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UniversiV M ic rm lm s International 300 N. Zeeb Road Ann Arbor, Ml 48106
8510547
Bauldry, William Charles
ORTHOGONAL POLYNOMIALS ASSOCIATED WITH EXPONENTIAL WEIGHTS
The Ohio State University Ph.D. 1985
University Microfilms I ntern StiO neI 300 N. zeeb Roaa, Ann Arbor, Ml 48106
ORTHOGONAL POLYNOMIALS ASSOCIATED WITH EXPONENTIAL WEIGHTS
DISSERTATION
Presented In Partial Fulfillment o f the Requirements for
the Degree Doctor of Philosophy in the Graduate School
of The Ohio State University
By
William Charles Bauldry, B.S., M.A., M.5.
The Ohio State University, 1985
Reading Committee: Approved By
Prof. Paul Neval
Prof. Bogdan BaishanskI
Prof. Ranko Bojanic Vs] ______
Prof. William Davis Adviser Department o f Mathematics ACKNOWLEDGEMENTS
1 would like to thank my adviser, Dr. Paul Nevai, fo r all the support, help, and wisdom he has imparted to me during my graduate training. It was his inspiration and guidance that made this paper possible. A vote of thanks also goes to Drs. Doron Lubinsky and Attila Mate fo r the time and knowledge they shared with me in developing this dissertation project.
Last of all, 1 wish to thank my family, especially my wife, Sue, fo r their moral support, encouragement and love. VITA
December 14, 1950 ...... Born - Mt. Clemens, Michigan
1975 ...... B.S. in Mathematics, Central Michigan University, Mt. Pleasant, Michigan
1975-1977 ...... Teaching Associate, Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan
1977 ...... M.A. in Mathematics, Central Michigan University, Mt. Pleasant, Michigan
1977-1980 ...... Lecturer, Department of Mathematics, The Ohio State University, Mansfield Campus. Mansfield, Ohio
1980-1985 ...... Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio
1983 ...... M.S. in Mathematics, The Ohio State University, Columbus, Ohio
PUBLICATIONS
"Estimates of Christoff el Functions of Generalized Freud-type Weights." Journal of Approximation Theory (to appear).
II) FIELDS OF STUDY
Major Field: Mathematics
Studies in Approximation Theory. Professors Paul Nevai and Ranko Bojanic.
Studies in Analysis. Professors Bogdan Baishanski and Gerald Edgar.
IV TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS...... ii
VITA ...... iii
INTRODUCTION...... 1
CHAPTER 1: ESTIMATES OF CHRISTOFFEL FUNCTIONS OF GENERALIZED FREUD-TYPE WEIGHTS...... 5
1. Introduction ...... 5 2. Notation ...... 7 3. ThB Main Results...... 9 4. Proof of the "Infinite to Finite Range" Inequality ...... 12 5. Proofs o f the Upper and Lower Bounds o f the Christoff el Functions ...... 17 6. Connections to the Orthonormal Polynomials Pp(w^;x) . . . 24
CHAPTER II: ASYMPTOTICS FOR THE RECURSION COEFFICIENTS a^ ANDb^ ASSOCIATED WITH THE ORTHOGONAL POLYNOMIALS WITH THE WEIGHT w(x) = exp{-Q(x)}...... 30
1. Introduction ...... 30 2. Notation ...... 34 3. The Main Results...... 35 4. Proof of the Existence Theorem fo r Asym ptotics ...... 38 5. The Asymptotic Series of the Recursion C oefficients 50 CHAPTER III: ASYMPTOTICS FOR THE ORTHOGONAL POLYNOMIALS ASSOCIATED WITH w(x) = exp{-Q(x)} ...... 62
L Introduction ...... 62 2. Notation ...... 65 3. The Main Results ...... 67 4. A Preliminary Estimate o f pp(w:x) ...... 70 5. The Asymptotics of P p (w :0 ) ...... 81
6. Asymptotics for P p , ( w ; x ) ...... 91
CHAPTER IV: PLANCHEREL-ROTACH TYPE ASYMPTOTICS FOR THE ORTHOGONAL POLYNOMIALS ASSOCIATED WITH w(x) = exp{-Q(x)}...... 105
1. Introduction ...... 105 2. Notation ...... 108 3. The Main Result ...... 110 4. Lemmata ...... 112 5. Proof of The Plancherel-Rotach Type Asymptotics 126
BIBLIOGRAPHY...... 140
VI INTRODUCTION
"It is the essence o f Mathematics that it concerns itself with those relations which lie so deep in the nature of things that they recur in the most varied situations. ... Among these are the formulations relating to the general analytical concept of orthogonality ..." - D. Jackson, [Jal, p. v].
The notion o f perpendicular is fundamental to our perception o f the reality o f the space in which we live. This concept has its generalization in the abstract as orthogonality. Once we define a criterion o f measurement on a collection o f objects, we can create a definition o f orthogonal elements. To wit; let $ and f be two objects o f a universal set with a measurement denoted by ($,9) taking real values. $ and 9 will be called orthogonal if and only if ($,»)=0. When the universal set is the collection o f functions and the measurement is given by an inner product, we have orthogonal functions. If we further specify that the functions are polynomials and the inner product Is an integral, we have entered the realm of orthogonal polynomials. The study of orthogonal polynomials originated from the theory of continued fractions ( SzegojSzl] ). In its infancy, the theory was developed from the point o f view o f continued fractions and their relation to the moment problem iShoTal]; however, as the importance o f the theory was recognized,the starting point was shifted to the property of orthogonality in the following manner. Let ji(x) be a nondecreasing, bounded function on the real numbers taking real values. Define an inner product on the class L^(dp:E) by
(*.9) =J $(x)Ÿ(x) djj(x), and we shall use our definition o f orthogonality given above. The Gram -
Schmidt process of orthogonal izat ion (see e.g. Jackson, [Jal, p. 151]) can be applied to any collection of ji-integrable linearly independent functions to produce an orthogonal set. The set o f powers o f x { $j(x) = x' : 1er, r c N }
( r may be either o f finite or infinite cardinality ) gives rise, through this process, to the polynomials { pp(d;i; x ) } orthogonal with respect to dp(x).
If we define a^ = where is the leading coefficient of Pp(d|i;x), then we immediately see that
ap(d|i) = r X PpCdp; x)pp_, (dp: x) dp(x); K we also define
br^(dp) = J _ X Pn^(dp: x) dp(x). E
All sets o f orthogonal polynomials satisfy a three term recurrence relation
X pp(dp; x) = 3n+| (dp) Pn+, (dp; x) + bp^(dp) Pp(dp: x ) + ap,(dp) Pn_j (dp; x) with api and b^ as above. We can easily see that the recursion coefficients
and bp, completely determine the system of orthogonal polynomials they are associated to. This property is the basis o f much o f this paper, further, much o f the current research is a result o f this observation. Richard Askey
[Asll noted in a recent review o f Chihara's An Introduction to Orthogonal
Polynomials. "General orthogonal polynomials are primarily interesting because of their three term recurrence relation." He goes further saying "... the deeper work o f Nevai and his coworkers did not start to appear until
1979. Nevai's work is one o f the real reasons there is a lot o f work being done on general orthogonal polynomials.” This paper is in two parts: Chapter I is an investigation into the
Christoffel functions of Freud - type weights having a singularity at the origin, while Chapters 11 through IV develop asymptotics o f the polynomials orthogonal with respect to the weight w(x) = exp{ -Q(x)} where Q(x) is an arbitrary polynomial of the fourth degree. CHAPTER 1
ESTIMATES OF CHRISTOFFEL FUNCTIONS OF
GENERALIZED FREUD-TYPE WEIGHTS
1. INTRODUCTION
Geza Freud Initiated investigations into the polynomials orthogonal with respect to W(x)=exp{-Q(x)} with Q(x) chosen as x^Vzk [Frl.Fr3-Fr6l
Nevai [NeI,Ne2,Ne4] and Sheen [Shl,Sh2] have successfully handled the cases k=2 and k=3, respectively, where, as in much o f Freud’s work, estimates of the Christoffel functions gave crucial information needed in bounding the orthogonal polynomials. Freud also used the bounds to find weighted
Markov-Bernstein type inequalities [Fr2] when Q is a Freud exponent (see
(1.2.1)). Recently Lubinsky [Lull, Mhaskar-Saff [MhSa2], and Zalik [Zal] have investigated similar weighted inequalities: further, Lubinsky [Lu2] and
Mhaskar-Saff [MhSal] have bounded the generalized Christoffel functions fo r a wider class of smooth weights. Both the bounds of the Christoffel functions and the weighted Inequalities are used In Magnus' proof [Mai,Ma2] of the Freud conjecture [Fr3].
In this chapter we will Investigate the Christoffel functions of
Freud-type weights that have a singularity at the origin, that Is, weights of the form:
Wp(x) = |x|r exp{-Q(x)} ( -«>< x<+oo, r>-1 ), with Q(x) being a Freud exponent. We Intend to use the estimates given below to find the asymptotics of orthogonal polynomials associated with these generalized Freud-type weights.
The organization of the chapter is as follows: In Section 2 we define our notation: Section 3 contains the statements o f the main results;
Section 4 Is the proof of the Integral Inequality; Section 5 contains the derivation o f the bounds; and. lastly. Section 6 relates q^, (see (1.2.3)) to the largest zero and to the ratios of leading coefficients of the orthogonal polynomials associated with these weights. 2. NOTATION
The following notations will be observed throughout. Q(x) will be called a "Freud exponent" when Q is an even function and satisfies;
i) Q'(t) >0. Q"(t) >0 for t £(0,oo).
(1.2.1) ii) Q"(t) is continuous on E.
iii) Q'(2t)/Q'(t) > Cq >1 forteCO.oo),
iv) t Q"(t)/ Q’(t) The weight function, w^(x), will then be w^(x) = |x|r exp{-Q(x)}. The polynomials orthonormal with respect to w^ are pp,(w^; x)= 2fpx'^+... ; denote the greatest zero of pp,(x) by X|p,(Wp) and let (1.2.2) 3r^(Wp)= (Wp) / 2fp,(Wp). Let be defined by the equation (1.2.3) dp Q'(qp) = n. By Ppj, denote the set o f all polynomials with real coefficients o f degree at most n. The generalized Christoffel functions of the distribution djj are 8 (see Nevai [Ne31, where they were first introduced) Xpp(d|i;x)= inf [ J |Tr(t)|Pd|i(t) / |Tr(x)|P ]. 7Td>n-i ® We note that, fo r the special case p=2, the following identity is well known (e.g. Freud [Fr7, Theorem 1.4.1] n~1 Xp 2 (dfi; x) = [ S Pk^(d]i; x) 1"^. k=0 Denote by C|, C2 , .... positive constants independent of x or n. 3. THE MAIN RESULTS The first result is the main tool with which the bounds were obtained. Theorem 1.3.1 . Let Q(x) be a Freud exponent and be as defined in (1.2.3), then fo r a fixed e>0, and p, r such that e 17t(x) Wp(x) I) < (1+c I) TT(x) Wp(x) I Lp(R) Lp(-Bqn.+Bqn) where Tr(x)6P^. Remark. The above inequality can be significantly sharpened using the techniques o f Potential Theory ( e.g. see Mhaskar-Saff [MhSa2] ). We have chosen the methods used fo r simplicity o f exposition since they do produce results sharp enough fo r the purposes of the following theorems. We also note that using q^ is nonzeroforn less than ng and standard compactness arguments we can extend the inequality to n = 1, 2,... . With this "Infinite to Finite Range" inequality in hand we can proceed to the main results, upper and lower bounds of the generalized Christoffel functions: Nevai [NeSl was the first to use the method of reducing weights 10 over the real line to compact Intervals in order to estimate the Christoffel functions. Theorem 1.3.2 . Let Q(x) be a Freud exponent with as defined in (1.2.3), let 0 E, 0 Wr"P(x) X^p(w^P;x) > A(dn/n) (l+ (q ^/n )/|x|)P r ( |x|< EBq^) where B is the constant of Theorem 1.3.1. Theorem 1.3.3 . Let Q(x) be a Freud exponent with q^ as defined in (1.2.3), let 0 8>0 and constant A', independent of x and n, such that Wf'P(x) \p(w /:x) < A'(q^/n) (l+(q/n)/|x|)Pr ( |x|< 8q^). We immediately obtain the following Corollary 1.3.4 . Under the conditions of Theorems 1.3.2 and 1.3.3 Wr'P(x) \ p ( w / : x ) ~ (q^/n) (l+(q ^ /n )/|x |)P r ( |x|< 8q^ ). Remark. We note that from the definition o f Freud exponent that Q” continuous is used fo r the lower bound but not fo r the upper bound while Q'(2t)/Q'(t)>CQ is used fo r the upper bound and not the lower. The relation of to the polynomials Pp|(Wp-, x) ( see Freud [Fr4l ) is seen in Theorem 1.3.5 . Let Q(x) be a Freud exponent with as defined in (1.2.3) and let r >-1; define w^(x) = |x|r exp{-Q(x)}. Let Xj^(Wp) be the greatest zero of p^(w^; x) and let a^(w^) be defined by (1.2.2). Then we have *ln(Wr)~qn ^nd a^(Wr)-qp. 12 4. PROOF OF THE "INFINITE TO FINITE RANGE” INEQUALITY. Following the method of Lubinsky [Lu2] we use Cartan’s Lemma. Lemma 1.4.1 (Cartan). If P(z)=(z-z,Xz-Z2 )-(z -^ ), then fo r any H>0 the inequality I P(z) I > (H/e)M holds outside at most n circles, the sum o f whose radii is at most 2H. Proof. See e.g. Baker [Bal, p. 174]. □ Proof (Theorem 1.3.0. If ir(x)=0 the inequality Is trivial. Let we can express m T f ( x ) = c ï ï ( x - X j ) : c%0, 0 Let dp, be defined by (1.2.3). Determine j>0 such that for l |xj|<3q2n/2 and for j< i lx-X||/|u-Xj| < 0 * |x|/|X||)/(l- |u|/|x||)<3(K2/3)(|x|/q 2r,)). i.e., 13 0.4.1) Ix-xj I/ 1u-xj I <5(|x|/q2n). If IXI>Bq 2 p. |u| 0.4.2) I x-x j I / 1 u-x j I <(|x| +(3/2)q2n)/ |u-Xj| < 2 |x| / |u-Xj|. Putting (1.4.1) and (1.4.2) together yields j m |Tr(x)/TT(u)| < n(2|x|/|u-x j| ) n (5|x|/q2n) = i=1 i=j+1 j = 2l 5^-J (IXr/(qgr^'^-J)) [ n|u-Xj| F’ i=l ] We shall now apply Cartan's lemma to {ÏÏ | u-Xj | ) to obtain i=1 I Tr(x)/7T(u) I < 5 ^ I 4 8 1XI /q 2 n P fo r IXI > Bq 2 n, | u | < q 2 n- and u*/6cE, where / is a set which can be covered by Intervals, the sum o f whose lengths is at most Q2n'^®- Let in = (-A2n''^^2n)^ then % has Lebesgue measure at least (15/8)q 2 ^. So forufU l, |x|>Bq 2 n 14 I Tr(x)Wp(x) I / 1 TT(u)Wp(u) I < 5*^ [ 4 8 1XI /q 2n Wp(x)/Wp(u). Let C| =min{ 1, (3/8)''} and = HL\(-(3/8)q2n, +(3/8)q2n). then I 'n'(x)Wp(x) I / 17T(u)Wp(u) I < 5^^ [ 481XI /q 2n vVp(x)/ <\2n 1 < [28 n/Cj ] I q 2n/1 x | f [ | x | 2nw^(x) / ( q 2n^^/o( 02n) ) ^ But, by the maximality of |Tr(x)W p(x)|/lTr(u)W p(u)| < [2®'^/Ci U q 2n /1 x | f" " '. i.e., fo r I XI >Bq 2 p, and UE%*^ |7T(x)Wp(x)| <[2®^/C|][q2p,/|x| |Tr(u)Wp(u)|. Therefore I Tr(x)Wp(x) I P < [2®^/Cj ]P [ q 2p/1 x | min | iT(u)Wp(u) | P, or 17r(x)Wp(x) I P < [2®^/C| }P [ q2n/1 x | ]^'^''')P (1/q 2^) J | TT(u)Wp(u) | Pdu HI’’ 15 "^2n < IzSn/c, )P [ q ;/ 1X | (t/q J | TT(u)w/u) | Pdu. ■P2n Whence +(l2n J17T(x)Wr(x) IP dx < 2 8 pn*' B-(n-'")P^I c, 'PtpCn-r)-!) ' ' J17r(u)Wr(u) | Pdu. l-®'l2n ‘ Thus fo r B suitably large and n>n^ '02n J I 'n '(x )W p C x ) I P d x < A { p j n /c ^ f [ p n ]" ^ J | T T ( u ) W p ( u ) | P d u . Nl-®^2n -Q2n Now J I TT(x)wr(x) I P dx = [ J + J 1 17T(x)Wp(x) I P dx E |x| J 17r(x)Wf.(x) IP dx < [l+(c, /(pn)) p ^l J | tt(x)w^(x) | P dx. E |x| So we have 1 tt(x) w/x) II <[1+(c ] /(pn)) II 7T(x) Wp(x) [ Lp(R) Lp(-Bqp,+Bqr,) choosingB possibly larger, since q2n<2qn ( Freud [Fr2,p. 22] ). Fix 0>O then fo r o<ô I tt( x) Wp(x) I < l1+(c, /(6n)) I tt(x) Wp(x) | Lp(R) Lp(“ Bq|^,+Bq[^) By the continuity o f | • | norms and the independence of the constants Lp uponp, the limit as p may be taken and the inequality holds fo r 0< e < p < 00. B 17 5. PROOFS OF THE UPPER AND LOWER BOUNDS OF THE CHRISTOFFEL FUNCTIONS. First, we shall require a technical lemma, n~l Lemma 1.5.1 . Let Rp,(x) = 'Z x^/k! then k= 0 (3/4) exp(x) < Rp(x) < (5/4) exp(x) ( | x | Proof. From Taylor's theorem, we have, for | x | I exp(x)-R^(x) I < (n!)"^ max { exp(x) | x | '^ ) < (n!)"' exp(cn) (cn)^. IXI son Applying the Inequality nl > (n/e)^ (n>l) gives I exp(x)-Rp(x)| In particular, for c=1/5, I l-exp(-x) Rn(x)| < (8/9)^». g A theorem o f Nevai's will be used to bound the Christoff el functions after we reduce to a compact interval. Theorem 1.5.2 (Neval). Let r > -l and 0 < £ < 1. If w(x) = | x | ^ on [- 1,1], then 18 \p (w ; x) ~ ( 1/n) ( IXI + 1/n) ^ (Ix| < € ). Proof. Nevai [Nb3, Theorem 6.3.25, p. 119]. B We shall now construct the polynomials that will be used to approximate W q ( x) (as in Freud lFr2]). Lemma 1.5.3 . Let Q(x) be a Freud exponent, be defined by (1.2.3), and fix xeE. There exists a polynomial S^(x:t) such that I) Spj(t) 6E2kn^^^ fo r each fixed x and some integer k=k(Q,B), ii) Sp,(x:x) = W q ( x), Hi) 0 < Sn(t) < (5/4)Wo(t) for 1 1 1 < Bq^. where B is the constant o f Theorem 1.3.1. Proof. Let Vp(t)= Q'(x)(t-x) + IcQn/(2qp,^)j (t-x) 2 for teE. Define S^(t) = Wo(x)R,^n^-Vn(t)) ( |t|< B q n ). then i) and ii) follow directly. Now to prove Hi): for |t|< Bq^ |Vn(t)| <|Q'(x)| 2 Bqn ♦ [Con/( 2 qn 2 )l < C| I Q'(q„) I 2Bqn * 2B^CQn 5 2B[c, +BcJn. Therefore, if k is a large enough positive integer, so that k/5 > 2B[C| +CqB], 19 then, by Lemma 1.5.1, Rkn( -^n(0 ) ~ G xp( -Vp,(t) ) ( 111 Sn(t) = Wq(x) Rkn("Vn(0) w jx ) exp(-Vp,(t)), and hence Sn(t)Wo"’ (t) ~ exp{ Q(t)-Q(x) - Q’(x)(t-x) - [cQn/CZq^^)] (t-x)^}. Since Q” is continuous, Q(t)= Q(x)+Q’(x)(t-x)+Q''(0 (t-x ) ^ / 2 fo r some ^ between t and x, but, since Q is a Freud exponent, | Q” (01 < c^n/q^^. and thus iii) holds, g We are now in a postion to determine the lower bound. Proof (Theorem 1.3.2). Let p>0, fix r such that pr>-l. Then \p ( ^ r ^ : x) = inf J 17T(t) | P w^^Ct) dt / [7r(x)]P TrePn-i K +Bqn > inf J I TT(t) IP WpP(t) dt / [Tr(x)]P T f^n -1 -Bqn 20 •Bqn > C| WqP(x) inf J I TT(t)S 2 |^r>(t) | P 11 1 P'' dt /[Tr(x)S2 |(n(x)]'^ Tr6Pn-i -Bdn +1 - ^ 2 irif J | R(tBqp,) | P 111 P^ dt / [R(x)]P +1 > C2 WqP(x) inf J | R“ (t) | P 111 P^ dt / [ R^Cx/fOq^]) ]P So that Xn.ptwr'^’ *) & % v/J>M qn'*’'*' %kY\p( I < I P^X[-i,,|| (Odti «/Bqn ). Using Nevai's result. Theorem 1.5.2, we have for | x | < cBq^ (0< £ Xnp(Wf.P;x)> A WpP(x) [q^/n] [ 1+B(q f^/n)(l/|x|) F . B Remark. Using the techniques found in Theorem 3.4.4 the inequality above can be extended to hold for all real x. Now we shall construct the polynomials to estimate W q ( x) for the upper bound. 21 Lemma 1.5.4 . Let xeE be fixed and let n>12. Then there exists a polynomial S^(x;t) and 8>0 such that fo r |x|< Sq^and |t|< Bq^, 0 S^Ct) e P [^ 2 j(t), ii) Sp(x;x) = (x), iii) 0 < 5p(t) WqO) < 5/4, where B Is the constant of Theorem 1.3.1 and q^ is defined by (1.2.3). Proof. Define S,^(x:t)= Wq"^ (x) Rpp( Q'(x) (t-x) ) where m=[n/2] and is defined in Lemma 1.5.1, then i) and ii) follow immediately. For IXI< 8 qp^and |t|SBq^, we have |t-x|<(B+ 8 )qp,; now, since Q' is increasing 1 Q'(x)(t-x) I < Q'(8 qn)(B+ 8 )qn = [ Q'( 8 qn)/Q'(qn) 1 (B+&). Since Q is a Freud exponent I Q’(8 qn)/Q'(qn) 1 < I Q'(qn2-")/Q'(qn) 1 s c^K Thus we can take 8>0 so small that 1 Q'(x)Ct-x) I < Cq"*^ n (B+ 6 ) < n/20 < m/5, therefore, by Lemma 1.5.1 and the convexity of Q, S^(t) < c Wq~^ (x) exp{( Q'(x)(t-x)) < c exp{ Q(t)} = c Wq'^ (t). B 22 Let us proceed to the Proof (Theorem 1.3.3). As before let p>0, fix r such that pr>-l. Then Xpp(WrP;x)= Inf J 17 T(t) | P WpP(t) dt / [tt(x) 1P < 7T^n-l B +Bqn < C| in f J I 7T(t) I P WpP(t) dt / [Tf(x)]P Trel>n-l "BAn Which, applying Lemma 1.5.3, is +Bqn < C| inf j I R(t) Sn(t)w J t ) I P 11 1 Pf dt / [R(x)Sr,(x)]P B^B[n/2 ] -Bqn "Bqn < C2 WqP(x) inf J I R(t) | P 111 P^ dt / [R(x)]P. B^B[n/2 ] -Bqn We apply the same change o f variables as in the derivation of the lower bound to obtain +1 < C3 WqP(x) dpjP^"^ inf J I R*(u) I PI u IP*' du /[R»*(x/Bq^)]P R"fB[n/2 ] 23 So that Xp^pCwrP; X) < Cj WqP(x) qpPr'l X[n/ 2 ),p( 111 «)dl; x/Bq^ ), Once more using Nevai’s result, Theorem 1.5.2, we have for |x|< SBq^ Xnp(WpP;x)< A' w^P(x) [q^/n] [ 1+B(q ^/nXl/1 x | ) F . Q 24 6 . CONNECTIONS TO THE ORTHONORMAL POLYNOMIALS While Freud originally used the property that Q'(q 2 n) maximized X Q'(x), there are other significant relations concerning Lemma 1.6.1 . Let x^p (Wp) denote the greatest zero of the orthonormal polynomial, PpCwpix), and q^ be defined by (1.2.3): then I imsup p_»oo X ,p (Wp)/ qp < const. Proof. From a well known result of Chebyshev (see e.g. Szego[Szl, p. 187]) we have X|n(Wr)= max [ J x tt2(x) Wp(x) dx / J 7 T^(x) Wp(x) dx ]. TfëPp E E Accordingto Theorem 1.3.1 +Bqn J |x| TT^(x) Wp(x) dx < ll+c 1 J |x|tt^(x) Wp(x) dx E "Bqp or "Bqp < 2 [\*c p2n*l 1 Bq,, J jr^(x) w,.(x) dx, -Bqn 25 and the result is seen to hold. S Lemma 1.6.2 . Let r>-1. Then 00 (Wp) = ( n+rAp)"^ Jpp(x) pp_| (x) Q’(x) w^(x)dx, A^=sin^(mT/2). -0 0 Remark. For Q(x)= |x|# Lemma 1.6.2 was proven to r r>0 and #>0 in Freud [Fr5l and fo r r>-1 and ^>1 in Nevai [Ne5]. Proof. First integrate directly 00 00 (1.6 .1) J Pn'(x) Pn-1 (x)Wr(x)dx = J (n x"^'^ +...)p (x)w,.(x)dx — OO - 0 0 CO = J (n(V^n -I )Pn-l Pn-I (x)w/x)dx = n -1 ) -0 0 where Tr^_2 (x)( P^_2 - The last equality holding by virtue o f orthogonality. Now integrate by parts 00 00 ( 1.6 .2 ) J Pp,’(x) pp|_| (x)Wp(x)dx = - J Pp(x) (Pn-I (x)w^(x))' dx —00 -00 26 00 = J Pr|(x) Pn-I (x)Q'(x)w ^(x)dx -00 00 - r J p^(x) Pn_i (x)x‘ ^ Wp(x)dx -00 Since w^(x) is an even weight, is an even/odd polynomial as n is even/odd respectively, therefore 00 J Pn(x) Pn-1 (x)x’ ’ Wp(x)dx = ( V ^ n -1 ) ^n- -0 0 Combining (1.6.1) and (1.6.2), the result follows, g Lemma 1.6.3 . Let r>-l, n>nQ, and 3 p(w^)= 3 ^ - 1 (w^)/3pj(w^), then Aqp,< ap(Wr), where A is an absolute constant. Proof. From Lemma 1.6.2 we have , 00 3p|(Wf)/3p^ _l (Wp) = ( n+rA^)"' J Pp(x) Pp,_, (x) Q'(x) Wp(x)dx -00 Since Q is a Freud exponent, forx>0 27 Q'(x) = Q’(qn) expf log(Q'(x))-log(Q'(qr,))} X = Q‘{%) exp{ J (Q”(t)/ Q'(t)) d t } % X < Q'Cqp,) exp{ J ( c / 1) d t } = Q'(q^) Ix /q ^ l^ % with c being the constant o f ( 1.2 .1,iv), whereupon +00 2fn(Wr)/î^n -1 (^ r) 5 c, ( n+rAp,)"^ Q’(dn) J IPn^x) p^., (x)| |x/q^l^ w/x)dx. —00 We now apply Theorem 1.3.1 to obtain +Bqn < C 2(n *rA n )"’ Q'(qp) J |P n (« )P n -iW | Ix/qnl'^w^W dx. -Bqn So that 'Bqn »n (W r)/» n-I (Wr) < C3 n"' Q'(qp) J |P n W P n -lW | ■Bqn i.e .. 28 < Cj n"' Q'(qn) = Cj / q^. The last equality follows from the definition o f q^. B Proof. (Theorem 1.3.5). The inequality const qp,_| < ap_| < max aj < Xj^ < 2 max aj < 2 Xj^, < const q^ l max aj < X|p, < 2 max aj l 1< q 2 n/An ^ ( Freud [Fr2, p. 22] ) the Theorem holds. B Remark. When Q is an even polynomial o f degree 2m then q^^ ~ and given that Ap(wp) = a^(wp)/(n^^^^ ) has a limit, it is an easy calculation to find the value. Following the method of Freud [Fr3] we integrate / Pn'(x)Pn-i (x)Wp(x)dx in two ways ( as in Lemma 1.6.2, above), we arrive at the recurrence relation fo r a^(Wp) m +00 n + r sin2 (nTT/2 ) = 2 a^ E kd2 k J pp^(x)p^_^ (x)Wp(x)dx k=l -0 0 29 where Q(x) = Z 4 2 ^ x 2 %; now, noting that the "order" of each of the integrals is ~ C 2 |^_j a ^ ^ k '), ( Cj j being the binomial coefficient ) we find llm an(Wr)/(n'^2 m ) = ( 2 m d2 m C2 m-I, m n -*«) which is consistent with the Freud conjecture [Fr3]. For the weight Wr m(x) = IXI ^expt - 1 x| ^ } Freud predicted that lim m ) = [ r(m+l)/( r(m/2 ) r(l+m /2 ) ) ; n -*00 Freud proved that the relation is true for m=2, 4, and 6 . Magnus [Mai] recently established the result for the casewg also Freud's conjecture was discussed for Wq ^(x) ( m >1) in [LuMaNel]. CHAPTER 11 ASYMPTOTICS FOR THE RECURSION COEFFICIENTS an AND bp, ASSOCIATED WITH THE ORTHOGONAL POLYNOMIALS WITH THE WEIGHT w(x) = exp{-Q(x)} 1. INTRODUCTION. Let dcx(x) be any distribution function. Then the polynomials { pp^(docx)} orthonormal with respect to d«(x) satisfy the recursion formula (2.1.1) X Pp(d(x;x) = 3p+| (dcx) Pn+; (docx) + bp(dcx) Pn(d(x;x) + a^(do<) p^-i (d Following the method of Freud [Fr3], this formula can be extended. Let (v)^= (vi, V2 , ..., v^) be a multi-index o f length X with the ordering c < (v)^ ( resp. <, = ) if and only if c < V|^ fo r k=l, 2 ..... X ( resp. < = ). Denote by 30 31 3(v)^(do<) := ay^(d(x) a^^Cdcx)... a^^(do<) and b(y)^(d(x) := b^^(do<) by^(d(x)... by^(d Then (2.1.2) x'^ pj-j(d(x;x) - S ri“ where (2.1.3) 2 a (d(x) b (doc) n-i<(v)^.(u)^ if we repeatedly multiply ( 2 .1.1) by x and iterate (2 .1.1) after each multiplication, we can determine the expressions Ap^^|^(do<). On the other hand, if we multiply ( 2 .1.2 ) by p|^(do<:x), then integrate both sides, we also have an expression for i Freud used the information contained in these two forms to obtain much information about the coefficients ap,(dcx) 32 and bp(d o f 3 p(dcx) fo ra large class of weights [Fr3]. For Q(x)= ( m, an integer ), w(x) is a symmetric weight function and therefore bp,(dtx)= o fo r all n, le. Pp,(dcx;x) does not appear on the right hand side of equation ( 2 .1.1) and the resulting formulas fo r |^(d«) are much less complex. In spite of this apparent simplicity it has required great ingenuity to show the existence of an asymptotic series fo r a^(w) fo r the case w(x)dx = exp{-x"^} dx. In 1982, J. Lew and D. Quarles. Jr. [LeQull showed that an asymptotic series existed fo r ap(w) of the form a^(w) = (n/12) [ I + 1/(24n + 0(n“ '^) ]. Their method is both long and deep: P. Nevai established the existence of this series by a different technique which was also complicated. Recently, A. Mate and P. Nevai [MaNell showed the existence o f an asymptotic series for a^(w), w(x) = exp{ -x ^ / 6 ], as an application of their theorem on asymptotics of smooth recurrence relations. Their technique immediately provides a template to handle weights of the form w(x) = exp{ -x ^ ^ }. In 33 this chapter we will extend their result on smooth recurrences so as to be able to cover the nonsymmetric weights. The existence o f the asymptotic series forap,(w), where w(x)=exp{ -Q (x)} ( Q, a fourth degree polynomial ), will be a direct application of the extended theorem. In a future work we hope to use our extension to show that asymptotic series exist forap(w) and bpj(w) where w(x)= exp{-Q(x)}, ( Q, a polynomial o f arbitrary even degree). The organization of Chapter 11 is as follows; Section 2 is notation: Section 3 contains the statements o f the main theorems: Section 4 is a proof o f the general existence theorem fo r asymptotics; and lastly, Section 5 gives the development of the asymptotics fo r a^ and b^. 34 2. NOTATION In Chapter II we will observe the following notations. For Q(x) = x'^/4 + q^x^/3 + q 2 X^ / 2 + q| x the weight function w(x) will be defined by (2.2.1) w(x) = exp{ -Q (x)}. The polynomials orthonormal with respect to w(x) are p^(w; x) = 2fp)(w) x"^ +... , let the recursion coefficients ap,(w) and b^(w) be defined by ( 2 .2 .2 ) a^(w) = (w) / ^^(w) and (2.2.3) bp|(w) = J X pp,^(w; x) w(x) dx. E By denote the set of all polynomials o f degree less than or equal to n. 35 3. THE MAIN RESULTS The main tool that we use to demonstrate the existence o f asymptotic series is an extension o f tIate-Nevai’s [MaNel] theorem on the existence of asymptotic series for the solutions of smooth recurrences. We also use the result o f Magnus [Ma2] on the first order approximation o f a^ and Theorem 2.3.1 . Let k>0 and m>! be integers. Let H| and H 2 be comp I ex-valued functions o f 2k+3 real variables ( X q ,...,x |^; yg y j,: € ) of which all partial derivatives o f order less than or equal to m are continuous in a neighborhood of the origin 0. Assume k k (2.3.1) E ( 3/9xj)H,(0) =«=0 and ^ 2 ) ( 3 / 8 y;)H2 ( 0 ) % 0 j = 0 j= 0 holds fo r all complex numbers z such that | z | = 1: also assume that (2.3.2) O /8yj)H i(0) = 0 and ( 8 / 0 Xj)H2 (O) = 0 , j=0, 1. ..., k. Let the numbers v^, and with (2.3.3) lim Up = o and lim = 0. n-»oo n -^00 36 (2.3.4) £^= ( (2-3.5a) H{( Up,Up^|,..., Vp ^n+k' ^n^ ~ (2.3.5b) H2 ( ^n’*^n+1 * ' ^n+k’ ^n* ■■■’ ^n+k ' ^n^ ~ ^ fo r n>1. Then there exist constants Cy ^ ,.... c ^ ^ ; c ^ ^ such that m (2-3-6a) "n = E =u,j 'n* * i=I and m (2.3.6b) v^ = E Cyj * o(E^^), j=l moreover, each c^j and Cy j depends only on the partial derivatives of both H,(0) and H 2 ( 0 ) o f order not more than j. The information concerning a^(w) and b^(w) is mostly found by exploiting the two recurrence relations o f Theorem 2.3.2. Let w(x) be defined by (2.2.1) and a^(w) and b^(w) be defined by (2.2.2) and (2.2.3) respectively. Then, fo r n= 1, 2,... , 37 (2.3.7) n = an^l Z * b^., bn^b^^ * qsCbp., «bp) ^ 2 1 and (2.3.8) 0 - (ap|^bp_| +2a^^bp + bp^ + 2ap+] ^bp+8p+| ^bp+j ) * We can now prove the existence o f the asymptotic series for 3 p(w) and for bp(w) by an easy appiication o f Theorem 2.3.1 which yields Theorem 2.3.3. Let w(x) be defined by (2.2.1) and 8 p(w) and bp(w) be defined by (2.2.2) and (2.2.3) respectively. Then m 3p(w) = (n/3)'/4 [ ^ Cj (n/3)"K^ + o((n/3)''^'^^) 1, j= 0 and m bp(w) = [ 2 dj (n/3)"j^^ + o( (n/3)"^/^) ] j=0 as n-»oo for each positive integer m with certain constants Cj and dj. In Section 5 we will the result that Cq = 1 and do = -qg/3 [LuMaNel] to find the first few coefficients of the series above. 38 4. PROOF OF THE EXISTENCE THEOREM FOR ASYMPTOTICS. To begin we need to generalize the lemma o f Mate-Nevai [MaNel] to the case o f two coupled equations; the proof follow s their pattern. Lemma 2.4.1. Let f and g be bounded, complex valued functions on the positive integers, with complex numbers Xj and Pj such that, for a fixed (x<0 , k k (2.4.1) ^ X j f(n+j) =o(n®^) + o (^[ |f(n+j)| + | g(n+j) | ] ) j= 0 j = 0 and k k (2.4.2) Z pj g(n+j) =o(n^^ + o (^[ | f(n+j) | + | g(n+j) | ] ), j = 0 j= 0 k k where P(z) = E XjZ^^O for |z |= l, and Q(z) = ^ pizis^O fo r |z|= l. j = 0 j = 0 Then f(n) = o( n^^ and g(n) = o( n*^. Proof. Since P and Q have no roots o f modulus 1, there exist Laurent series fo r 1/P(z) and 1/Q(z), 39 00 (2.4.3) I/P(z) = ^ i=-oo and 00 (2.4.4) t/Q(z) = Z i=-oo absolutely convergent In the annul us r“ ' <|z |< r (r >1). By the uniclty of the Laurent expansion, I if i=0 (2.4.5) j = 0 and 1 if>e=o (2,4.6) Z P jb ^ .j- j= 0 0 if For negative k set f(k) and g(k) equal to 0 to extend f and g. First, consider f(n). From (2.4.1), for large positive n. k k I Xj f(n+i+j) =o(n^^ +o(S[ |f(n+i+j)| + |g(n+i+j)l 1 ) j=o j=o 40 fo r every Multiply by a j and sum both sides fo r -«>o 00 OO k (2.4.7) f(n) = ^a^o(n^ + o( E [ |f(n+j)| + |g(n+j)| 1 ). X=-00 i=-oo j=0 The first sum on the right of (2.4.7) is o( n°9; the second, fo r | i | > n/2, is 0 ( (r+l) / 2 = o( n° 9 , using the boundedness o f f and g and the convergence Laurent series. Therefore, fo r any £>0, there exists an Ng, such that, forn>N j, I f(n) I < £ n^ + (£/2 ) max |f(n+i)| + ( e/ 2 ) max |g(n+i)|. “ (n/2 ) Define F(x):= sup{ | f ( i) | : Jt>x, xeE} and G(x>= sup{ |g(4)| : Jt>x, xeE}; hence F(x) <£X^ + (e/2 ) F(x/2 ) + (e/2 ) G(x/2). Similar analysis gives G(x) Let M(x)= max{ F(x), G(x)}. Then, with x>N,, 41 M(x) < E x^+ € M(x/2). Iterating this inequality, with x replaced by x/(2'*) fo r 0 q = IIlog 2 (x/Ng)D ( with BtD being the greatest integer function), gives q M(x) < ^ (x/2 '*)^ + E^*) M(x/2 ^*^ ). i = 0 Since M is a decreasing function for x>0, 00 M(x) < (% [' 2 ' ^ ) £X« + (4+1 11(0 ). Â=0 Now, since e is arbitrary, fo r a fixed £, we see that e^+1 = 0(x'®92£), and so M(x) = o(n*^ and the result follows, g We follow the method o f Mate-Nevai IMaNel] closely for the Proof (Theorem 2.3.1). Hj(0) = 0 is seen by applying (2.3.3), (2.3.4) and (2.3.5a, b). Therefore, by Taylor’s formula, with (u:v:E^) = (Un> ^n+k’ ^n' •••' ^n+k' 0 - Hj( Upj ^n+k’ ^n' ^n+k' ^n ^ ” 42 m- 1 k = Ï (l/i!) UnO/StM (Un»jO/8 i(j)* Vn*jO/3yj))]-*Hi(0) i= l j = 0 k + (1/m!) [cp,( 8/0E)+E ( Un+j( 3/9Xj) + Vp,^jO/9yj) ) Hj(eu: 0 v;ee^) j=0 fo r some 0 6 (0 ,1), and provided that n is so large that (u; v: £^) belongs to a convex neighborhood of 0 where Hj has continuous partial derivatives of order m. So, we see that m k Z 0/-»!) [£n(a/3e)‘Z(Un*j(9/3xj).Vn*j(8/3yp)]^H|(0) i =1 j=o = (I/ml) ( 9/9E)+^ ( Un+j(9/9xj) + v^+j( 8/9yp ) P j=0 x[Hj(0; 0; 0) - Hj(0u; 9v; 0£^) ], i.e., m k (2.4.8) I (l/il) [en(3/3£)*Z:(Un,j(3/3Xj)^Vn*jO/0aj))]'‘ Hi(O) i =1 1 = 0 43 k = o(En^ + Z[ kn+jT]). 1=0 where the error function depends on k, m, and the order partial derivatives of Hj near 0. We will now use induction onm to establish the asymptotics. Assume that (2.3.6a, b) hold with m replaced by m-1, i.e. m-1 (2.4,93) Un = I c^i i Cn' * i=1 m-1 (2.4.9b) Vn = z Cv,i En' * «v,n' i=l and where (2.4.10a) £„ n = o( En'^"' ), (2.4.10b) Since u^, v^ 0 as n -» « , (2.4.9a, b) and (2.4.l0a, b) hold fo r m=1. We substitute (2.4.9a, b) into (2.4.8) to obtain 44 (2.4.11) m k m-1 m-1 ^(1/ ^!)[E^(9g)+2!{ (S^u.i ^n+j '^^u,n+j '‘^^v.n+j )(%,] ^ i =1 j=0 i = l i = l k m-1 m-1 X H | ( 0 ) = o(£n'>'*Sl |Scu,i£n+i'*5u,n*j T * l2cv,iV j'**v,n *J l"" > ) j=0 i=l i=l k = o ( ) • o( i=0J: [ |5un*j I ♦ |5v.n*jl > '>■ Where the follow ing abbreviations are used: 9^= 9/9e ; 9^ j = 9/9xj: a n d 9y j = 9/9yj. The second term of the right hand side o f equation (2.4.11) is seen to be needed only when m=1. Rearrange (2.4.5) to yield (2.4.12) m k k m-1 2(1/ i!) [£n(9e)+Z(S j n+j ^x.j ^^v,n+j ^ ^n+j ' (^ ,i ®x,j *^v,i% j Jt =1 j=0 j=0 i=1 k « H|(0) = o( E^m ) . o( ^ ( 15^^^. I . 1 I 1 ) j=0 45 Denote the differential operators A^+jj == (Syn+j \ j * ^v,n+j 9y j ) and fij = (Cu,i\j ♦CviSyj) to obtain m k k m-1 (2.4.13) I (1/ iOlEpOt) ‘SAn*j,j G r,j Cp*! ')!'*’■ H|(o) À =1 j=0 j=0 i=1 k = o( ) + o( ^ [ I Sy.n+j I * I ^v,n+j I ^ ) 1=0 Noting that (2.4.10a, b) gives both . 8^ ,^+j = o(1) and both 8y , £n Sy n+j = o(€^'^). also noting that any term with or 8^ a power greater than 1 is o(€^^), we can find j depending only onT|^ j and the k^^ order partials of and H 2 for k ^'m.i- Then m-1 k (2.4.14) Z Ep^ * C'p, i Cp^ ^ Z Ap*J,j H|(0) = o(Ep"') i= l j=0 Applying (2.3.2) to (2.4,14), we see that 46 m-1 k (2.4.l5a) E * C'm,, S '" * % «umj \\ ", <«) i= l j=0 k = o( ) + o( ^ [ I Sy pi+j I + I&Y n+j I 1 ) j=0 and m-1 k (2.4.l5b) E G^,2 * ®'m,2 S '" ' Z 5v,n.j % ,j " 2 C) i= l j=0 k = o( ) + o( ^ [ j 8y n+j I + 1 Sv,n+j 1 ^ )' j=0 But Sypi+j andS^f^+j are o(£p*^"’ ). so it follow s that j=0 for 1< i k (2.4.16a) ^n"" + Z \n + j9 x ,jH l(0 ) = j=0 k - o( ) + o( ^ [ 15y p+j I + 1 Sy p+j I 1 ) j=0 47 and k (2.4.15b) T3 ^ 2 ^n*^ ^ ^ ^v,n+j ^y,j ^2^^) j=0 k = o(E^^) + o(El 18u.n+j I " l^v.n+j I ^ )• j=0 Define k *^,m " " ^ m,l ^ 2 9xj Hj (0)} j=0 and S/.m " ■ ^ ’m.2^ ^ S ®y,j *^2^®^ j=0 and define m " ^u,n ‘ ^,m^n and g(n) = - Cy ( The above denominators are not zero by hypothesis (2.3.1). ) 48 As was seen in Mate-Nevai [MaNei], the theorem follow s if we can establish that f(n) and g(n) are o(E^^). Using our definitions (2.4.16a) gives k E I * c, m( En.j"’ - ) B x j H,(0) J=0 k - 0( ) + 0( ^ [ I Sy p+j I + I p+j I ] ) j=0 With Xj := 9^ j H,(0) and (2.3.4) (which implies Ep+j^ - £p*^= o(£p^)), the above equation is k k Z Xj f(n+j) = o(Ep"i) + 0 (2 : [ 15y n+j I + l&v,n+j I ^ )• j=0 j=0 In an analogous manner, with ]ij ;= 9y jH2(0), we arrive at E Mj 9(n*J) = * 0(2 [ |«u,n.j I * I &v,n.j I I )• i-0 j=0 In view o f the definition o f f(n) and g(n), 49 k k o(Z[ I&u,n+j I + 18v,n+j M ) = + °(Z[ If(n+j)| + |g(n+j)| 1 ); j=0 j=0 thus k k Z f(n+j) = o(E^"^) + o( 2 [ I f(n+j) I + I g(n+j) | ] ). j=0 j=0 and k k Z jij g(n+j) = 0 (E^^) + o(Z [ I f(n+j) I + I g(n+j) | ] ). j=0 j=0 Recall that the conditions on 2 9x i Hi(0) and 2 9u ; H2(0) give us the j=0 ’ ]=0 k k property P(z)= ^ Ajzî and Q(z)= ^ |ijz i have no zeros or moduius I. j=0 j=0 Now we apply Lemma 2.4.1, the generalization of the lemma of Mate- Nevai, to the above equations to obtain f(n) = o(E^"^) and g(n) = o(e^^): and the theorem is proven. S 50 5. THE ASYMPTOTIC SERIES OF THE RECURSION COEFFICIENTS. Let w(x) be defined by (2.2.1) fo r x real and where Q(x) = x^^/4 + qjX^/3 + q2^^/2 + q| x. The polynomials orthonormal with respect to w(x) have the three relations found in Property 2.5.1 . By recursively applying the recursion fo r Pp, we have X Pp,(w; x) = Pn+] (w; x) + bp Pp(w: x) + a^ Pp_, (w; x), and Pn(w: x) = Pp+2 (w: x) [ 8^+, 3^+2 1 Pn+i [ 3p+| 3p+2 ( l^n "^^n+l ^ ^ ^Pp(w:x) [ap2.ap„2.bp2] + pp_,(w: x)[3p(bp_, +bp)] + Pn-2(w: x) [ 3n-l ^n 1- and «3 p^(w; x) = Pn*;(w; x) I apt, a ^ 2 ^0 * 3 > * 51 * Pn.2(w; «) 13n*l ( "n * * b^2> > * Pn*, (w; X) [an*, (an^^ap*, V n *l * ‘>ml • Pp(w; x) |an^(bn_, *2Pp) »bp3 ^bp., 2(2bp»bp*, )1 * Pn-i (Wi X) lap (ap_, 2*ap2*ap*, 2^ bp., 2* bp ., bp* bp2)l * P p _ 2(w ; x) I ap _2 ap_, 3p ]. The recurrence relations fo r a^Cw) and b^(w), (2.3.7) and (2.3.8), are Theorem 2.3.2. These are the "Freud equations" o f Lubinsky-Magnus-Nevai [ L u M a N e l ] . Proof. (Theorem 2.3.2). We w ill integrate Jpp,'(x)Pp^_^ (x)w(x)dx in tw o ways. First, since Pp'(x)=n2fp,x'^"^ + TTp_ 2(x) where TTp _2 e P p _ 2> w e h a v e J Pn'(*)Pn-l (x) w(x)dx = (n/a^) J[P p_] (x) + 7Tp_2(x)] Pp_, (x) w(x)dx. R £ Now, by the orthogonality o f Pp_, to Pp_ 2 , 52 J Pn'WPn-1 w(x)dx = (n/a^) ^(x) w(x)dx = n/a^. E E Second, WG shall integrate by parts. J Pn'(x)Pn-i W w(x)dx = J p^(x) (Pp_, (x) w(x))' dx. E E = - JpnW Pn-1 '(*) w(x)dx - Jpn(x) Pp-i W w'(x)dx. E E The first integral of the right hand side is 0 by orthogonality. Since w'(x)=-Q'(x) w(x) we see that J Pn'(>()Pn-l (*) w(x)dx = J Prj(x) Pp., (x) w'(x)dx. E E = J x ^ p^(x) Pp^_, (x) w(x)dx + d3 J x ^ Pp^(x) p^_, (x) w(x)dx E E + ^2 / ^ Pn(^) Pn-1 w(x)dx + q, Jpr,(x) Pn_; (x) w(x)dx. E E The equation (2.3.7) follow s from Property 2.5.1 applied to the integrals. 53 Now, we will prove (2.3.8) by integrating J(pp2(x))’w(x)dx in two ways. Directly J (Pn^(x))’ wCx)dx = 2jpf^(x) Pp’(x) w(x)dx E E = 2 J Pp,(x) TTp., (x ) w (x)dx = 0 E As before, integrate by parts to obtain J (Pn^(x))’ w(x)dx = Jpn^(x) Q'(x) w(x)dx E E = Pn^(x) w(x)dx + q 3 J x ^ Pn^(x) w(x)dx E E + (\2 ^^ Pn^(x) w(x)dx + q, Jpp^(x) w(x)dx. E E As before, we conclude (2.3.8) from Property 2.5.1. B Remark. Freud [Fr3] was the first to apply the recurrences above to gain information about a^, and bp for exponential weights; he found the relations 54 using the method we followed above. However, using the technique of Mate-Nevai-Zaslavsky [MaNeZal], we can find the expansion o f x^p^(w: x) directly fo r any integer k without iterating the recurrence. We have need of a result of Lubinsky-Magnus-Nevai [LuMaNel] that gives a first approximation to the behaviour o f a^(w) and b^(w). Theorem 2.5.2 (Lubinsky-Magnus-Nevai). Let w(x) be defined by (2.2.1) and a^(w) and b^(w) by (2.2.2) and (2.2.3) respectively. Then a^(w) = (n/3)’/ '’ [ 1 + 0(n"i/2(log n)^/^) I and b^(w) = -q^/3 + 0 (n“ ’'^2 (|og n)'^^ 2 ) V/e will first transform (2.3.7) and (2.3.8) into a form suitable for the application o f Theorem 2.3.1. Lemma 2.5.3. Let a^(w) and b^(w) be defined by (2.2.2) and (2.2.3). Put u^:= a^(w)/(n/3)'/4_ b^(w). and E^:= (n/3)"'/2 Then (2.5.1) 0 = -3 * ( Un-i ^Un^Ci-1^2/3)'^2 * ♦ Un^u^., Z(|. ’ V l V ''n ^ ) > * V l *Vn ) * 55 and (2.5.2) 0 = Un^v^,, « 2 u ^ \ ♦ EnV * \(l' * “ n*1 ^V l ('* ^n^/3)'/2 * qj( * u^,, E^^/;),/; ) * «n< < '2 V A| )' Proof. Elementary calculation. D Remark. Recall the weight w(x) is defined by w(x) = exp{-Q(x)} = Bxp{ -(x ^/4 +qgX^/3 + qj ) ). Without loss of generality, the coefficient q^ of Q(x) can be taken to be zero. The translation x = t-(4/9)q^ eliminates q^ and gives R(t) = t^ /4 + r2 t^/2 + r; t where r^ and r 2 are constants determined by q |, q 2 , and q^, and with rg = “ (qj/243)[ (128/27)q - 24q2q^ 108q ^ 1, then Pn( exp{-Q(t)}; x) = exp{-rg/2} Pn( exp{-R(t)}; x+(4/9)q3 ). Thus we shall take q^ = 0 in the proofs,but we shall keep q^ in the statements of results for completeness. 56 We are now in a position where we can prove Theorem 2.3.3 ( cf. [MaNel] ). Proof. (Theorem 2.3.3). Put (u: v; E^) := (Up, Up+|^: v^,,..., Vp,+|^; 6^,). Let H,( u; V; E^j) be given by the right hand side of (2.5.1) and H 2 ( u; v; Ep,) by (2.5.2). it follow s immediately that u^, v^, and Ep^ solve the recurrences H,( u; V; Ep) = 0 and H2 ( u: v: £^) = 0., where we have shifted from the origin 0 to the point p = (I, I, 1: 0, 0, 0: 0) since u^-»l, Vp,-+0, and £p|-»0 as n-*w according to Theorem 2.5.2. Let Xj ;= Up,+j. yj " Vp,+j for j=0, l, or 2: and let € :: €|^. From (2.5.1) and (2.5.2) we see that the conditions (2.3.1) and (2.3.2) are satisfied at the point p as we have 2 'Z (9/9xj) H,(p) dXj = 2dXo + 8dx, + 2dX2 j=0 and 2 Z (9/3yj) H 2 (p) dyj = dyo + 4dy, + dy 2 ^ ]=0 with the coefficients of dx, and dy, greater than the sum o f the others. 57 Also we note that (9 /9 y j) H,(p)=0 and (9 /9 x j) H2(p)=0, j=0, 1, or 2. Thus, by Theorem 2.3.1, and have asymptotic series, i.e. fo r each integer m = (n/3)’'^"’ [ Co + c,(n/3)"i/^ + C2(n/3)"’ +...+ c^(n/3)"'^^^ + o(n“^^^)] and b„ = [ do + d,(n/3)"'^2 ♦ dzCn/J)"' d,n(n/3)‘ '^''2 ♦ o(n‘ '"''2) ). g Now that we have proven the existence o f these series we can set about the task of determining the values o f q and dj. Lemma 2.5.4. Let u^, and v^ be solutions of (2.5.1) and (2.5.2) with e^=(n/3)"'-^2 Up = 1 + (l/12)(q 3Z_3q^)(n/3)-i/2 + (i/288)(q 3^-3q2)^(n/3)"i + 0(n"3/2) and Vp, = -q^/3 - (1/162) [ 2q^^ - 9q| q 2 +27q] ] (n/ 3 ) " ' ' ' 2 + (1/1458) [ d3^-3q21 [ 2q^^ - 9q| q 2 +27q] ] (n/3)"’ + 0 (n"^''2 ) Proof. Rewriting (2.5.1) with = (n/3)"’/2 gives 0=-3+ U^2(Up,_; 2(|-l/n)1/2 + u^2 + 2(j+;/^y/2 + ^+V^_, Vp+Vp,2))d 58 + (n/3)"i/2 qg(v^_, +v^) + (n/3)‘ ^/2 Set (2.5.3a) Up, = 1 + c, (n/3)"’/2 + C2 (n/3)"* + 0(n"3/2) and (2.5.3b) Vp, = -q^/3 + d, (n/ 3 ) " ’ ^ 2 + (n/3)"' + 0(n~2/2). Then we note that Up,+| -Up. Up+; ^-Up^. Vp+| -Vp, and Vp+, ^-Vp^ are all 0 (n~2 / 2 ) since [(n+j) ~®-n“ ®l= 0(n"®"^ ). So 0 = -3 + Up'^I (1-1/n)i/2 + 1 + (i+i/n)i/2 ] + Up2 (n/3)"’/2 [ 3 v ^ 2 + 2d3Vp + d 2 1 + 0(n"3/2). Now (l±l/n)'/2 = iii/(2 n ) + 0(n~2), thus 0 = -3 + 3Up"^ + Up^ (n /3 )"^/2 [ 3v^2 + 2q3Vp + q2 1 + 0 (n"^/2 ) If we use (2.5.3a, b), then 0 = -3+ 311 +4c, (n/3 ) " ' / 2 + (6Cj2+4C2) (n/3)"' + 0 (n~^'^2 ) ] + [ 1+ 2c, (n/3)-’/2 + (2c2+c,2) (n/3)"' + 0(n"3/2) ] x [ (n/3)‘ ’/2 ] X [ 3{ q 3 2 / 9 - (2/3)d,d3 (n/3)‘ '/2 + (d,2-(2/3)q3dz) (n/3)'» + 0(n‘ 3/2)} + 59 + (-2/3)q3^ +2diq3 (n/3)"^/2 + 2d2q3 (n /3 )" ’ + 0 (n"^/2) + ] Simplifying 0 = 4c, (n/3 ) " ' / 2 + (6c,2+4C2) (n/3)"’ + [ (n/3)"’/^ +2c, (n/3)"’ ] x X [ (qg- (l/3)q 3 2) + 3d,2 (n/3)"’/2 ] + o(n"3/2). Comparing terms of like orders, we have that c, = (1/12) (q3^-3q2) and 02 = (1 /2 8 8 ) (qgZ-Sqg)^. We will now turn to the determination of d, and d 2. Putting = ( n / 3 ) "’'^2 jp (2.5.2) yields 0 = U p^Vl * 2 U p \ + (n/3)"’/2v^3 + 2Up+| ^Vp(l+l/n)’/2 + Up+j ^Vp+, (M /n )’/2 + q^( u^2 + (n /3)"’/ 2v ^2 + 2 ) + (n/3)"’/2q2Vp + (n/3)"’/2q^ . As noted above, Up+j -Up, Up+j ^-Up^, and Vp+; -Vp are all 0 (n"^/2) hence 0 = Up2 [ 3Vp(l+ (1+l/n)’/2 ) + qg(|+ (i+i/n)’/2 ) ] + 60 + (n/3)"i/2 [ + q| 1 + 0(n"2/2) Using (l+l/n)’'^^ = i + l/(2n) + 0(n“ 2), we obtain 0 = I 3Vn (2 + l/(2n) ) + (2 + l/(2n) ) ] + + (n/3)"’/2 [ v^3 + q^v^2 + q^y^ + q, ] + 0 (n’ ^ / 2 ) Apply the equations (2.5.3a, b) to see Vn^ + qsVp^ + q 2 Vp + q; = i (2/27)q3^ - q2q3/3 + q, ] + d, (q2-q32/3)(n/3)‘ i/2 + dg (q2-q3^/3Xn/3)-' + 0(n'3/2) and 3Vp(!+ (1+l/n)'/2) + qg(i+ (i+i/n)’/2) = 6d, ( n / 3 ) -1/2 + 6dg (n/3)"' + 0(n"3/2). Thus 0 = 6 (n/3)->/2 [ d, + (2c,di+dg) (n/3)“ l/2 ] + I(2/27)q3^ - q2q3/3 + q; ](n/3)"’/2 + d,(q2-q3^/3) (n/3)” ’ + 0 (n"^^2 ) Comparing coefficients and recalling c, gives d, = -(1/162) [ 2d3^ - 9q, qg +27q, ] and 61 d2 = (1/1458) [ q3^-3q2 ] [ 2qg^ - 9q^ q 2 +27q; ]. B Remark. It is easy to see that, by extending the series fo r Up, v^, and (1+1/n)^/2 _ wg can find any number of terms of these expansions by simple (if somewhat tedious) calculations. In Sheen [Shi] a computer algorithm was discussed in the case o f w(x) = exp{-x^/6) fo r the approximation of the coefficients found above. Chapter III ASYMPTOTICS FOR THE ORTHOGONAL POLYNOMIALS A550CIATEDWITH w(x) = GXp{ -Q(x) } 1. INTRODUCTION. There are three facts about w(x) = exp{-Q(x)} that are obvious when QCx) is an arbitrary polynomial. First, the support of w(x) is not compact: second, the weight w(x) is not even, moreover w(x) is not ( except in special cases) symmetric about any point on the real axis. These facts together make up the significance of the results o f this chapter - the results are for a nonsymmetric weight having noncompact support. A large amount of information is known about the classical polynomials on infinite intervals, Hermite, Laguerre, etc., (e.g., see Rainville [Rail and SzegolSzl] ) but this information is mainly due to the existence of specific differential equations, and generating functions, involving special functions fo r these polynomials; all being items that do 62 53 not generalize. The present investigations were initiated by the results o f P. Nevai [Ne4]. In 1983 Nevai found asymptotics for the polynomials associated to w(x)=exp{-x‘^} by improving the Liouville-Steklov method ( see e.g. [Szl] ): Theorem 3.1.1 (P. Nevai.) Let Pp be the orthogonal polynomial with respect to w(x)=exp{-x4) and let a^ = /2fp where 2fp>0 denotes the leading coefficient of pp(x). Then there exists a positive number A such that on every fixed interval A exp{-x^/2)pp(x) = A n "’/8 sin{4ap[ap_| 2+ap2)i/2 2^q^2]\/2 X [ X + nx^/(24ap[ap_) [ap+; ^+ap^]'/2) ] - (n-l)TT/2 } + o(n"’^®). After the determination o f the asymptotics for a^ by Lew and Quarles [LeOul], Nevai was able to refine his result to the form [Ne4, (28) p. 2801 cxp{-x'’/2}pp(x) = An"i/8 sin{ (64/27)^/4n3/4 ^ + (n/12)’/'^ x^ - (n-1)Tr/2 } + n"^/® o(1). In 1984 R-C. Sheen, in his Ph.D. dissertation [Shi], derived asymptotics for the polynomials associated to w(x) = exp{-xV6} using the same methods. 64 Both cases relied on the evenness of the weight function. In this chapter we will show that Nevai's improvement o f the Liouville- Steklov method can be extended to the nonsymmetric cases as well and obtain Theorem 3.1.1 as a special case o f Theorem 3.3.1 below. The organization of the chapter is as follows: Section 2 contains our notation; Section 3, the statements of the main results; Section 4, the derivation of the preliminary estimate o f Pp(x): Section 5, the proof of the asymptotic of pp(0); and lastly, Section 6 contains proofs of the asymptotics fo r Pp(x) when x is in a fixed interval. 65 2. NOTATION AS in Chapter 2, let Q(x) = x'^/4 + q^x^/3 + Q2X^/2 +q; x and define (3.2.1) w(x) = Bxp{ -Q (x)}. The polynomials orthonormal with respect to w(x) are Pp(w: x) = 2fp,x'^ +... , and satisfy the three term recurrence (3.2.2) X pp,(w: x) = a^+; p^^+i (w: x) + p^(w: x) + a^ pp^_, (w; x) where the recursion coefficients a^(w) and b^(w) are defined by (3.2.3) ap(w) = V l (w) / V w ) and (3.2.4) b^(w) = J X Pp^(w: x) w(x) dx. E We will often use the result o f Lemma 2.5.4; for each integer m (3.2.5) = (n/3)'/"* 11 + Ci(n/3)"'/2 + c2(n/3)"' + ... + c^(n/3)“*''^^^ and 66 (3.2.6) = (n/3)“^'^2 [ çjj + d2(n/3)"^'^2 + + d^(n/3)"^^^ + ) ] where the C; and dj are constants. The Christoff el function of w(x) is Xp,(w: x) which is the reciprocal of the kernel K,^(x). that is. n~l (3.2.7) I X^(w: x) = Kp(w; x) = ^ P^^(w; x) k=0 We shall follo w the usual custom of supressing arguments such as p^ = Pp(x) = Pp(w:x) when the meaning is clear from the context. 67 3. THE MAIN RESULTS In order to estimate Pp(w; x) at x=0 first we need to bound p^(w: x) for anintervai. The endpoints o f this interval are essentially plus and minus the greatest zero o f Pp(w; x). Theorem 3.3.1 . Let 0<€ (3.3.1) I pp(w: x) I < c n"'/8 w” ’/2(x) for n=l, 2 ..... and |x|< 2e(n/3)’/ ^ x real. We next find the behaviour o f pp(w: x) at x=0 using Nevai’s method from Mate-Nevai [MaNe2] and Mate-Nevai-Totik [MaNeTol], This technique was used to find asymptotics when the recursion coefficients a^ and bp had finite limits. Here we extend the method to the case where 3p and bp have asymptotics series. Theorem 5.3.2 . Let pp(w: x) be the orthonormal polynomials associated with the weight w(x) defined by (3.2.1). Then there exists constants A>Oand c, independent o f n. such that 68 (3.3.2) pp(w; 0) = Acosl(mT/2)- 6di (n/3)^/"^ + c ] + n"^'^®o(l) where d, = (9/162)[ qj q 2 - 3q| ] ( the coefficient o f (n/ 3 ) ' ’ ^ 2 jp the asymptotic series ofb^, cf. (3.2.6) ). The next step in the analysis is to generate the differential equation that describes Pp(w: x); Shohat [Shol] first used this method, it was then rediscovered by Nevai in [Ne4l. The differential equation is derived from the recurrence formula. Define (3.3.3) fp(x) = + ap+; ^ + bp^ tPpq^ + q 2 +xbp + xq^ and (3.3.4) 'j'p(x) = bp_j + bp + q^ + x. Theorem 3.3.3 . Let z(x) = Pp(w: x) [ w(x)/ (3.3.5) z"(x) + [ (-3/4)[ - (Q ’(x) / 2 ) 2 + [fp ”(x)/ + ap2li+(p p_^ (x)fp(x) - 'pp(x)(Q'(x) +fp'(x)/ Now we are ready to state the main estimate o f this chapter - the asymptotic expression forp^Cw; x) fo rx in a fixed interval, (but first recall that q and dj are from the expansions o f a^ and b^, (3.2.5) and (3.2.6) respectively, while Qj Is from Q(x) ). The technique used is the improved Liouville-Steklov method developed by Nevai [Ne4l. Theorem 3.3.4 . Let p^(w; x) be the orthonormal polynomial with respect to the weight function w(x) = exp{-Q(x)} defined by (3.2.1), and let a^ and b^ be the recursion coefficients o f Pp,(w: x). Then there exist constants A>0 and c ( the constants o f Theorem 3.3.2 ) such that on every fixed interval A w’/ 2 (x) p^(x) = A n “ ’/® sin ^ 2 x(n/3 ) ^ / 4 + [(6 c,+q 2 )x +x^ / 2 - 6 d,] (n/3)’/'^ + njT/ 2 + c } + n"''^®o(l) where iimp,_»j^ o(l) = 0 uniformly for x ç A. 70 4. A PRELIMINARY ESTIMATE OF D^fw: x). S. Bonan's result that the polynomials orthogonal to w(x) = exp{-x'^} form a generalized Appel) sequence was one of the motivating factors in the research of polynomials with exponential weights on infinite intervals (see [Ne4] ). For the present case we have Lemma 3.4.1 . Let w(x) = exp{-Q(x)} be defined by (3.2.1). The polynomials p^(w; x) orthonormal with respect to w(x) satisfy (3.4.1) Pn'(w: x) = (n/a^) pp_| (w: x) + a^_; ar^(bp_ 2 +b^_, ^bn+qj) Pn_ 2 (w:x) * ^n-2^n-l ®n Pn-3^'^’ Proof. Expand pp'(x) in a Fourier series n-1 Pn'(x) = E C|(Pk(x) k=0 where = J Pn’(x)p|((x) w(x)dx. We integrate by parts to obtain 71 C|( = J Pn(x)P| I.e. C|ç = J Pn(x)P|ç(x) w(x)dx + p^(x)P|((x) x^ w(x)dx R E + ^2 J x' w(x)dx + q, J pr,(x)p|^(x) w(x)dx. E E For k The main estimate o f this section requires that we express Pp,'(x) in terms of Pp(x) and p^_] (x). To do so we will apply the recurrence relation to (3.4.1) to derive Lemma 5.4.2. Let p^(w; x) be the polynomials orthonormal with respect to w(x). Then fo r n=l, 2,.... (3.4.2) Pn'(w: x) = t^(x) p^(w: x) + ar^‘pp,(x) Pp,_, (w; x), 72 where (3.4.3) «l^pCx) = bn_, + + X and (3.4.4) 9n(x) = " 4n+) ^ ^ ^2 " " *^3 Proof. We apply the recurrence relation aj^+j P(^ = - ^k+Z^k+Z * (x-b|^+j ) P|^+) to (3.4.1), first with k = n-3, then with k = n-2, and see Pn'(«) = -an^I*=n-l PpW * -"n -, **1 Pn-I (*) Replacing n/a^^ according to (2.3.7) gives the result. B We need a theorem o f Lubinsky [Lu2] that bounds the Christoffel functions o f w(x) (cf.. Chapter I), but first, the relation o f the greatest zerox] p, o f Pp(w; x) to is required ( also see Freud [Fr5] ). Theorem 3.4.3 . Let the greatest zero of p^(w: x) be X| . Then (3.4.5) limp_)ca Xj p, / (n/3)^/"* = 2. Proof. Multiply both sides o f the recursion (3.2.2) by P|^(x), then sum fo rk from 0 to n: setting x=x^ ^ obtains 73 n n-1 2/u Z (*l.n -D|() Pk^(X|,n ) " 2 E ak-i Pk+, («i.n > Pk(*l,n ) k=0 k=0 By the Cauchy inequality n n-1 n-1 11/2 2( Z(X|,n-«k)Pk (X|,n)^2 Z P k 4 ^P k 4 ^(X |,n ) Z Pk^(*t,n ) k=0 k=0 k=0 which Is n n-1 Z(«l,n-VPk^(*l.n)^2 max3^„ EPk^(«|.n)- k=0 k Hence n n-1 X|,n ZPk^(*t,n)^ (max h kl *2 man aj.,, 1 EPk^(ni,n). k=0 k Xj p, < max |b|^ I +2 max a|^+| k ( cf. Nevai [Ne3, Lemma 1, p. 201 ). Now we have, according to Theorem 2.3.3, (3.4.6) lim s u p r^„ x^ / (n/3)^'^'’ < 2. 74 Fix a positive integer m. Define R(x) as in Nevai [Ne4, p. 276], a nonnegative polynomial o f degree 2n-2, by n-1 R(x) = 1 Pk(x) k=n-m Apply Gauss - Jacobi quadrature: (3.4.7) J X R(x) w(x)dx < X| p, J R(x) w(x)dx = m X| £ £ On the other hand, expanding R(x) from its definition. J X R(x) w(x)dx = £ n-1 n-1 n-2 = Jx[ 1 Pk^(x) + Z P|(WPk-| (x) + Z PkWPk+1 (x) ] w(x)dx, k=n-m k=n-m+l k=n-m which n-1 n-1 n-2 Sb|< ^ Z a j, * Z a k .i k=n-m k=n-m+1 k=n-m Theref ore (3.4.8) JX R(x) w(x)dx > m min b^ + 2(m-1) min £ n-m Put (3.4.7) and (3.4.8) together and divide by m; > min b|( + 2(1-1/m) min a,^ . n-m Again in view oT Theorem 2.3.3, lim infp_>oo X| p / (n/S)’/*^ > 2(l-l/m); let m approach infinity. Then (3.4.9) lim infp_^oo ^ / (n/3)'/"* > 2. The result follow s from (3.4.6) and (3.4.9). B We are ready for Lubinsky's result on the Christoffel functions of w(x). It is stated here in the form specific for our weight, see [Lu2l for the statement o f the general theorem. The kernel function Kp(w; x) is defined by (3.2.7). Theorem 3.4.4 (Lubinsku). There exists a constant C independent o f x and n such that Kp(w: x) w(x) < C n^^^ ( xeE ). Proof. We set ( using Lubinsky’s notation ) j=0 and p=2 and note that for w(x) = exp{ -Q (x)} we have n'^^. Then as a special case o f [Lu2, Theorem 3.1 (i) 1 there exists an 0<£ (3.4.10) Xp(w; x) = [ K^(w: x) f ' > A n"3/4 w(x) ( |x|< eB(n/3)^^"^ ) fo r an absolute constant B. Let k be a positive integer, if tt( x) is a polynomial o f degree at most n-1, then the definition of the Christoffel function Xp,(x) yields I 7T(x) I ^ w(x) < (x) w(x) J TT^(u) w(u)dU E which is (3.4.11) I t t( x) I 2 w(x) < A ^ k n^/4 J ” tt2 ( u) w(u)du E fo r IXI < cB(kn/3)^/'^ by (3.4.10) above and since k^^'^ < k. Choose k so large that k4 > i/ 1 , i.e., EB(kn/3)i/4> B(n/3)i/4 f or n=1, 2,..., then (3.4.11) yields (3.4.12) fiTT^(x)| < A"^ k n^/4 j Tr^(u) w(u)du Loo(w;[-B(n/3)^/4B(n/3)i/4]) R But the right side of (3.4.12) is independent of x. Then Theorem A of Lubinsky [Lull may be applied to give 77 Itt^Cx) ! < 2A“ ' k j 7T^(u) w (u)du U (w ;E ) E fo r large enough n. Hence fo r all polynomials tt( x) of degree less than n and fo r n suitably large J Tf^(u) w(u) du / 7T^(x) > [A/(2k)j w(x) ( xeE ) E and the theorem is established. S We are now in a position to derive the main estimate o f this section. Proof (Theorem 3.3.1). Following the remark o f Section 2.5 we take d3 = 0. The Christoffel-Darboux formula ( see e.g. SzegoiSzl, p. 43] ) is Kr^(x) / a^ = pp’(x) p^_, (x) - p^_, '(x) Pp(x). Replace Pp,’(x) and Pp_| '(x) according to Lemma 3.4.2 and Kn(x) /8p = - ap2f^(x)pp(x)pp_, (x) + S n W ^n -l + 3p_, (x)Pp(x)Pp_, (x) - 3p_| (x)Pp_ 2(x )P p (x ). Rewrite 3p_| Pp_2 (x) = -apPn^^) + (x-bp_; )Pp_; (x) to have Kp(x) / 3p = 3p (pp_j (x) Pp2(x) + 3p + t 3n-1 (x) ■ an^4'n(x) " x Elementary calculation using the definfitions of 3n-i (x) - 3n^+n(x) " *fn-1 (x) " V i V i W = V i ^’^n-l (0) -^n^'^n(O) "X^Pn-l (0) +bn-1 V i +x(2ap2+x2+q3x+q2) Denote D(0:n) := |ap_, (0) " ^ V i V i (°) I ■ then, since xy<(x 2 +y2 )/ 2 , (3.4.13) 3Kn(x)/n > [0n"^/(n/3)] [Fo(x:n) + F,(x;n) p^., 2(x) ] where F|(x; n) = ipn_|W/an^ - D(0;n)/(2a^^) 4|x|/(2an^)l( |x2|,|x| |bn_|tbn|«2an2«|bn_,2.bn2*q2| ). Rearrangement and simplification gives Fo(x; n) = [ 2+( IXI /a ^)2]-I l- |x | /(2a ^) ] + ( a^-] ^/a^^-1 ) + F and F,(x: n) = [ 2+(IXI /ap)2]-[ l-|x | /(2a ^) ] + ( a^+^ ^/a^^-l ) + F where 79 F = I /(2an^) + ( xb^ - D(0:n)/(2a^^) - [ IXI /a n 1[ I bp_| +bp| II |x| /(2a|^) - 1/a p ] Replace x by its largest value: £(i6n/3)^^'^. Then, since F = 0(n"'/^). (3.4.14) Fj(x; n) > Fj( e(16n/3)i/^; n) > 2(1- E(n/3)'/4/a^) + ( 9n±l ) + 0(n*’/2). The right hand side o f (3.4.14) approaches 2-2c since ap/(n/3)’^'^ approaches I as n gets large: hence, there exists a positive integer Ng such that, forn>Ng, ( ap/(n/3)^/4 ) Fj(x; n) > 3(1- £)/2, and therefore, for n>Ng, Pp2(x) < Pp2(x) + pp_j 2(x) < [ 2 /( 1- £)] Kp(x)/n. By Theorems 3.4.3 and 3.4.4, the inequality above gives us |Pp(x)| <[2B/(1- £)] n"’/8 w"»/2(x) 80 forn>Ng and |x| < eClôn/S)’/"*. Taking B possibly larger, the result holds for all n. B Remark. A simpler method is given in Nevai [Ne2] where the weight is an even function and so is 0 for all n. 81 5. THE ASYMPTOTICS OF D^fw: Q). In order to solve the differential equation of p^(w; x), we need data fo r an initial point of the solution. We shall choose the point x=0, in part, because this choice gives the simplest form o f the recurrence fo r p^(w: x). We shall also immediately assume q^=0 following the Remark after Lemma 2.5.3. This proof extends the technique of Mate-Nevai-Totik [MaNeTol]. Proof (Theorem 3.3.2). Recall the recurrence relation fo r Pp(x): (3.5.1) a^+i Pn+1 (x) + (bp-x) Pn(x) + a^-l Pn-1 (*) = 0- The characteristic equation o f (3.5.1) has the roots (3.5.2) tj^(x) = (x-b^)/(2a^+, ) ± i [ a^/a^+j -(x-bn)^/(4a^+| ^) which. forn>no, have non-zero imaginary part: also note that fo r x=0 (3.5.3) | t | _ j2 = an/an.| ()=1.2). Define 82 (3.5.4) := Ppj(x) - n (x) P^-i (x), then ^n+1 ~^2,n^n~Pn+l "(h,n+l ^^2,n) (^n'^^n+l ) Pn-1 - Thus (3.5.5) ^n+1 " ^2,n ^n ’ (^l.n "^l,n+1 ) ^n- n Upon dividing both sides o f (3.5.5) by TTt2 j and defining j=1 n (3.5.6) »n*|W= V | W ^ 1=1 we obtain n (3.5.7) '^n+1 ■ ^n (^1,n "^1,n+1 ) ^n ^^2 ,j- j=1 Successive applications of (3.5.7) yield n k ^n+1 =Z ((tik -tik+i )Pk / TTtgj }. k=0 j=l Setting x=0 results in 83 n k (3.5.8) (0) = Z((t|,k (0)-t,k+i (0))Pk(0)/ Ïït2j(0)}. k=0 j=l To estimate hi,n ^1,n+l (o)| we recall, from Theorem 2.3.3, that = (n/3)i/4 [ 1 + c,(n/3)-i/2 + ain/zV + 0 ( n"3/2) ] a n d = d,(n/3)-’/2 + dgCn/S)'» + 0( n‘ 3/2). Therefore the following expressions hold: (3.5.9 a) a|(/ak+i = [k/(k+1)K4 [ i + o(k'3/2) ] = 1 - l/4k + o(k"3/2), (3.5.9 b) ak+{ /ak+2 = [(k+i)/(k+2)]'/4 [ i + o(k"3/2) ] = I - l/4k + 0(k'3/2), (3.5,9 c) b /aK *, =d,(k/3)"'^2[l*O C '^2)l « (( (k *l)/3 )'''‘'(t*0Ck -'/2 ))|-' = (k/3)-:/'' [ d, * (d2-c,d,)(k/3)-i/2 ♦ 0(k’ ') ), (3.5.9 d) b^.,/aK*2=d|(k/3)''-^2(„o(-i-'2)lx[((kti)/3)i/4(|.o(k -i/:))]-' = (k/3)-3/4 [ d, ♦ (drC,d,)(k/3)-'/2 * 0(k’ ') 1. (3.5.9 e) b|(^/(4aK,, 2) = |d,2(k/3) * '(t* 0(k'3^2)) 1X 84 X 14( (k+1)/3)i/4 (1+ o(k'i/2))]-1 = d,2(k/3)-3/2[Ho(k -1/2) I and (3.5.9. f) 4<),2 (k/ 3)-I (,* o(k-3/2)) ] X I4( (k'2)/3)i/4(„0(k -1/2))]-' = d|2(k/3)‘ 5^2[,.o(k -1/2)] Thus l^j.k ^l,k+l - I b|(+j /3|(+2 - b|(/a)(+] ) + ‘ ( (ak/^k+i - bk^/(4a|(+j 2)]1/2 _ /a,(+2 " ^k+l ^/(4ak+2^)^'^^ ) I wnich using (3.5.9 a-f) Is = I (d,/2)(k/3)-3/4[o(k-')] *1(11 -l/(4k) *0(k-3/2)]i/2 - [1 -|/(4k) .o(k‘ 3/2)]i/2 ) I, i.e., = I 0(k-i'/4) * i ( [t -|/(8k) *0(k-3/2)|. [I -i/(8k) *0(k-3/2)| ) | which is l>i,k W-'i,k*l (0)| = I 0(k-'/"). i o(k-:/2) | = o(k-2/2) Also, from (3.5.3), 85 k k [ TT 112 j I V = [ ï ï 3j+| /3j = ( a^+j /3| j=l ' j=1 Using the above estimates, we have k=I From Theorems 2.3.3 and 3.4.5, n I*n4 (Wl S A' % k-3/2 , k=l and so we have that Ÿ^(0) is an absolutely convergent sum. The definition of $p(0) and (3.5.5) imply tn„(0) = t2,n(0)*„(0) ^ ^ ^ ^^),n (^)"^l,n+l (0)]/[t2_n(())'"(3nPn-l Pn+I )- Since 111 p, (0)-t| p+| (0) | = 0(n"^/^) and since l^2,n(°)"(^nPn-l (°) V(an+] Pn+I (0) ) I ^ I Wt 2,^(0)} U S > 0 for n>N, it follow s that 86 00 TT (o)/[t2k(o)$k(o)l = s%o k=N or 00 TT (0)/»|((0) = S=:0. k=N That is, Ÿp|(0) can be written as an absolutely convergent infinite product and so we have shown n §Pi+l (0) [ TT t2 j (0) ] * -> const. ^ 0 (n->oo). ]=1 Multiply (3.5.8) by the product o f the t 2 j 's and take the imaginary parts of the resulting equation to have n (3.5.10) lm(# (0)) = lm( T T tjj (0) «* ^ 4 (0)) J=0 recalling that n k *n4= (0)-t|,k4 (O)lPk(O) ' Ïït2j(0)l; k=0 j=l 87 but, from the definition (3.5.4) of $^(x), (3.5.11) lm{ (0) ) = 4 (a,,., ‘ "n .l Pn(0). Hence, putting (3.5.10) and (3.5.11) together. n (3.5.12) l(a^+, /an+2)"^n+1 Pn(0) = Ï ït2 j(0) (0)} j=1 Now consider fo r a moment the product o f the t 2 1 's. n n n ï ï t 2 j( 0 ) = ï ï |tç i(0)| X exp{ I ^ arg{ t 2 j( 0 ) } } = j=1 ’ j=l j=I n = (a; )’/2 exp{ i ^ arg{ t 2 j (0 )}}. j=l Therefore (5.12) is [ (an., /an.2> - b^., 2/(4an.2^) Pn(0) = n = lm{ (a, /8p+| )’/2 |» ^+ j(o )| exp{ i 2 arg( t 2 j (0)} + arg(»^+| (0)}}}. j=l i.e., I (3n+1 /9n+2) ■ bn+1 ^/(49n+2^) Pn(^) = 88 n = (3| /a^+i 1»^+^ (0)| sin{ ^ arg{ t 2 . (0) ) + argiŸ^+i (0)} I j=1 From above, recall that I (an*l - ‘>n*l ^/Man.2^) = 1 - 1/(8n) * OCn':/:), and, also noting anti = 3'/» n-i/a t l-(c,/2)(n/3) "'/z <4C2-2c,2-i)(n/3) ♦OCn'^-"?)), we obtain (3.5.13) Pn(O) = - 3 1 / 8 a, 1 / 2 n"'/8 | ^ (0) | [ l-(c,/2)(n/3) ‘ ’ / 2 n + (4 c 2 - 2 c,^-l)(n/3 ) " ’+0 (n“ 8 / 2 )] x sln{ ^ arg{ t 2 ■ (0)} + arg{Ÿp+] (0)}}. j=l Let us consider arg{ t 2 |^(0)}. From definition (3.5.2), 2cos{arg{t2|^(0)}}= b|^/(a|^a^+l y / 2 ; thus we need to analyze bj,/(a|^a|^+) ) ’ ^ 2 a^ak+i = (k/3)i/2 o+;/k)i/4 [i+ c,(k/3)‘ >/2 + c2(k/3)’ » +0(k"3/2)]2 which = (k/3)’/2 (|+i/k)i/4 [1 + 2c,(k/3)"'/2 +(2c2+c,2)(k/3)-« +0(k"3/2)], i.e., since (l+I/k)’/4 = ; + ;/(4k) + 0 (k"2 ), 89 31(3^+, = (k/3)i/2 11 + 2c,(k/3)-i/2 +(2c2+c,2+i/i2)(k/3) +0(k“ 3/2)]; also b|( = d, (k/3)-'/2 4 d; (k/3)-' *0(k-3/2). Thus bK/(a| arg{ } = cos'» { (l/2)(k/3) "3/4 [ o, + (d 2 -c,d,)(k/3 ) ' ’/ 2 + o(k‘ D 1}. or ^ (3.5.14) arg{ t 2 ,^(0)} = tt / 2 - (d,/2 )(k/3 ) '3/4 _ ((dg-c,d,)/2) (k/ 3 ) “ 5/4 + 0(k“ 7/4). Putting (3.5.14) into (3.5.13) and using the Euler - flaclaurin formula for sums ( e.g. see [BrI, p. 40] ) we arrive at Pp(0) = -3'/8 a, > / 2 n-i/8 14- (o) | [ l-(c,/2)(n/3) “'/z + (4c2-2c,^-l)(n/3) + 0(n“ 3/2)] x sjn{ n7T/2 - 2 33/4 n^/4 + const + 0(n“ '/4)+ argi'i'n+i (0)}}. Since (0) converges to 9^0, then 19 (0) | = | Ÿ | +o(l) and arg{'?p+^ (0)} = arg{»} + o(l). So 90 t h a t Pp(0) = -An"^/8 sjp( pTf/2 - 2 3^/'^ d, + c } + n“*/®o(l) w h e r e A = 3^/8 i /2 | \p | jg g positive real number. A fter absorbing the minus sign Into the sin{9} term we obtain Pp(0) = A n"^/8 cos(nTT/ 2 - 2 3 ^/4 pi/4 + c} + o( 1) . B W e notice that c depends onargfî'} and on the sum o f the arguments of t h e t 2 | ç ( 0) ’s . 91 6. ASYMPTOTICS FOR D^Cw: x). In order to find an asymptotic expression forpp(w; x), we will make use of Nevai's [Ne4] improvement of the Liouville-Steklov method by applying it to the differential equation of p^(w: x). First, the second order differential equation will be established. Lemma 3.6.1 . Let ppj(w: x) be the orthonormal polynomial with respect to the weight w(x) defined by (3.2.1). Let a^ and bp be the recursion coefficients of (3.2.2) and (3.2.3) respectively: define (3.6.1) (3.6.2) tp(x) = bp_| + bp + qg + X. Then, fo r n=1. 2,.... Pp(w; x) satisfies (3.6.3) Pp"(x) ~[Q’(x) + fp'(x)/fp(x)] pp'(x) +ap2[l+(p p_^ (x) Proof. The differential equation is essentially the result o f (3.4.2). Differentiate (3.4.2) and then apply (3.4.2) to pp,_| (x) to see Pn" * Pn^+nPn' * ^n^Pn " W n '/fn '^ n -l ) V n P n -l " Pn-I «nfn-I 'fnPn-2- Accordingto (3.4.2), an^n^n-l “ ^n* * ^n^'^n^n* ^'^o, from the recursion formula, 2 ^ - 1 Pn- 2 = -Sffn ^ (x-bn-l )Pn-l • using these identities transforms the equation above to (3.6,4) Pn" -t 'Pp-I ^'f'n-1 *(Pn-l ' V C p l Pn' ♦an^[I*iP n-i t ' I'n'l'n'/fn 'f'n *(Pn-l -xJf’n-l fpl Pn = Elementary calculation involving the definitions of The standard transformation to eliminate the first order term in (3.6.3) is our next step. Proof. (Theorem 3.3.3.) By well-known formulas ( see e.g. Szego, [Szl, p. 171 ) the differential equation fo r z(x) (3.3.3) follow s directly from 93 the formula (3.6.3). B We shall begin the Liouville-Steklov method by putting (3.3.3) into a more usable form. Lemma 3.6.3. Fix an interval A c E. For all xeA and n sufficiently large, the equation (3.3.5) is asymptotically equivalent to (3.6.5) 2 "(x) + a^^t Dfl(n) + Di(n) x + D2 (n) x^ 1 z(x) = 0(2^^) z(x) where Do(n) = V l (0) “ ^1 " ^n^ D,(n) = (bn * q j) n/ + a^,, ^ ( bn_, • b^ * b^*, t q^ ), and D2(n) = n/a^^. Proof. Fix xeA. The terms Q'(x)/2 and Q"(x)/2 are both 0(1) as x is bounded. Recall the definition (3.6.1) o f f^(x) y x ) = 3^2 ♦ an., 2 * bn^ * bnqj * q j * * xbn * xq,; thus fp'(x) = 2x + bp + q^ is 0(1) for xeA since x is bounded, q^ is constant, 94 and approaches a constant as n gets large. If n is sufficiently big, then l/(p p|(x) is 0(1) since a^ ~ So we have z" + a^i^l I +9n-l "Pn " "^n(Q' +^n^^'n +an^*n) 1 ^ = 0(0 z. We noted above that is 0(1); putting this together with 'f'^(x) =b^.^ + + bn qs + X, which is 0(1) fo r neà, we see that z" * 'fn-1 »n ‘ <’nW * > z =0 (%2) z. Now let us consider the coefficient o f a^,^. V l (^)^n(^) - = ( V l ("^) + + bn_) X + qgx )( 9^(0) + + bpX + q^x ) -( fn(°) + x)(x^ + q^x2 + qgx + q, ) - + x )2. Perform the multiplication and collect like terms to find fn -i (x) = [ + X [ qsC Vl (0 > W 0 ) ) +bn_, ^ 0 ) ^b^fn-i (0) -q2'^n(°) “^1 -Zan^'^n(O) ] + 95 " I V l (0) *$n(0) 'bn_, bn +q3(bn_, *bn) ' -qs+n^O) ‘ ^ 1 »n-l ' bn+ 43 - '■ The coefficient of is zero according to the definition of 'l'p(x). In view of the recurrence relation (2.3.7) and the definition of ^p(x) V l (0) "Pn(0) = n/Sn^ * * <\2 ' »n-l "n. and by the definition of 'Pp(x) C3A'n-rV* q3( V i (0)+V0) ) +bn_i V ° ) ^^n V i (°) "^1 -23n4n(0) = (bn+A])! V i (0)+V0) ] + V O ) (bn-1 ‘ bp) - V ° ) ^ ^2 +2an^) . which, using the expression above for ( V i (O )^VO ) )• is = (bn-qj)! n/a^^ - ‘>n-l *>n ' "bn) (bn^qjX n/an?! * (D n^q^Xan^'q; -b n ., bn 1 ♦ f>n(°) (bq-l ' bn) - 96 - 4"n(0)[ A2 +23n2] - q ; . Once more using the definitions of -»n-I ' fr/O ) (t>n-l ' >>n) ‘ '('n(0)l - 3n+l ^^n-l * ^n+I * Which, by the recurrence (3.3.8), gives (»n* = bp.|^(bp-| *l3n*Dp,| ‘ qjX n We are now ready to prove the main theorem o f this chapter, the asymptotics of Pp^(w: x) fo r x in a fixed interval A. Proof (Theorem 3.3.3). Assume, without loss of generality, that q3=0 and that the point 0 is in the interval A. Define (3.6.6) fp(x) :=[Do + D,x + DgxZ F 2 - set (3.6.7) y(x) := z'(x) + I a^ fp(x) z(x). 97 Then y’(x) = z ”(x) + i f^Cx) z'(x) + i a^ f^'Cx) z(x), soy(x) satisfies the differential equation y’(x) - i a^ fp,(x) y(x) = [ z"(x) +a^^f^^(x) z(x) ) + i a^ f^'(x) z(x), which, under the aegis of Lemma 3,6.3, is y'(x) - i fn(x) y(x) = 0 (^ ^ ) z(x) + i f^Xx) z(x). We claim that f^'(x) is 0(1) since, from (6.6), f^'(x) = ll/ f ^(x)] [D, + 2 D2 X] 12, which, accordingto Theorem 2.3.3, is asymptoticaliy f^Xx) ~ [l/n '/^] [n’^^j forxfA. Thus (3.6.8) y'(x) - i a^ f^(x) y(x) = 0(a^^) z(x). Define X (3.6.9) jj(x) := exp{ J - i fp(t) dt } 0 and X (3.6.10) 6p^(x} := J fp(t) dt. 0 98 Then, solving (3.6.8), we have X y(x) = l)j(x)]"’ I J 0 ( ^ 2 ) 2 (t) |i(t) dt + y(0) 1. 0 I.e., y(x) = y(0) exp{ I 0p,(x)} + 0(2^^) max { | z(x) | : xeA } Recalling the definition z(x) = Pp,(x) [w(x)/ see that max { |z(x)| : xeA } < A n "’/® / [ since max { I z(x) I : xeA } = 0( n"^/® ). Theref ore y(x) = y(0) exp{ I 6p(x)} + 0(g^^) n"^/®; which is (3.6.11) a(x) = l2-(0) * i a„ f n(0) z(0)l exp{ i ) * 0 (%Z) n'S/e Equating the imaginary parts o f (3.6.7) and (3.5.11) yields an fn(x) z(x) = an fn(0) z(0) cos{6n(x)} + z ’(0) sin{ 6n(x)} + 0 (^ ^ ) n"^/®, 99 o r z(x) = [fp( 0 ) /y x )) z(0 ) cos(%(x)} +[z'( 0 )/(a^r^(x))] sin{9p,(x)} + 0 (ar^^)n 3 / 8 Reintroducing p^(x), we obtain w’/ 2 (x) p^(x) = [fnCx)]-’ [y x ) /( fn ( 0 ) ] ' / 2 [ yo)Pn(O) cos(%(x)} + ap,"’ [p^'(O) + (pn(0)/2)Iq| +bp,/ Define C, and (2 as follow s, ( 1 := fn(0) Pn(0) and ^ 2 := tPn'(O) + (pp^(0)/2)[q^ +bp,/fp^(0)l I Then CiCos( 6 p^(x)}+C2 Sin{ 8 ^(x)} = [ ^ , 2 + (g^t'/^sinf e^(x)+tan " '( ( 2 /( 1) ). Therefore we conclude that w’/2(x) p^(x) = [fn(x)]-i [‘Pn(x)/ X [(,2 + C2^]i/2sin{ 6n(x)+tan-'(f:2/(,)} + n'i/Bod). Before proceeding we will develop several asymptotic formulas that 100 will be needed. First recall from (3.2.5) and (3.2.6) that a^ = (n/3)i/4 [ 1 + Cl (n/3) - i / 2 + cg (n/3)"< ^ 0(n‘ 3/2) ] and bp = (n/3)->/2 [ d, + dg (n/3)-'/2 ] + o(n“3/2) According to (3.3.3) this is ■PnCo) = Z(n/3)'/^ [I *2c,(n/3)-'/2 *(zc2*c,^*l/6)(n/3) *0Cn‘ ^''2) 1 * ( d,Z(n/3)-i * 0(n-3/2) I * q ; or (3.5.12) (pp(0) = 2(n/3)>/2 [i +(2c, +q2/2)(n/3)'i/2 + (2cg+c,2+i/6)(n/3) "> +0(n"3/2) ]; similarly (3.6.13) cpp_, (0) = 2(n/3)i/2 [1 +(2c, +q2/2)(n/3)">/2 + (2cg+c,2-l/6)(n/3) “ > +0(n"3/2) ]. Hence fp(0) = [ fp(0) = 2 (n/3)i/2 [i+(4cfq 2)(n/3)'i/2 4(4cg-^2c,2 +(2c,+q2/2)2)(n/3)“i + 101 + 0(n"3/2) ]1/2; as «fpCo) and are 0 (n "i/ 2 ) and thus (3.6.14) fn(0) = 2 (n/3)i/2 [i+(2c,+q 2/2 )(n /3)'’/2 + (2c2+q^)(n/3)"i + 0(n"^/2) ] Now (p^(x) = (3.6.15) fn(x)/9n(0) = I + x(x+b ^) 2 (n/3 )"i / 2 [,+ o ( n - i/ 2)] = 1 + 0 (n "^/2) (xeA). Recall that D,(n) = ( b^n/a^,^ ) + a^+] ^( bp,_| +b^ +bp+] ) therefore D,(n) = 3d, [1 + 0(n"»/2)] + (n/3)'/2[1+ 0(n‘ >/2)] 3d, (n/3)"'/2 [|+ o(n"’/2)] = 6d, [1 + 0(n"*/2)] Thus we have (3.6.16) D,(n)/[ fn -i (O)fn(O) 1 = (3/2) (n/3)"' [1 + 0(n‘ >/2)]. Since = f^(0) Pp,(0), we apply (3.6.14) and (3.3.2) to yield ^, = 2 (n /3)^/2 [] + o(n"i/2)] ^ n"i/8 [ cos(cx+nTr/2)+o(l) ] where ex = ( -2 3^/'^ d, n^^'^ +c ) with c being the constant o f Theorem 3.3.2, thereby (3.6.17) C, = 2A (n/3)^/® [ cos(«+nir/2)+o(l) ] 102 As ( 2 = I Pp'(O) + Pn(0)(q| +bn/ = -ap|tn(0)Pn(0) + (0) + Pn(0)( Considering (3.3.2) and (3.6.12) (2 = 0(n‘ ^/®) +2(n/3)^/^ [l+0(n"'/2)] An"*/®[ cos(cx+(n-l)Tr/2) +0 ( 1) ] +0(n-3/8); that is (3.6.18) ( 2 = 2A(n/3)2'^8 [ cos(cx+(n-l)Tr/2)+o(l) ]. Hence (3.6.19) = 2A(n/3)3/« [ i'o(t) I and as ( 2 / ( 1 = cos((x+(n-l)7T/2) / cos(cx+nTr/2)+ o(l) = tan(cx+nTr/2) +o(l) we obtain (3.6.20) tan“ ^ (( 2 /^;) = nTT/2 - 3 ^/^ r^i/4 + o(l). Returning to the proof, we note from (3.6.6), (3.6.10), and (3.6.16) X @n(«) = y fn - l (0)fn(0)K y ([l*nt '^1(2?^^ (0)^0))] ‘0(r|-’) )dt 0 so that 0p(x) = ap[ Thus w '/2(*) Pn(x) = [fn W r' X sin( 0n Ifn-I (0)$n(0)l'"'^ I * * (O)'Pn(O)) 1 ‘ tan"' (Çj/Ç,) ) + n"’/® o(l). Accordingto (3.6.14) and (3.6.15) f^(x) ~ 2n'/^ and [ (3.6.21) w’^2(x) p^(x) = A n~’/® >c sin( a^ [fn - l (0)'Pn(0)]’''2 [ x + nx^/(6a^^ + nTT/2 - 2 3^/4 (jj r,i/4 } + n"^/® o(l). We note that if w(x) = exp(-x"^}, then (3.6.21) is Theorem 11 o f Nevai [Ne4]. Consider for a moment 3n [fn-1 (0)fn^0)]’/2 [ x + nx^/(6ar^^ fn-1 (O)fn^O)) ]. Using (3.6.12), (3.6.13), and Theorem 2.3.3, ap^fn-i (0)f^(0)]'^2 is equivalent to an [f n-i (0)fn(0)K2 = (n/3)'/^[l +c,(n/3)-’/2 +o(n-')] X 2(n/3)'/2[i <2c,+q2/2)(n/3)“»/2 +0(n"’)] 104 o r f V l = 2(n/3)3/4[i +(3c,+q2/2)(n/3)~’/2 +o(n"')] and so (an (V l (0)VO)('''^r' " (l/2)(n/3) * 0(n-'/2)|. Thereby we see that an (V l (0)V0)1'^^ I « * nn3/(6an2 f^-l (0)V » » 1 = 2x (n/3)^'''^ ♦ [(5C|*q2)x *x^/2) (n /3 )'/'' and SO (3.6.2!) takes the form w’/2(x) p^(x) = A n ” ’/® sin[2x (n/3)^/4 + [(5c,+q2)x + x^/2](n/3)'/^ + nTT/2 - 2-33/4 d, n’/4 +c} + n” ’/®o(l) and the result is proven. B CHAPTER IV PLANCHEREL-ROTACH TYPE ASYMPTOTICS FOR THE ORTHOGONAL POLYNOMIALS ASSOCIATED WITH w(x) = expf-Q(x)) 1. INTRODUCTION. In Chapter 111 we found an asymptotic formula fo r p^^Cw: 0) and then, using this expansion as an initial point, we found an expression fo r p^(w; x) that was valid fo r x in a fixed interval A. The method we used to find the equation o f Pp(w; O), due to Nevai, could be extended to work fo r any fixed value o f X ( see [MaNeTol] where this technique was introduced for weights with recursion coefficients ap, and tending to finite limits, cf. (4.2.5) and (4.2.6) ). We immediately note that n was dependent on A, hence on x: this means that the asymptotics were valid for any fixed x, however n may need to be very large when x is great. 105 !05 In this chapter we will find Plancher el-Rotach type asymptotics ( see [PIRol], also[5zl, p. 201, 203] ) for p^(w: x) using the method o f Nevai [Nell. That is, we will find an asymptotic formula for p^(w; x) where x depends on n, rather than n on x, as was the case in the theorems o f Chapter III. This equation will hold uniformly fo r |x|< cx^p, where x^^ is the greatest zero of p^(w; x) and 0 < c < 1. From Theorem 3.4.3 we know that limn->oo ^i/n = 2, hence we will consider the asymptotics for the interval |x| < 2c(n/3)^'^'^. Nevai’s technique was also employed by Sheen [Shl,Sh2] to find asymptotics fo r the polynomials orthogonal to w(x) = exp{-x^/6). Asa consequence of Nevai's method we can determine the value of the constant A that appears both in the f ormula of Theorem 3.3.2, Pp(w: 0) = A n "i/G cos[ (nTT/2) - 2*3^^"’ di n^^"’ + c ] + n"^'^®o(l) ( where d, = (9/162)1 q^ q 2 - 3q| ], the coefficient of (n/ 3 ) " i / 2 the asymptotic series of bp, cf. (4.2.6) ), and in Theorem 3.3.4, 107 w i/ 2 (x) p^(x) = An"i/® sin ^ 2 x(n/3 p / 4 + [(6 q+q 2 )x +x^ / 2 -ed^l (n/3)'^"^ + nTT/2 + c j + n'''^®o(l). We will find that A = 3 ^/^ 7 t~i/ 2 Chapter IV is organized as follows. In Section 2 the notation is introduced; Section 3 contains the statement o f the asymptotic theorem; Section 4 has the lemmas used in the proof of the theorem; and, lastly. Section 5 is the derivation of the asymptotic formula. 108 2. NOTATION We will observe much the same notation as in Chapters II and 111. As in Chapter 11, let Q(x) = x*^/4 + q^x^/3 + +q| x and define w(x) = exp{ -Q (x)}. Without loss o f generality, we immediately take q^ = 0, therefore w(x) takes the form (4.2.1) w(x) = exp{ -( x^/4 + q2X^/2 +q| x ) } The polynomials orthonormal with respect to w(x) are Pp(w: x) = y^x^ * ... . and satisfy the three term recurrence (4.2.2) X Pp^(w; x) = 3p,+| Pn+] (w: x) + b^ Pf^(w; x) + a^ Pp,_| (w; x) where the recursion coefficients a^(w) and b^(w) are defined by (4.2.3) a^(w) = (w) / y^(w) and (4.2.4) bp(w) = J X p^^(w: x) w(x) dx. E 109 We will often use the result o f Lemma 2.5.4; fo r each integer m (4.2.5) = (n /3 )'/'’ [ I + c,(n/3)"’/2 + c2(n/3)"’ + ... + c^(n/3)"'^^2 + ) ] and (4.2.5) bp = d,(n/3)"'/2 + d2(n/3)"' +... + djp(n/3)'^^^ + ) where the Cj and dj are constants (see Lemma 2.5.4 for the exact values ). The Chrlstoffel function of w(x) is )vp(w:x) which is the reciprocal of the kernel Kp(x) where n-1 (4.2.7) Kp(w; x) = ^ P|^^(w; x) k=0 As in the previous chapter we shall follow the usual custom o f supressing arguments, such as in Pp = pp(x) = Pp(w:x), when the meaning is clear from the context. 110 3. THE MAIN RESULT. Theorem 4.3.1 . Let w(x) = exp{-Q(x)} be defined by (4.2.1), let Ppj(w; x) be the polynomials orthonormal with respect to w(x), and let 0<€ p^(w; x) w^^^(x) = 3^'^® n"^^® sin"^^^ (6) X cos[nr(e)/3 + m r/2 - 2-3^'^^ d| n'^^ + c I + n"^^® o(l) holds uniformly for € < 9 < tT-e where c is a constant and r(e ) is defined by r(e ) = [ 39 - sln(29) - sin(49)/4 - 3tt/2 ] + (n/S)"'^^ I q2(29 -7T+sin(29)/2) - 3” ^^^ (I2C{ +q 2 ) tan"' (3'^^ cot(9))] + (n/3 )"^/^ [ q2(1-e) +(6d, - q2) X { ,(24)-'/2|og| [(tan2(0/2)-(8/3)'/2tan(9/2)+l) /(tan2(9/2)+(8/3)'/2tan(9/2)+l) ]x[ (2+(8/3)'/Z) / (2-(8/3)'/2)] + 121/2 [ tan"' (3i/2tan(e/2) -2^/2) + tan"' (3i/2tan(0/2) +2'/2) - tan"' (3’/2 -21/2) + tan"' (3«/2 +2«/2) ] ) ] + (n/3)"' [ (2-q2)(9-Tr/2)/4 + 2(4c,+2c2+l5c, ^+4c,q2+(q2/8)+(l2^/'^) Ill X 3"^/^ tan"^ (3^/^ cot(e)) + ( 6 c,+q 2 ^ / 4 ) li - 2 q 2 (6 q+q 2 / 2 ) I2 -( 6 c,+q 2 / 2 ) 2 I3 ] with Ij defined below. 1, = 2 3 " 3 / 2 tan"^ ( 3 " ’ / 2 tan(e)) - (3 tan(0))"^ + sgn( 0 - 7 T/2 ) 3 ” 3 / 2 jt; and I2 = [ 2 3 "3/2 tan^(0) tan"' ( 3 " ' / 2 tan(0)) + 2 -3 " ' / 2 tan"' ( 3 ~ ’ / 2 tan(0)) - ( 3 tan(0) )"' - (1/3) tan(8 ) ] / ( 3 + tan^(e) ) - Sgn(0-TT/2)3"3/2 7T; also I3 = [ 5 3"3/2 tan^(0) tan"' ( 3 "’/2 tan(0) ) +10 3"3/2 tan^(e) tan"' ( 3"’/2 tan(e) ) + 5 -3"’/2 tan"' ( 3 "'/2 tan(0) ) - ( 3 tan(e) )"' - (4/9) tan'^(e) - (11/9) tan(e) ] / ( tan^(e) + 3 )^ - (5/2) sgn(0-TT/2)3"5/2 jT. 112 4. LEMMATA First a lemma that will assist in determining the constant A in the statement of the asymptotics fo r p^(w; x) of Chapter 111. This lemma is the extension o f Nevai's [Nel] Lemma 2 and uses the same method o f proof. Lemma 4.4.1 . Let 0<£ { xeE : COSE < (|x|/2 )(n /3 ) < 1 }. There exists a constant csuch that fo r all £ lim sup J [1-( IXI /2) ^(n/3)"^^^ ] Pp,^(w; x) w(x)dx < c(l-cos £). n->oo A Proof. We follow [Nell. During the proof o f Theorem 3.3.1 we found h“ l the inequality involving the kernel function K^(w; x) = ^ P|,^(w: x) k=0 ar^^(n/3)“ ^ [ (Fq+F) p^^(w; x) + (Fi+F)Prj_i ^(w: x) 1 < 3 K^^CxVn where Fo = [2+( IX I /(2 a n))2] [l-( | x | /(2 a ^)] + (-1+ a^., ^/a^^ ). F, = [2+( IXI /(2a n))2] []-( | x | /(2a ^)] + (-1+ a^., ^/a^^ ). 113 and with F = [ (q2/3n^) - 1 q2X I /(2a] + [bp^/ap^] + [xbp/a^^^] - ID(0;n)/(2ap3)] - |bp_;+bp|(x^/(2ap3)- |bp_,+bp|(x/ap3) for D(0;n) = I a^,., 2 4.^., (0 ) - >).„(0) * b^., $ ^ - 1 («) I It was shown that F is 0 (n"'/'2) in the proof of Theorem 3.3.1. Since 3 p/(n/ 3 )^/'^ = 1 + 0 (n "i/ 2 ) there Is a postive Integer no such that fo r n > no [ -1+ 3p+| ^/3p^ 1 = l/(2n) + 0(n‘ 2) > 0 and [ -1+ ap_) ] = -l/(2n) + 0(n"2) > -(l-cos e). Now [1-1XI /(2a p)l > [1-(n/3)(/4/a =o(n“ ’/ 2 ) for n>ng and | x | < 2(n/3)’/^ , therefore we have F,>2ll-lx|/(2ap)] + F = 0(n''/2) and hence F, > -(l-cos e) ( n>no ). 114 Also, since ap(n/3)"^/'^ approaches 1 as n goes to infinity, there is an n, (>rio) such that n > n, implies apCn/S)"’/"* > 3/4. Thus we have Kp(x)/n > (1/4) [ 2 [1-1XI /(2a ^)] p^^^(x) - (l-cos c)[Pn-i ^(x)+Pn^(x)] ] or, rearranging the inequality (4.4.1) [1-1X1 /(2a p)] Pp^(x)l < 2Kp(x)/n + (l-cos OtPp-i ^(x)+Pp^(x)]/2. Because cosc < 1, there is an n 2 (>n,) so that ap(n/3)"’'^"’ > cose fo r n>n 2 , and then we see [l-|x|/(2a p)]> [ l-(|x|/(2 cose)) (n/3)"’/'*] = = (1/cos e)[ l-(|x|/2 ) (n/3)"’/"*] - (l-cos e)/cose. As(l- y) > (1- y^)/2, [1-1XI/(2a p)] > (l/(2cos e)) [ l-(|x |/2 Ÿ (n/3)"’/^] - (l-cos e)/cose, i.e., [ l-( IXI /2 Ÿ (n/3 )"’/ 2 ] < (2cose) [1-1 x | /(2a p)] + 2 (l-cos e) < 2 [1-1XI /(2a p)l + 2 (l-cos e). 115 Therefore [1-( IXI /2 Ÿ (n/3 )~’/ 2 ] < 2[1-1XI /(2a p)) Pn%) + 2(1-cos E)pp^(x), which, after the application o f (4.4.1). is [ 1-( IXI /2 )2 (n /3)-1/2] p^2(x) < 4Kn(x)/n + (l-cos £)( Pn-1 ^(x) + Pn^(x) ) + 2 (l-cos :) p^^^(x). According to Theorem 3.4.4 Kp,(x) < C n^/4 w‘ l (x), hence [ l-( IXI /2 Ÿ (n /3)“’/ 2j p^2(x) < 4c n '’/^ w“ ’ (x) + (l-c o s £)( 2 Pp_, 2(x) + 3 p^2(x) )_ We multiply by w(x) and Integrate over A to obtain J [ 1 - ( IX 1 /2 Ÿ (n /3) “ ’'^2] w(x)dx < A < 4C n"^/4 Jdx + (l-cos e) J ( 2 pp,_| ^(x) + 3 p^^(x) )w(x)dx A A If we compute the first integral of the right hand member and extend the second integral from A to E before evaluating, then 116 J [ 1-( IXI /2 (n/3)"i/2] Pn^(x) w(x)dx < [ 8C + 5 ](1-cos e) for n large enough and | x | < 2(n/3)'^'* and the result holds. B We will now define the function h(0;n) that will be used in the proof of the asymptotics of Pp(x). First a few preliminary items: recall ( with ^^=0 ) Q(x) = %4/4 + q2%^/2 + x, and the expansions for a^ and bp, are ap, = (n/3)'/4 [ K c,(n/3)"’/2 + c2(n/3)“ ’ + 0(n"3/2) ), and bn= (n/3)->/2 [ d,+ d2(n/3)'»/2 + o(n‘ >) ]. Set B,(e) = [ 6c, + q2-2q2COs4e] [ 2sin^(G ) (l+ 2cos ^(8) ]" 1, 62(8) = cos( 0) [ 3d, - q; cos^(e) ] [ 2s i n ^ ( 8 ) ( l + 2c o s ^ ( e ) Ÿ 1" ^ . 63(8) = [ ( 1/ 4 )+ (2-q2)cos^(e)/4 + 4 c , + 2 C 2 + 15c, ^ +4c,q2 +(q2^/4) ] X [ 2s i n ^ ( 8 ) ( 1+ 2C 0 S 2( 8 ) Ÿ 1" ’ - 6, 2( 8 ), 6 4 (0 ) = cos(e)[12(d2+c,d,) - q, q2l [2sln2(e) (i+2cos2(e) )2]"i - B,(8) 62(8). 117 Bs(6) = [q2 -qj ^ /2 + (l+2cos^(e))"^ (4coë^(0)(4q+q2) +2c, + 2q2Cos^(e)) + K ] I8sin 2(e) (|+ 2c o s 2(e) )2]‘ 1 -26,(6)63(6) - 62^(6) + 6, 3( 6), 6 g ( e ) = I 2d , c o s ( 6) ( 2+ 3c o s ^ ( 6) ) (1+2C0S ^ ( e ) ) " ^ + cos(6)(l+2cos2(e))~* ( q] -2d,cos(e)+ 8d,cos'^(6) (l+2cos^(6))“' )] X [8sin^(6) (l+2cos2(e) )2]"1 - B ,(6)64(6) - 62(6)63(6) + 36,2(9)62(6), a n d 67(0) = - 6,(e)65(e) - 62(6)64(6) + 3 6,2(6)63(6) + 36,( 6)62 2( 6) - 632( 0)/2 - 156, " * ( 6 ) / 8 . W ith these functions in hand w e turn to h(6;n); let ( 4 .4 .2) h(6:n) = 4 s i n 2( e ) [ l + 2c o s 2( e ) ] X [I + 6,(6)(n/3)"'/2 + 6 2 (6 )(n/3 )- 3 / 4 + 6 3 (6 )(n/3 )"« + 6 4 (6 )(n/3 )' 5 / 4 + 6s ( 6) ( n / 3)- 3/2 + 6g ( 6) ( n / 3)- 7/4 + B 7( 6) ( n / 3)- 2j. Remark. The expressions 64, 65, 65. and 67 do not affect the com putations o f the proof: they appear here fo r the sake o f com pleteness. The next lemma w ill allow us to approxim ate the coefficient o f z(x) in the differential equation o f z(x); it corresponds to Lemma 3 in [Nel]. As preparation define ( from ( 3.3.5) ) 118 (4.4.3) fn(x) = [Q” (x)/2 - (Q '(x)/2)42i_ - [ + [ [ + [3 ^2 (0 0 + D,x + Dgx? ) ] + 3^2 [1+ -<|'n(x) Do = D, = nbn/an2*an„2(Dn-| ). and Ü2 = n/a^j^. Lemma 4.4.2. Let 0<£ (4.4.4) 4(n/3)'/^ sin2(e) f^(x) = (n/3)^ [ h(6:n) + 0(1) uniformly for € < 9 < tt-c where x = 2 (n/3)^/'^ cos(e), moreover, fo r n>ng (4.4.5) 0 < 2 sin % ) < h(9;n) < 18. Proof. We need to establish the asymptotic behaviour of f^(x), this will be done in five parts. 119 Part 1. By the definition of Q(x) Q "(x)/2 - (Q’(x)/2)2 = -%6 / 4 _ - q, x^/2 + (3 -^2^/2) xZ/2 - d; d2/2 + (dsrd; ^/2)/2. Setting x = 2 (n/3)’'^'^ cos(e)yields (4.4.6) Q"(x)/2 - (Q'(x)/2)^ = -15 (n/3 ) ^ '^ 2 cosP(e) - 8 q 2 (n/3) cos'^(e) - 4q| (n/3)^/^ cos^(e) + ( 6 -q 2 ^) (n/S)’'^^ cos^( 9 ) - d| d2 (n/3 ) ' / 4 cos(e)+ (qg-q, ^ / 2 )/ 2 . Part 2. The definitions of Q(x) and f^(x) give (Q’(x)/2)( If we again let x = 2(n/3)’^'^ cos(e), then (Q'(x)/2)(9n'(x)/ + (8 d, co^(9)+4q| cos(e))(n/3)i/4 + 0(n"l/4) ] / Now, from (3.6.12) and the definition o f X (n/3 )" 3 / 4 + (2 c2 +Ci^+l/6 ) (n/3)"’ + d2 cos(0)(n/3)"^/'^ + 0(n"^/^) ], 120 therefore (4.4.7) (X) = (1/2) (n/3)"’/2 [ (1+2 COs2(e))-> - (2cfd2/2) (1+2 cos^(e))-' (n/3)-i/2 - d, cos(e)(1+2 cos?(e))"^(n/3)"^/^ + O(n~0 ]. So we have - (Q'(x)/2)(fn'(x)/fn(x)) = -0/2) [ 32(n/3) cos'^(G) + 4q2(n/3)i/2 cos^O) + (8dt cos^(e)+4q| cos(e))(n/3)i/4 + o(n'^/^) 1 x ( 1/ 2 ) (n /3 )"i/: X [ (1+2 cos2(e))-1 - (2q+q2/2) (1+2 cos^O))"^ (n/3)"1/2 - di cos(e)(l+2 cos2(e))-2(n/3)-3/4 + o(n‘ i) ] or (4.4.8) - (Q'(x)/2)(fn'(x)/fn(x)) = -8cos^(e) (1+2cos 2(e))'’ (n/3)’/2 + cos2(0) ( 4cos2(e)(4c,+q2) -q 2 ) (l+2cos 2(8))-^ + cos(e)[8d,coé^(9)(l+2cos2(e))-^ -2d,cos2(e) +q| ](l+2cos2(0))'^ (n/3)^/'^ + 0(n-i/2). 121 Part 3. WE analyze [ l9n''(x)/ - (3/4) [2(n/3)i/4 cos(e)+ 0 (n i/2)]2 x [2(n/3)’/2 (U 2 cos 2(e)+o(n"’/2) ]~2 which is (4.4.9) !fn"W/'l'n(*W2 - (3/4)(fn'(*)/'Pn(«)l^ = 0(n‘'/2). Part 4. First (4.4.10) nx^ = 12 (n/3 ) ^ / 2 cos^( 6 ). Second, a^^ D, x = I 3(n/3)bp + a^^a^+; ^(b^_, +b^+b^+j ) ] 2 (n /3 )'/^ cos( 0 ) which by Theorem 2,3.3 is ap2 D, X = {3 (n/3)i/2 [d, + d; (n/3)-'/z ^ dj (n/3)'’ ♦ 0(n'3/2) ] * 3 (n/3)'/2 (l*l/n)'^2 [I ♦ c, (n/3)''''2 ♦ C; (n/3)'' * 0(n'3/2) | « [d, ‘ dz (n/3)'i/2 ♦ dj (n/3)"' • 0(n'3/2) |} 2cos(9)(n/3)'^''. Simplifying, we see (4.4.11) D, X = 6 cos(0)(n/3)^/4 M 2d, + 2 (d2 +c,d,) (n/3 ) " ’ / 2 + (2 d3 +2 c,d2 +2 c 2 d, +c,2d,) (n/3)"’ + o(n” 3/2) ]. 122 Last, Do = (3.6.12) and (3.6.13) Do = 4 (n/3) [ 1 + ( 4 C]+q 2 ) (n/3 ) " ’ / 2 + (4 C2 +2 c,2 +(2 c,+q 2 / 2 )2 ) (n/3)"’ + 0 (n"3 / 2 ) ]. Hence (4.4.12) 3^,2 Do = (n /3 )’/2 [|+ 2c, (n /3 ) " ’/ 2 + (zcg+cA (n/3)"’ + 0(n"2) ] 4 (n/3) [ I + (4 c,+q 2 ) (n/3)"’/2 + (4 c2 +6 c,^+2 c,q 2 +q 2 ^ / 4 ) (n/3)"’ + 0 (n"^/2 ) ] By putting (4.4.10), (4.4.11), and (4.4.12) together we have (4.4.13) ( Dq + Dj X + nx^/a^^ ) = 4 (n/3)^/2 ( 1+3 cos^(0) ) + 4 (n/3) (6 c, + Q2 ) + 4 (n/3 )3 / 4 (3 d, cos(9)) + 4 (n/3)’/2 ( 2c2+4c,+15c, 2 +4 c,q 2 +q 2 ^ / 4 ) + 4 (n /3 )’/ 4 3 (d2 +c,d,) cos ( 6 ) + K + 0(n"’/2 ) fo r K an absolute constant. 123 Part 5. The last term to consider is a^^ [1 - fn(^)fn(x)/fn(x) ]. From the definition o f fn'(x) = 4 (n/3 )i/ 4 cos( 8 )+ (n/3)'>/2 [ d, + dg (n/3)"’/2 + o(n"’) ] and so with (4.4.7) X [ (1+2CGs2(9))-’ - 4COS(e)(2C,+q2/2) (l+2COs2(e))"' (n/ 3 ) "1/2 - d, cos(e)( 1+2 co s 2 (e) ) - 2 (n / 3 ) "3/4 + o(n“ ’) ]. i.e., ‘PnW - 4cos(e)(2q+q2/2) (l+2cos2(e))-1 (n/3)"’/2 + d, (1- 2 co s 2 (e)) (1+2 c o s 2 (e) ) - 2 (n/3 )" 3 / 4 + o(n"i) 1. Since 8p^'^p2(x) = 2 (n/3)3/4 [1 + 2c, (n/3 ) - ' / 2 + o(n"')] X [ cos(e)+ d, ( n / 3 ) " 3 / 4 + o(n"5/4) ] = 2 (n / 3 )3 / 4 [ cos( 8 )+ 2c, cos( 0 )(n / 3 ) ’ ’/ 2 + d, (n/3)'3/4+ o(n'>)]. then 124 - ‘Pn’W/^PnCx) = ~ (n/3 ) ^ / 2 [ cos(e)+ 2c, cos(e)(n/3)"^''2 + d, (n/3)"^'^'^+ 0(n"^)l X [ 4cos(e)(1+2cos^(6))~^ - 4 cos(e)( 2 c,+q 2 / 2 ) (1+2 cos^(e))"' (n/ 3 ) " ^ ^ 2 + d, (l-2cos^(ô)) (1+2COS % ))"^ (n/3)"3/4 + o(n“’) ]. If we perform the multiplication and add the expansion for we obtain (4.4.14) 3 p2 [1 - '('p(x)f p'(x)/‘Pp(x) ] = (n/3)’/2 [1-2C0S 2(6)1 / [1+2cos 2(e)] + [ 2 c, + 2 q 2 COs2 (e) / [l+2 cos 2 (e)] ] - (n/3 ) " ’/ 4 [ 2d,cos(e)[2+3cos2(e)l / [l+ 2 cos2(e)]2 ]. To proceed with the proof we collect the expressions (4.4.6), (4.4.7), (4.4.9), (4.4.13), and (4.4.14) from Parts 1 through 5 obtaining (4.4.15) fp(x) = 4 (n/ 3 )3 / 2 ( | + 3cos2(e) - 4co^(6) ) + 4 (n/3) ( 6 c, + q 2 - 2 q 2 cos‘^(ô) ) + 4 (n/3 )3 / 4 cos(e)( 3d, - q; cos2(e) ) + (n/3 ) ’/ 2 ( (6 -q 2 2 )cos( 0 ) + l-4cos 2(e) + 8 C2 +16c, +60c,2 +16c,q 2 ) + 125 + (n/3)i/'4 cos(e)( 3 d2 + 3cA - q; q 2 ) + ( qz - q| ^/2 + (1+2COS ^(6))"^ (4cos'^(e)(4Ci+q2) +2c, + 2q2COS^(0)) + K ) + (n/3)“ ^/"* [ 2d,cos(e) (2+3cos^(e)) (1+2cos ^( 6 ))"^ + cos(e)(i+ 2 cos^(e)r^ ( qj - 2 d,cos(e)+ 8 d,cos^(e) (i+zcos^Ce))"^ )l We multiply (4.4.15) by 4 sin^(0) (n/ 3)'/2 and compare the result with (4.4.1) to yield 4 (n /3)i/2 sin^(0) f^(x) = (n/3)^ [ h(0;n) + 0(1) and the first part of the lemma is proven. Because h(0;n) = 4 sin^(0) (l+2cos % )] [1 + 0(n"’)], there is an ng such that if n>ng, then 2 sin^(0) [1+2COS 2(8)1 < h(6;n) < 6 sin2(e) [l+2cos 2(e)]. Replacing 8 by E for the lower bound and sin(e) and cos( 8 )by 1 2 sin 2 (e) < h(8:n) < 18 (n>ng) and the p ro o f is complete. S 126 5. PROOF OF THE PLANCHEREL-ROTACH TYPE ASYMPTOTICS. We use the method found in Nevai [Nell fo r the Proof (Theorem 4.3.1). Without loss o f generality take 9 3 = 0 so that w(x) = exp{-Q(x)} = exp{- (x‘^/4 +q 2 X^ / 2 +q; x )}. The proof will be done in three parts ( similar to the four stages in Sheen [Shi] ). Part I. We transform the differential equation z +f^z =0 ( cf. (3.3.5) ) by setting x = 2 (n/3 )i/ 4 cos( 6 ) and u(6 ) = z( 2(n/3)^/"’ cos(e)), then z'(x) = Uq • (-2 (n/3)'/4gin(6) )"i and z"(x) ( 4 (n/3 ) ' / 2 sin^(e) ) = Ugg - cot(e) u@ so that (3.3.5) is (4.5.1) Ü0 0 - cot(e) Ü0 + 4 (n/3 ) ' / 2 sln2 (0 ) f^(x) u(8 ) = 0. We make the further transformations (4.5.2) v(e) = u(e) h~'^^ (0;n) sln"^^^ (e) and 127 e (4.5.3) r(e ) = j h(t;n) dt 1T/2 and note that Zq = h(8 :n). Then (4.5.4) u(ô) = v(e) sin^^^ ( 8 ). Differentiate (4.5.4) (4.5.5) UQ = ( Zq sin(0) )^^^ + [ sin’^^ (8 ) Zq'^^^ Jq v, differentiate again to have Uqq = z ^ ''^ sin ^ / 2 (0 ) + [ ( -Cq sin(e) + v.j,rQ [ sin(8 ) ]q + [ sin*'^^ ( 8 ) Iqq v or (4.5.6) Uq 0 = sin^^^ (8 ) + V.J. Tg^^^ sln^^^ ( 8 ) cos( 8 ) + [s in ^ /^ ( 8 ) Tg"^/^ ]ggV. From (4.5.5) we have (4.5.7) Ug cot(e) = v^ Tg^^^ sin“ *^^ ( 8 ) cos( 8 ) + V cot(e) [ sin " * / 2 (e) rg "’^^ ]g. 128 We insert (4.5.4), (4.5.6), and (4.5.7) Into (4.5.1) to obtain sin^^2 (0 ) + V {[ sln^/^ (e) ] 0 - cot(e) [ sln"^^^ ( 6 ) } + V { 4 (n/3 ) ’ / 2 sin 2 (e) f|^(x)} sln^^^ (e) = 0 . According to Lemma 4.4.2 we replace [ 4 (n/3 ) ^ / 2 sln2(e) fp(x)l by [ (n/3)^x X h^( 6 ;n) + 0(1) 1: also from Lemma 4.4.2, [ (e) 1 = [ h^^^(e:n)x X sin^'^2 (e) ] is bounded away from 0 for c < 0 < tt-ï ( cf. (4 .4 .5 ) and (4.5.3) ), then we can divide by [ sin^^^ (0) ] to get (4.5.8) + V { I(n/3)^ r 0 ^+O(1)][r q " ’^ ^ gjnl/2 (e))/[rQ^/2 gin)/2 (g)]} + V { [sln ^ ^ 2 (0 ) ] 0 0 -cot( 0 )[sin^ ^ 2 (0 ) ^ 0 "'^ ^ ]0 } / [ r 0 ^^^ sin^^^ (0 )] = 0, i.e., + V { [(n/3)2 + 0(1)] r 0 ~ 2 } + V { [sin^^^ (0) l00 -cot(0)[sln'^2 (0) ]0}/[r0^^^ sin^'^^ (0)] = 0. 129 Again, Tq Is bounded away from zero, so the last term o f (4.5.8) is 0(1) and f or £ < 0 < Tf-£, also Is 0(1) f or £ < 0 < tt-£. Thus (4.5.9) + (n/3)^ v = 0(1) v uniformly for e < 0 < TT-f where 0 < £ < tt/ 2. In Theorem 3.3.1 we proved w"'/2(x) |Pn(x)| =0(n"^/Q ) uniformly for € < 0 < tt-£. Hence, from (4.5.9), (4.5.10) v^-p + (n/3) 2 V = 0( ) uniformly for £ < e < tt-e where 0 < £ < tt/2 . Part 2. In this portion of the proof we solve (4.5.10)considering it as a second order nonhomogeneouslinear differential equation. Thus v ( r ) = v ( 0) c o s ( n r / 3) + 3( v . ^ ( 0)/n) sin (n r/ 3) r +J { [ cos(nr/3) s i n ( n t / 3) - s i n ( n r / 3) c o s ( n t / 3) ] 0 X [-n sin^(nt/ 3)-n cos?(nt/ 3) l } 0 ( n " ^ ^ ^ ) dt. Elementary trigonometric identities give 130 v(r) = v(0) cos(mr/3) + 3(v~(0)/n) sin(nr/3) ) Jsin( n(t-Tr)/3 ) dt, i.e., (4.5.11) v(r) = v(0) cos(nr/3) + 3(v.j.(0)/n) sin(nr/3) + 0( ). We now need to find v(0) and v.^(0). From (4.5.3) r=0 implies e= 7 T/ 2 which in turn gives x=C, so v(0) = u(tT/2) h^^2 (TT/2:n) sin^^^ (TT/2) = Pn(0) (0) (TT/2;n). From (4.4.2), h*^^ (Tr/2: n) = 4[ l+0(n ) ], and by Theorem 3.3.2 Pp^(o) = A n ” *'^® cos(nTr/2- 2-3^^'^ d, + c ) + n"’'^®o(i), ( di from (4.5.6) ); thus we have (4.5.12) v(0) = 2A n"*'^® cos(nTf/2+(x) (0) + n"^''^®o(l) with (X = - 2 3 ^ / 4 + c, also we used To analyze v^(0) first consider v^. Since = Vg , we have sin"^/^ (e) (re)'^ + u(e) (1/2) Zqq sin"^/^ (e) + u(e) (- 1/ 2 )sin '^ ^ ^ ( 0 ) cos(e)(r 0 )"^. Therefore v^(0) = u@(TT/2) Zq~^^^ (TT/2) + u(iT/2) (1/2) ZQ~^^^iTr/2) ZQQin/l). Since U0 ( t t / 2 ) = z'(0) (-2(n/3)^^'^ ), v^(0) = ztO) (-2(n/3)'/^ ) r 0 " ’ / 2 ( 7 ^ / 2 ) + z (o )(i/ 2 ) ZQ~'^^^{n/2)zQQ{Tr/2). Because Y0 (Tr/2 ) = h(TT/2:n) = 4 [l+o(n ) ] and h 0 ( 7 r / 2 ) = 0(n"'/^ ). v^(0) = - (n/3) 1/4 z'(0) [l+0(n "1/2 ) ] + z(0) 0 (n’ 1 / 2 ), which, from the definition o f z(x), is v^(0) = -(n/3)'''^ Pn'(O) V '''^ W ♦ (n /3)'/"" Pn(0) Observing that (0) is ), we can conclude that v^( 0 ) = - (n/3 ) ' / 4 p^'(o) / 2 (0 ) + o(n - ' / 8 ). Recall (3.4.2); Pn'(O) = fp (0) pn(0) + a^^ (pn(0) p^-; (0) = a^^ %(Si) Pr^_| (0 ) + 0 (n "'/^ ), so that, with (3.6.12) and Theorem 3.3.2, Pn'(O) = 2 I1+0(n” '''2 )] [ cos((n-l)Tr/2 +o<) +o(1) ] = 2A3"^^^^ n®^® cos((n- 1)TT/ 2 +o< ) + n®^®o(l). Hence v.^(0) = -2 A 3 “ ^ n^/® cos((n-l)TT/2 +o< ) (0) + n®/®o(1), that is, (4.5.13) v.p(0)/n = -(2/3) A n"^^® sin( nTT/2 +cx ) (0) + n"^^®o(l). If we put (4.5.12) and (4.5.13) into (4.5.11), then v(r)=2An"'^® XI cos(nTT/2+o<) cos(nT/3) - sin(nTT/2 +o<) sin(nr/3) 1 + n"^/®o(1), i.e.. 133 (4.5.14) v (r) = 2A (0) cos(nr/3 +nTT/2 +o<) + n"^^®o(l). By the definition (4.5.2) o f v. (4.5.14) is Ppj(x) (x) = An"^^® [ 4((p^(x)/(p^(0))(sin^(e)/h(e;n)) X sin"^^2 (0) cos(nr/3+nTT/2+«) + n“ ^'"®o(l) uniformly for e < 0 < tT-c. First y x ) / fn ( 0 ) = 1 +[4(n/3) 1/2 cos2(0) +0(n‘ 1/‘^ )] (l/2)(n/3) 1/2 [,+Q(n "1/2 )] = 1 + 2c o s 2(0) + 0(n"1/2 ), and second, s i n 2(e) / h(0:n) = (1/4) [h2cos2(0)]-1 [h O (n " 1/ 2 )], hence I 4((p^(x)/fp(O))(sin2(e)/h(0;n)) ]1/2 = [ l+ 0 (n " 1/ 2 ) ]. Then (4.5.15) Pp(x) w1/2(x) = An”1/® sin“l/2(e) cos(nr/3+n7T/2+«) + n'^/®o(l) uniformly for E < 0 < tt-e. By integrating rg we obtain r(0) = (30 -sin(20) - sin(40)/4 - 3 7 T/2 ) + 134 + (n/3)“^^2 [ q2(2e -TT +sin(20)/2) - 3"^^^ (12q +Q2) tan'^ (3^^^ cot(e)) ] + (n/3)"^^^ [ q 2 (1-e) +(6 di - q 2 ) X { (24)‘ »'^2|og| [(tan%/2)-(8/3)i/2tan(e/2)+1) /(tan2(e/2)+(8/3)’/2tan(e/2)+l) 14 ( 2+(8/3)'/2) / (2-(8/3)'/z)] | + 1 2 1 /2 [ t a n " ’ ( 3i / 2t a n ( e / 2 ) - 2 i / 2 ) + t a n " ’ ( 3i / 2t a n ( 0 / 2 ) + 2 ^ /2 ) - tan"’ (31/2 -21/2) + tan"’ (31/2 +2'/2) ]} ] + ( n / 3)"’ [ (2-q2)(0-7T/2)/4+ 2 ( 4 c ,+ 2c 2+ l5 c , ^ +4 c ,q 2 + (q 2 / 8 M 2 ^ ^ '^ ) ^ X 3 ' 1/2 tan"’ (3I/2 cot(0))+(6Ci+q2^/4) I, -2q2(6q+q2/2)l2 -(6c,+q2/2)^ I3] + 0(n"5/4) 0 ith Ij ^ = J sin“ 2 (t) ( 1 + 2 cos^(t))"i dt.with V / 2 Part 3. In this section o f the proof we will determine the constant A that appears in (4.5.15) above and in the asymptotic formulas (3.3.2) and (3.3.6) forpp(x). Before we continue we will consider a short remark. 135 Suppose f(e) is a continuous function and T(e) is cos(e) or sin(e) then TT-€ T(Tf-E) J f(e) T(2nr/3 +nTr +2 7T-Î (4.5.16) lim J f(e) T (2 n r/3 + n T T +(x) de = 0 n-»oo E From J Pn^(x) w(x)dx < J Pn^(x) w(x)dx < x|< 2cos(E)(n/3)'/'^ K and (4.5.15) we see E 1 > J [a^ n“ *^^ sin“ ^ (0 ) cos?(nr/ 3 +n7 r / 2 +o<) TT-E + 2An"^'^® sin'^^^(e) cos(nr/ 3 +n7 T/2 +o<)n‘ ^'^® o(l) + o(l) ] X [ - 2 (n/3)'/'^ sin(0 ) Ide 136 or 7T-€ (4.5.17) J cos^(nr/3+n7r/2+o<) de < 1 + o(l). e Since cos^(t) = (1/2) + (1/2) cos(2t), (4.5.17) can be expressed as TT-E tt-£ J de + A^ 3"^/^ Jcos(2nTr/3+n7T+2(x)de < 1 + o(l), z z letting n-»oo and applying (4.5.15) a 2 3-^/") (tt-2e) < I: since £>0 is arbitrary and A is positive ( cf. Theorem 3.3.2 ), (4.5.18) A < 3 ’/® Apply the recursion formula to J [l-(x 2/4)(n/3)~'/2 ] Pp,2(x) w(x)dx E ■ J [ ■ (i/4)(n/3) x^Pp2(x) ] w(x)dx E to obtain 137 = 1 - (1/4)(n/3) ^ ]. Because |x|> 2(n/3)^^'^ implies that [1-(x 2/4)(n/3)"^'^21 < 0, it follow s that J [l-(x 2 / 4 )(n/3 )- 1 / 2 ] p^ 2 (^) w(x)dx E ^ J [l-(x 2/4)(n/3) ] Ppi^(x) w(x)dx |x|< 2(n/3)’/ ^ and theref ore 1 - (l/4)(n/3) (a^+i ^ + b^,^ + < J [1-(x 2/4)(n/3)"’'^2 ] p^2çx) w(x)dx |x|< 2( n / 3) ' / ^ We split the integral into two parts: A, = { x: | x | < 2(n/3)^^'^ cosCe) ) and A 2 = { X: 2(n/3)^^'^ cos(£) < IXI < 2(n/3)^^'^ }, and have (4.5.19) 1 - ( l/ 4 )(n/3 ) - 1 / 2 ( 3 ^ ^ 2 ^ p^ 2 + g ^ 2 ) < < I J + J j [l-(x 2 / 4 )(n/3 )- 1 / 2 ] p^ 2 (^) w(x)dx A| A 2 From (4,5.15) with x = 2(n/3) cos( 0 ) (then 1-(x 2 / 4 )(n/3 )" 1 / 2 =sjn2(e) ) 138 cornes J [l-(x ^/4)(n/3)"^^^ ] Pn^(x) w(x)dx x|< ZCn/S)’^"^cos(£) TT-E = J sin^(e) Pn^(x) w(x) 2 (n/3)^/^ sin(e) dô TT-E = j sin^(0) [An"*'^® sin (0)cos(nr/3+nTr/2+o<)+ri'o(l) X 2(n/3)^^'^ sin(0)d0 which upon squaring gives TT-€ = Jsin^(0) sin"k 6 ) cos^(n'r/3+nTr/2+o<) E + 2An“ ^^® sin"^'^^(0) cos(nr/3+n7T/2+{x)n"^'^® 0 ( 1) + o (l) ] X ( 2 (n/3)^'^^ sin(0 ) ) de TT-E = Jsin^(e) [2A^ cos^(nr/3+nTr/2+o<) ]d6 + o (l) 139 again noting that cos^(t) = (1/2) + (1/2) cos(2t), we have 7T-£ 7T-€ = P? { j sin^(e)de + j sin^(e) cos^(2nr/3+mT+2o<) d9 }+ o(1). Once more we apply (4.5.16) to conclude (4.5.20) lim sup J [1-(x ^/4 )(n/3)"’^^ 1 Pp^(x) w(x)dx n-+oo |x|< 2(n/3)'/4cos(E) < a 2 3 - 1 / 4 gY/Z. Since (1-(1/4)(n/3) (ap,+| * a^^) ] -» 1/2 as n-»“ , we have,in view o f (4.5.19) and (4.5.20), (1/2) TT/2+c(1-cos(£) ). Letting £-»0 we see (1/2) < a2 3-^/4 tt/2 . I.e., since A is positive, (4.5.21) A > 3 ^ / 8 .^-1/2 When we take (4.5.18) and (4.5.21) together we have the desired result A = 3 1 / 8 7T-1/2 and the proof is complete. B BIBLIOGRAPHY lAsl] R. Askey, A Review o f Chlhara’s Orthogonal Polynomials, J. Approx. Theory ( to appear ). [Ball G. Baker, Jr., Essentials of Pade Approximants, Academic Press, New York, 1975. [BrI] N. G. de Bruijn, Asymptotics Methods In Analysis, Dover, New York, 1981. iFri] G. Freud, On polynomial approximation with the weight exp{-x 2*^/2}, Acta Math. Acad.Scl. Hungar. 24(1973), 363-371. [Fr2] G. Freud, OnMarkov-Bernsteln-type Inequalities and their applications, J. Approx. Theory 19(1977), 22-37. 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