s s ds = 5/ ρt t/s / s s ds, and θ = π cos 9 t, we rewrite.9 as 9 p n x n + exp n + t { cos π 9 + νπ ν t cos 9 t 5/ ρ = x n t t exp x 5/ ρ ] t t/s 9 cos ν s s ds t cos Comparing.5 and the above formula, we have πx Ĉ x = 9/ exp x πγ 5/ Γ πx Ĉ x = 9/ exp x πγ 5/ Γ ] 9 t νt 5/ ρ + ν 9 sin x cos x ]}..5 ] x 5/ ρ. πγ 5/ Γ, πγ 5/ Γ t/s / } s s ds. This completes the proof of the theorem.

25 Constr Approx : 85 Remark. To get the asymptotic formula for t< from., one needs to consider the value t ± iε and take the limit as ε. Interested readers may compare the formula. with the Bessel-type asymptotic expansion for the Laguerre-type orthogonal polynomials in 7, Eq..5]. Remark.5 From our expansion., it is also possible to study the smallest zeros of Q n x. As in Corollary., we have, as ν, x πγ exp 5/ Γ x Q n x = K n νπ πγ sin πγ 5/ Γ x J s cos 5/ Γ x Y s + O ν ] uniformly for a bounded real number s, where x = 8 ν s. Then approximations for the smallest zeros can be obtained from the above formula and Lemma.. However, due to the sine and cosine terms, the formula is not as elegant as.. We leave it to interested readers. A similar situation happens for the smallest zeros of Conrad Flajolet polynomials in the subsequent two sections. Conrad Flajolet Polynomials I We can easily get a bound for the largest zero for Q n x from Theorem.. Proposition. Let x n,k be zeros of Q n x such that x n, >x n, > >x n,n. Then we have the following bounds for all n : x n, < 8n + 8cn, where c is the positive constant given in.. Proof Recall the recurrence coefficients of Q n x in., and choose c n = / in Theorem.. Then the result follows. Note that we don t provide a bound for the smallest zero in this case. The reason is that, although the chain sequence method can give us an estimation, it is not useful. In fact, we know that all the zeros of Conrad Flajolet polynomials should be positive, but Theorem. can only give us x n,n > n. For a similar reason, we only consider the largest zero in Proposition 5., too.. Difference Equation Method As usual, we introduce Q n x = n K n p n x with c K n := Γn+ + Γ n+ c + 5.

26 8 Constr Approx : to arrive at the standard form. Note that K n+ n + c = K n n + c +. Then the recurrence relation. with. reduces to the following standard form: p n+ x A n x + B n p n x + p n x =.. As n, the recurrence coefficients satisfy the following expansions: A n = K n λ n K n+ n θ α s n s, B n = λ n + μ n K n λ n K n+ β s n s,. with θ =, α = 7, α = c + 7,. and β =, β =, β = 9..5 Let x = n+ c+ t. Then the roots of the characteristic equation. coincide when α t ± + β =±, with α and β given above. This gives us the two transition points: t + = 8, t =. These transition points are similar to what we have for Berg Letessier Valent polynomials. So the two Airy-type and Bessel-type expansions follow. Proposition. When n is large, p n x in. can be expressed as p n x = C xp n x + C xq n x, where C x and C x are two n-independent functions, and P n x and Q n x are two linearly independent solutions of. satisfying the following Airy-type asymptotic expansions in the neighborhood of t + = 8: P n ν t Ut Ai ν Ã s U Ut tt 8 + Ai ν ] B s U Ut,. +

27 Constr Approx : 87 and Q n ν t Ut Bi ν à s U Ut tt 8 + Bi ν ] B s U Ut..7 + Here ν = n + c+, the leading coefficients are given by à U =, B U =, and Ut is defined as ] Ut = t B 8 t Ut ] = cos t 5 5 see.8 for the definition of B x a, b., log t 5 + tt 8, t 8,.8 5 t B 8 t, <t<8;.9 Proof With τ = α t + + β β θ = c + and ν = n + τ, our results follow from the main theorem in 9]. Proposition. When n is large, p n x in. can be expressed as p n x = C x n P n x + C x n Q n x, where C x and C x are two n-independent functions, and P n x and Q n x are two linearly independent solutions of. satisfying the following Bessel-type asymptotic expansions in the neighborhood of t = : P n ν t ν U t J νu t t8 t + J νu t à s U ] B s U,.

28 88 Constr Approx : and Q n ν t ν U t W νu t t8 t + W νu t à s U ] B s U.. Here ν = n + c+, W αx is defined in., the leading coefficients are given by and U t is defined as à U =, B U =,. U t ] t/s = ds log 5 t + tt 8, t ss 8 5 t,. U t ] t = B 8 t, 5 t cos, 5 <t<8;. see.8 for the definition of B x a, b. Proof With the asymptotic expansions for A n and B n in., our results follow from the main theorem in ]. Remark. Again, the integral in. can be written as a hypergeometric function, t t/s ss 8 ds = t F,, 7,t. Ut and U t are two monotonically increasing functions in the neighborhood of 8 and, respectively, with the following asymptotic formulas: Ut= t Ot 8 as t 8,.5 and U t = t 7 + O t as t... Determination of C x and C x According to Lemma., we have the following result.

29 Constr Approx : 89 Proposition. Let x = x n := n + c + / t with t C \, 8]. We have as n, p n x n Γ c + /Γ c + / t n n n c+ π/n / / 8/t / + 8/t ] n++ c { n + c + exp }. 8s /t ds Proof Recall the rates given in. and K n defined in.. By Stirling s formula, we have K n / n / and n k= λ k = n Γ c + / + nγ c + / + n Γ c + /Γ c + / n n c+5/ Γn Γ c + /Γ c + / n n c+5/ π/n / n/e n Γ c + /Γ c + /. It thus follows from.8 that n π n x = K n λ k p n x n n c+ π/n / n/e n / Γ c + /Γ c + / p nx. k= Since π n x satisfies the asymptotic formula.7 with p =, b = 7, u = c + 5/, and v = c + /, our result follows from the above formula. Again, using an argument similar to that in the proof of 8, Theorem.], we obtain the following result in the oscillatory region. Proposition.5 Let x = x n := n + c + / t with t = 8 cos θ,θ δ, π δ]. We have as n, p n x n c+ Γ c+ Γ c+ 7/ π / 8t t / { π cos n + n t c c + θ + exp n + c + t / ρ 8 n + c + 8 t t/s / 8s s ds },.7 where ρ = du u = u u πγ Γ 5..8 Then we get the Airy-type expansion in the interval containing t + = 8.

30 9 Constr Approx : Theorem. Let ν = n + c+. With K n and Ut defined in.,.8, and.9, respectively, we have Q n ν t n K n c+ Γ c+ Γ c+ 7 πν c t c πγ exp Ut Γ 5 νt Ai ν Ã s U Ut + Ai ν Ut uniformly for <δ t<. tt 8 ] B s U.9 + Proof When t<8, recall the asymptotic expansions for the Airy function in.8 and.9 again. We have from. and.7, p n x = C xp n x + C xq n x ν C x cos Ut π t8 t] π ν C x sin Ut π t8 t] π t 5 = C x cos ν cos t8 t] π 5 8 t/s ds π t s8 s t 5 C x sin ν cos t8 t] π 5 8 t t/s s8 s ds π. Comparing the above formula and.7, we have C x = c / Γ c + /Γ c + / / πx c C x =. x exp 8 πγ Γ 5, This completes the proof of the theorem. Remark. As in Remark., we can obtain the following limiting zero distribution for Q n 8n x from the recurrence relation.: π x / ds, x, ].. xs x

31 Constr Approx : 9 Since the above theorem is uniformly valid in the neighborhood of the large transition point 8, we have the following asymptotic formula for Q n x. Corollary. Let ν = n + c+ and K n be given in.. Uniformly for a bounded real number s, we have, for x = 8ν + 5 8sν 7, x c exp πγ Γ 5 x Q n x = n c Γ c+ K Γ c+ ν n Ais + O ν ] π as ν.. Proof Let t = sν in.9, and recall the asymptotic formula for Ut in.5. We have after some computations, x c πγ exp Γ 5 x Q n x = n K n c+ Γ c+ Γ c+ 7 π ν Ais + O ν ] O ν This above formula proves our corollary. Remark. Again we can put the left-hand side of. into its orthonormal form. Let Q n x be the orthonormal Conrad Flajolet polynomials I. Then. can be rewritten as x c πγ exp c + Γc+ ν Γ 5 x Q n x = Ais + O ν ] 5 as ν.. According to the above formula, it again seems reasonable to conjecture that at least one of the weight functions for the Conrad Flajolet polynomials I should behave like x c π Γ exp Γ 5 x as x. We have the following approximation. Corollary. Let x n,k be the zeros of Q n x such that x n, >x n, > >x n,n, and a k be the zeros of the Airy function Aix in descending order. Then we have for fixed k and large n, x n,k = 8ν + 5 8a k ν 7/ + O ν 5/, where ν = n + c+.

32 9 Constr Approx : Proof A combination of Corollary. and Lemma. givesustheresult. We also get the Bessel-type expansion in the interval containing t =. Theorem. Let ν = n + c+. With K n and U t defined in.,., and., respectively, we have Q n ν t K n c+ Γ c+ Γ c+ ν c { sin 5/ πt c πγ x c Γ 5 π ] + J νu t B s U πγ cos x c Γ 5 π + Y νu t uniformly for <t M<8. ]} B s U πγ exp U t Γ 5 νt t8 t J νu t Y νu t Proof When <t M, we have from. and., p n x = n C xp n x + n C xq n x n ν Ã s U Ã s U U t C t8 t xj νu t + C xw νu t ]. Note the following asymptotic expansions for Bessel functions, as x, J ν x cos x νπ πx π, Y ν x sin x νπ πx π.. As p n x is real, we choose C x = Ĉ x + iĉ x and C x = Ĉ x such that p n x n ν U t t8 t n π t8 t Ĉ xj νu t + Ĉ xy νu t ]

33 Constr Approx : 9 Since { t Ĉ x cos ν t + Ĉ x sin ν 8 t t/s 5 t ds cos s8 s 5 t/s 5 t ds cos s8 s 5 t/s / πγ ds = 8s s t t Γ 5 and θ = π cos 5 t 5, we rewrite.7as p n x c+ Γ c+ Γ c+ 7/ π / 8t t / x c x exp 8 5 π ] 5 π ]}. t/s / 8s s ds / ρ { cos π 5 t + νπ ν cos 5 πγ ν t t t/s / } + ν ds Γ 5 8s s = c+ Γ c+ Γ c+ n x / 7/ π / exp ρ x c 8t t 8 t t/s 5 t cos ν ds cos s8 s 5 5 π πγ Γ 5 x + c + Comparing. and the above formula, we have ] π. Ĉ x = c / Γ c+ Γ c+ exp x πγ / πx c πγ Γ 5 sin Ĉ x = c / Γ c+ Γ c+ exp / πx c x πγ Γ 5 cos x c Γ 5 π πγ Γ 5., x c π. This completes the proof of the theorem. 5 Conrad Flajolet Polynomials II Again we use Theorem. to get a bound for the largest zero for Q n x as follows.

34 9 Constr Approx : Proposition 5. Let x n,k be zeros of Q n x such that x n, >x n, > >x n,n. Then we have the following bounds for all n : x n, < 8n + 8cn. Proof Recall the recurrence coefficients of Q n x in., and choose c n = / in Theorem.. Then the result follows. 5. Difference Equation Method As we have done before, we introduce Q n x = n K n p n x with c K n := Γn+ + Γ n+ c + Γ n+ c + Γ n+ c to arrive at the standard form. Note that K n+ n + c n + c = K n n + c + n + c +. Then the recurrence relation. with. reduces to the standard form.. As n, the recurrence coefficients satisfy the following expansions: A n = K n λ n K n+ n θ α s n s, B n = λ n + μ n K n λ n K n+ β s n s, 5. with θ =, α = 7, α = c + 5, 5. and β =, β =, β = Since α and β are the same as those in. and.5, we get the same transition points as follows: t + = 8, t =. Again we get two different types of asymptotic expansions near t + and t.dueto the similarity between the current case and that in Sect., we get the same Airy-type expansion as that in Proposition. except that in this case, ν = n + c+. Because the higher-order coefficients in 5. and 5. are different from those in. and.5, we have a corresponding difference in the order of Bessel functions. The Bessel-type expansion near t is given as follows.

35 Constr Approx : 95 Proposition 5. When n is large, p n x in. can be expressed as p n x = C x n P n x + C x n Q n x, where C x and C x are two n-independent functions, and P n x and Q n x are two linearly independent solutions of. satisfying the following Bessel-type asymptotic expansions in the neighborhood of t = : P n ν t ν U t J νu t t8 t + J 5 νu t ] B s U, à s U and Q n ν t ν U t W νu t t8 t + W 5 νu t ] B s U. à s U Here ν = n + c+, W α x is defined in., the leading coefficients are given by à U =, B U =, and U t is the same as defined in. and.. Proof With the asymptotic expansions for A n and B n in 5., our results follow from the main theorem in ]. 5. Determination of C x and C x According to Lemma., we have the following result. Proposition 5. Let x = x n := n + c + / t with t C \, 8]. We have as n, p n x n Γ c + / Γ c + /t n n n c+/ π/n / 5/ 8/t / + 8/t ] n++ c { n + c + / exp }. 8s /t ds

36 9 Constr Approx : Proof Recall the rates in. and K n defined in 5.. By Stirling s formula, we have K n 5/ n 5/ and n k= λ k = n Γ c + / + n Γ c + / + n Γ c + / Γ c + / n n c+/ Γn Γ c + / Γ c + / n n c+/ π/n / n/e n Γ c + / Γ c + /. Then from.8, we have n π n x = K n λ k p n x n n c+/ π/n / n/e n 5/ Γ c + / p n x. Γ c + / k= Because π n x satisfies.7 with p =, b = 7, u = c + /, and v = c /, we get our proposition from the above formula. Again, based on the above proposition, we obtain the following result in the oscillatory region. Proposition 5. Let x = x n := n+c +/ t with t = 8 cos θ,θ δ, π δ]. We have as n, p n x n c+/ Γ c+ c+ Γ / π / 8t t / { π cos n + where ρ is defined in.8. n t c c + θ + Then the Airy-type expansion can be obtained. exp n + c + t / ρ 8 n + c + 8 t t/s / 8s s ds }, 5.5 Theorem 5. Let ν = n + c+. With K n and Ut defined in 5.,.8, and.9, respectively, we have Q n ν t n c+ Γ c+ c+ Γ K n πν c t c Ai ν Ã s U Ut + Ai ν Ut uniformly for <δ t<. πγ exp Ut Γ 5 νt tt 8 ] B s U +

37 Constr Approx : 97 Proof When t<8, recall the asymptotic expansion for the Airy function in.8 and.9 again. We have from. and.7, p n x = C xp n x + C xq n x ν C x cos Ut π t8 t] π ν C x sin Ut π t8 t] π t 5 = C x cos ν cos t8 t] π 5 8 t/s ds π t s8 s t 5 C x sin ν cos t8 t] π 5 8 t t/s s8 s ds Comparing the above formula and 5.5, we have π. C x = c Γ c + /Γ c + / 5/ πx c C x =. x exp 8 πγ Γ 5, This completes the proof of the theorem. Remark 5. Again, we can obtain the limiting zero distribution for Q n 8n x from the recurrence relation.. It is the same as that in.. Since the above theorem is uniformly valid in the neighborhood of the large transition point 8, we have the following asymptotic formula for Q n x. Corollary 5. Let ν = n + c+ and K n be given in 5.. Uniformly for a bounded real number s, we have, for x = 8ν + 5 8sν 7, x c exp πγ Γ 5 x Q n x = n c 7 Γ c+ c+ Γ K ν n Ais + O ν ] π as ν. 5.

38 98 Constr Approx : Proof The proof is very similar to that of Corollary.. Remark 5. Again we can put the left-hand side of 5. into its orthonormal form. Let Q n x be the orthonormal Conrad Flajolet polynomials II. Then 5. can be rewritten as x c exp as ν. πγ Γ 5 x Q n x = Γc+ ν 5 Ais + O ν ] 5.7 According to the above formula, it again seems reasonable to conjecture that at least one of the weight functions for the Conrad Flajolet polynomials II should behave like x c π Γ exp Γ 5 x as x. Since 5. is very similar to. except for some constants on the right-hand side and powers of x changed on the left-hand side, we have the same approximation for the zeros as in Corollary., that is, x n,k = 8ν + 5 8a k ν 7/ + O ν 5/, where ν = n + c+. The Bessel-type expansion is also obtained as follows. Theorem 5. Let ν = n + c+. With K n and U t defined in 5.,., and., respectively, we have Q n ν t c+/ Γ c+ c+ Γ K ν c n / πt c { πγ sin x c Γ 5 π ] + J 5 νu t B s U πγ cos x c Γ 5 π + Y 5 νu t uniformly for <t M<8. ]} B s U πγ exp U t Γ 5 νt t8 t J νu t Y νu t à s U à s U

39 Constr Approx : 99 Proof The proof is similar to that of Theorem.. Remarks on the Moment Problem The general solution of an indeterminate moment problem is described in terms of four entire functions Az, Bz, Cz, Dz such that AzDz BzCz =. The functions are called the Nevanlinna parametrization; see, ]. It is known that these functions have real and simple zeros. The following theorem, see, Thm...], describes all solutions to an indeterminate moment problem. Theorem. Let N denote the class of functions {σ }, which are analytic in the open upper half-plane and map it into the lower half-plane, and satisfy σz = σz. Then the formula R dμt; σ z t = Az σzcz, z / R,. Bz σzdz establishes a one-to-one correspondence between the solutions μ of the moment problem and functions σ in the class N, augmented by the constant. A theorem of Herglotz, Lemma.] asserts that any σ in the above theorem has the representation + tz σz= c z + c + dαt,. R z t where α is a finite positive measure on R, c and c are real constants, and c. The next theorem is in ] and ]. Theorem. Let σ in. be analytic in Im z>, and assume σ maps Im z> into Im σz<. If μx, σ does not have a jump at x and σx± i exist, then dμx; σ dx = σx i+ σx+ i + πi Bx σx i + Dx.. A consequence of Theorem. is that the polynomials associated with this moment are orthogonal with respect to the weight function wx = γ/π γ B x + D x. for any γ>. Berg and Pedersen ] showed that the four entire functions A,B,C,D have the same order, type, and Phragmén Lindelöf indicator. The formulas for the orthonormal polynomials in.,.,., and 5.7 seem to indicate an interesting pattern about the powers of ν a constant plus n, the

40 Constr Approx : degree of the polynomial. Indeed it suggested to us to formulate the following conjecture. Conjecture. Let { P k x} n k= be a sequence of polynomials orthonormal with respect to the weight function wx on an unbounded interval and t n be the large transition point of P n x. Without loss of generality, we may further assume the interval to be, or,. If the weight function wx has the following behavior: log wx = O x m as x,.5 then we have t n = O n m as n. Moreover, uniformly for a bounded real number s, we have, for x = t n + sn, wx P n x ĉsn k + o as n, where ĉs is a uniformly bounded function in s and the constant k is given by k = m.. Here m>, but is not necessarily an integer. Note that the above conjecture is true for Laguerre polynomials, Hermite polynomials and polynomials orthogonal with respect to the Freud weight e x β.forthe asymptotics of Laguerre polynomials and Hermite polynomials, see, Sect. 8.]. For the asymptotics of polynomials orthogonal with respect to the Freud weight, see 5, Eq..9]. When the corresponding moment problem is indeterminate, m in.5 equals the order of any of the entire functions A,B,C, ord. This seems to be the case for the weight function.. It also agrees with the known weight functions for the Chen Ismail polynomials. Remark. Note that the large transition point t n is closely related to the largest zero of the orthogonal polynomials, The asymptotics for the largest zero have been studied in the literature, which are related to.7 in our conjecture. For example, Levin and Lubinsky obtained the asymptotics for determinate exponential weights in, Thm..]. They also derived similar results for indeterminate weights and some weights close to indeterminate ones in 9, Cor..] and, Cor..], respectively. However, the interesting relation between k and m in. seemstobenewinthe literature. In the rest of this section, we shall show that the moment problems corresponding to Conrad Flajolet polynomials I and II do not have unique solutions; that is, the moment problems are indeterminate. The following proposition is in ]or].

41 Constr Approx : Proposition. Let P n x be the orthonormal polynomials satisfying the following recurrence relation: x P n x = b n+ P n+ x + a n P n x + b n P n x..7 If the moment problem has a unique solution, then the series P n z.8 n= diverges for z/ R. If the above series diverges at one z/ R, then the moment problem has a unique solution. Recall that a birth and death process leads to polynomials Q n x defined in.. We have a n = λ n + μ n, b n = λ n μ n in.7, and the corresponding orthonormal one is Q n x = n Q n x n λ k. μ k For the orthonormal Conrad Flajolet polynomials I, we obtain the following result from Theorem.. Corollary. Let ν = n + c+ and x = ν t. With K n and U t defined in.,., and., respectively, we have Q n ν t n n k= λ k πγ exp Γ 5 νt { πγ sin μ k K n c+ Γ c+ k= Γ c+ ν c 5/ πt c U t t8 t x c Γ 5 π ] + J νu t B s U πγ cos x c Γ 5 π + Y νu t ]} B s U J νu t Y νu t à s U à s U uniformly for <t M<8. Here λ n and μ n are defined in..

42 Constr Approx : Let z = ν t be a fixed complex number. Then t = z/ν = On. It follows from. that U t = Ot = On. Recalling the asymptotic formulas of Bessel functions at small arguments, we obtain J νu t = O n /, J νu t = O n /, Y νu t = O n /, Y νu t = O n /. Since à s U = and B s U = by., we observe that the expressions involving Bessel functions and trigonometric functions are of order On /.Next,we obtain the following asymptotic estimate from.,., and Striling s formula, n λ k = O n 5/, K n = O n /. μ k k= A combination of the above estimates yields Q n z = On 5/. This establishes the convergence of the series in.8. Thus, the corresponding moment problem is indeterminate. For the orthonormal Conrad Flajolet polynomials II, we obtain the following result from Theorem 5.. Corollary. Let ν = n + c+. With K n and U t defined in 5.,., and., respectively, we have Q n ν t n n k= λ k πγ exp Γ 5 νt { πγ sin c+/ Γ c+ K n μ k / πt c U t t8 t x c Γ 5 π ] + J 5 νu t B s U πγ cos x c Γ 5 π + Y 5 νu t ]} B s U Γ c+ ν c J νu t Y νu t à s U à s U uniformly for <t M<8. Here λ n and μ n are defined in..