Asymptotics for Entropy Integrals associated with Exponential Weights

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1 Asymtotics for Etroy tegrals associated with Exoetial Weights Eli Levi Mathematics Deartmet, The Oe Uiversity of srael, P.O. Box 3938, Ramat Aviv, Tel Aviv 639, srael. D.S. Lubisky, School of Mathematics, Georgia stitute of Techology, Atlata, GA , USA. December 8, Abstract We establish a rst order asymtotic for the etroy itegrals log W ad log ( W ) W where f g = are the orthoormal olyomials associated with the exoetial weight W. The Result Let = (c; d) be a real iterval, where c < < d ; ad let Q :! [; ) be covex. Let W := ex ( Q) ad assume that all ower momets x W (x)dx; = ; ; ; 3; ::: are ite. The we may de e orthoormal olyomials (x) = (W ; x) = x + :::; > ;

2 satisfyig m W = m ; m; = ; ; ; ::: : the last twety years, there has bee a remarkable develomet of quatitative aalysis aroud exoetial weights W, ad i articular aroud asymtotics for (x). See [3], [5], [7], [8], [], [3], [5], [7] for refereces ad reviews. this aer, we comute rst order asymtotics for the etroy itegrals ad E ( ) = E ( ) = log W log ( W ) W ; as!, for the most exlicit class of weights i [8]. These etroy itegrals arise i a umber of cotexts, for examle quatum mechaics ad iformatio theory. A extesive survey of recet develomets is give i [6]. For examle, if > ad oe cosiders the Freud weight W (x) = ex jxj ; x R; it is kow that there is the very recise asymtotic E ( ) = + + l () C + o () ; where C is a exlicit costat. See [], [8] ad also [], [6]. this aer, we treat a far more geeral class of weights tha Freud weights, but obtai oly rst order asymtotics, with a relative error of order c, some c >. To de e our classes of weights, we eed the otio of a quasi-decreasig/ quasi-icreasig fuctio. A fuctio g : (; b)! (; ) is said to be quasiicreasig if there exists C > such that g(x) Cg(y); < x y < b: articular, a icreasig fuctio is quasi-icreasig. Similarly we may de e the otio of a quasi-decreasig fuctio. The otatio f(x) g(x) meas that there are ositive costats C ; C such that for the relevat rage of x, C f(x)=g(x) C : Similar otatio is used for sequeces ad sequeces of fuctios. Throughout, C; C ; C ; ::: deote ositive costats ideedet of ; x ad olyomials P of degree at most.

3 De itio Let W = e Q where Q :! [; ) satis es the followig roerties: (a) Q is cotiuous ad Q() = ; (b) Q exists ad is ositive i fg; (c) lim Q(t) = lim Q(t) = ; t!c+ t!d (d) The fuctio T (t) := tq (t) Q(t) ; t 6= is quasi-icreasig i (; d), ad quasi-decreasig i (c; ), with (e) There exists C > such that T (t) > ; t fg; Q (x) j Q (x) j C j Q (x) j ; a.e. x fg: Q(x) The we write W F C. f i additio, there exists a comact subiterval J of such that for some C > ; the we write W F C +. Q (x) j Q (x) j C j Q (x) j ; a.e. x J; Q(x) We ow motivate this (comlicated!) de itio with some examles. Let ex (x) := x ad for j, recursively de e the jth iterated exoetial Let k; ` be oegative itegers. () Let = R ad for ; >, let ex j (x) := ex ex j (x) : Q(x) = Q`;k;; (x) := ex`(x ) ex`(); x [; ) ex k (jxj ) ex k (); x ( ; ) : articular, () Let = ( Q ;;; (x) = ; ) ad for ; >, let x ; x [; ) jxj ; x ( ; ) : Q(x) = Q (`;k;;) (x) := ex`(( x ) ) ex`(); x [; ) ex k (( x ) ) ex k (); x ( ; ) : 3

4 both cases, the subtractio of a costat esures cotiuity of Q at. t is fairly straightforward to verify that these examles of exoets corresod to W = ex ( Q) F C +. See [8] for further orietatio. aalysis of exoetial weights, a crucial role is layed by the Mhaskar- Rakhmaov-Sa umbers a t ; t R. For t > ad W F C +, c < a t < < a t < d are uiquely de ed by the equatios t = = at xq (x) a t (x a t )(a t x) dx; at Q (x) a t (x a t )(a t x) dx: t is a fairly basic result that a t is a icreasig fuctio of t R, with lim a t = c; lim a t = d: t! t! Oe of the roerties of a t is the Mhaskar-Sa idetity: for all olyomials P of degree at most ; k P W k L()=k P W k L[a ;a ] : Moreover, a are essetially the smallest umbers for which this is true [], [], [], [5]. We use the otatio t = (a t + ja t j) ; t > : This is ot to be cofused with the uit mass at t, or Dirac delta! Our result is Theorem Let W F C +. () Assume that for each " >, The there exists > such that E ( ) = T (x) = O (Q (x) " ) ; x! c + or x! d : () = = a Q (x) a (a x) (x a ) dx + O () log dt + O t s s ds + O : (3) s () E ( ) = log + O () : 4

5 Remarks (a) For the Freud weight ex jxj o = R, where >, we have s = a s = C s = ; s > ; where C may be exressed exlicitly i terms of the gamma fuctio. this case, () ad (3) become E ( ) = E ( ) = ( + O ); log + O () ; for some >. This is of course weaker the the result quoted above. (b) We ote that the growth coditio () o T esures that for each " >, the itegral o the right of () grows faster tha " as!. t is eeded i our roofs, but the result should hold without it. t is satis ed for all regularly decayig weights o the real lie, i articular for Q`;k;; for k; ` ad ; >. t is also true for weights that decay su cietly raidly ear the edoits of a ite iterval. For examle, if = ( ; ) ad > ad Q (x) = x ; x ( ; ) ; the i ; T (x) x = Q (x)= ad () is ot true for " <. But if Q (x) = ex x ex () ; x ( ; ) ; the () is true. Thus () is essetially a lower growth restrictio o Q if is a ite iterval. More geerally, () is true for Q (`;k;;) for `; k ad ; >. (c) the case of eve weights, it follows from (3) below that E ( ) ds T (a s ) : Whe T is ubouded, it may well hae that E ( ) = o (), i cotrast to the so-called Freud case where Q is of olyomial growth, ad where T. (d) Our methods ermit oe also to obtai rst order asymtotics for the more geeral itegrals jxj j( W ) (x)j jlog j (x)jj q dx; as!, rovided ; q, ad < 4. For 4, the domiat cotributio to the itegral comes from the growth of ear a, ad our 5

6 methods fail. The ecessary asymtotics to hadle these itegrals are available oly for secial weights, such as ex jxj ; > [7]. (e) The result actually holds for a larger class of weights tha that i De itio, amely it holds for the class F li + i [8]. However, the de itio of that class of weights is more imlicit, so we sare the reader the details. We reset the mai arts of the roof i the ext sectio, deferrig techical details till later. Sectio 3, we record some techical estimates, ad estimate some of the itegrals for Theorem. The remaider are doe i Sectio 4. The Proof of Theorem We begi with (), which admits a short roof: The Proof of Theorem () We aly Jese s iequality with the uit measure d (x) = ( W ) (x) dx o. This gives E ( ) = log j W j d log j W j d = log j W j 3 : Similarly, Jese s iequality gives E ( ) = log j W j d log j W j d = log j W j : We record i Lemma 3.(iii) below, the estimate j W j ; ; valid for ay xed < 4. Alyig this with = 3 ad = i the uer ad lower bouds, gives (remarkably!) E ( ) = log + O () : We shall give the mai art of the roof of () i several stes, ad defer all techical details till later. We shall use a arameter (; ) that is ideedet of ; x; ad may be di eret i di eret occurreces. However, as it is 6

7 used itely may times, ad i all uses the statemets hold true for smaller tha the give ; we ca simly kee reducig it. Proof of Theorem () Ste : Reduce to the mai art of E ( ) We write E ( ) = ( W ) log ( W ) + ( W ) Q = : E ( ) + J : Here from what we have just roved, ad () i Lemma 3.(i), E ( ) = O (log ) : Next, we slit J ito a mai art ad a tail: a J = ( W ) Q + a = : J + J : [a ;a ] ( W ) Q Desite the raid growth of Q ear c; d, oe may still aly the ideas of restricted rage iequalities (or i ite- ite rage iequalities) to show that J = O ; some (; ). This will be doe i Lemma 4.(b). The i summary, we have E ( ) = J + O : (4) Ste : Aly asymtotics of orthoormal olyomials We let = (a + ja j) ad = (a + a ) ad let L deote the liear ma of [a ; a ] oto [ ; ], ad L [ ] iverse, so that L (x) = x ad L [ ] (u) = u + : deote its The asymtotic for the orthoormal olyomials f g i [8] may be cast i the form = ( W ) L [ ] (cos ) si r = cos 4 + () + O (" ()) ; 7

8 where the order term is uiform i ad [; ] ad the error fuctio " satis es su su j" ()j < [;] ad for some (; ), ad some C > ; sufj" ()j : [ ; ]g C : The fuctio may be exressed i terms of a trasformed equilibrium measure. For the momet we just ote that () = (t) dt; [; ] ; (5) cos where is a ositive fuctio o ( ; ), with = : All this will be made more recise i Lemma 3.. A little trigoometry the shows that that uiformly i ad ; ( W ) L [ ] (cos ) si = + si ( + ()) + O (" ()) = + si cos () + cos si () + O (" ()) : The the substitutio x = L [ ] J = = = a a ( W ) Q ( W ) L [ ] (cos ) + +O Q L [ ] (cos ) d + Q L [ ] (cos ) (cos ) ad this last asymtotic shows that si Q (cos ) d L [ ] cos si () d j" ()j Q L [ ] (cos ) d Q L [ ] (cos ) si cos () d = : J ; + J ; + J ;3 + O (J ;4 ) : (6) The mai art here is J ;. Lemma 3.(v), we shall estimate J ; below, ad show that for each " >, there exists C > such that J ; C " : (7) 8

9 Moreover, i Lemma 3.(v), we shall use the estimates for " ad uer bouds for Q to show that for some (; ) ; J ;4 C : (8) Ste 3: Aly Jackso Theorems to estimate J ; ad J ;3 We shall describe the estimatio of J ;. That for J ;3 is similar. Now by its de itio i (5), () = ad () = ad so the substitutio = () gives () = (cos ) si > i (; ) where g () = Q L [ ] J ; = cos [ ] g () cos d; () si [ ] () = [ ] () ad of course, [ ] deotes the iverse of. Now if g was ideedet of, the Riema-Lebesgue Lemma would show that J ; = o () ;! : Sice g is ot ideedet of, we must roceed di eretly. The orthogoality of cos to trigoometric olyomials S of degree less tha gives jj ; j if deg(s)< jg () S ()j d: The Jackso tye Theorems [4, Theorem.3,. 5] show that jj ; j C su juj= jg ( + u) g ()j d: Usig estimates for Q ; ; we shall estimate this su ad show i Lemma 4.3 that for some (; ) ; J ; = O : As we oted, a similar estimate holds for J ;3. The (6), (7), (8) ad these last estimates show that for some (; ) ; J = Q L [ ] (cos ) d + O : So (4) gives E ( ) = Q L [ ] (cos ) d 9 + O :

10 Now the reverse substitutio x = L [ ] (cos ) gives the rst form of the asymtotic i Theorem. Ste 4: The other forms of the asymtotic We shall use a little otetial theory. Let deote the equilibrium measure of mass for the exteral eld Q. This is a o-egative measure o [a ; a ] with total mass, that has a umber of extremal roerties. The oe that we shall use ivolves the otetial V (x) = log jx yj d (y) : t is kow that where V (x) + Q (x) = c ; x [a ; a ] ; (9) c = log t dt: For a statemet of this, with this reresetatio of the costat c, see [8, Theorem.7,. 46] or [3]. For a detailed discussio of the otetial theory, see [5]. The a Q (x) a (a x) (x a ) dx = a (c V ) (x) a (a x) (x a ) dx " a a = c + a = c + log a a = c + log = # log jx yj a (a x) (x a ) dx d (y) d (y) log t dt: the third last lie we used the fact that logjx yj is bouded above i the itegral, to allow iterchage of the itegrals (Fubii s Theorem). Moreover, i the secod last lie, we used the classical equilibrium otetial for a iterval [a; b] [5, ]. So we have the secod form of the asymtotic. Fially, it was roved i [8, Lemma.,. 5] that a t are absolutely cotiuous fuctios of t, ad so the same is true of t. The log s s s dt = ds dt = dt ds: t t s s So we have the last form i (3). Of course s exists a.e. ad is o-egative, so the iterchage is justi ed.

11 3 Techical Estimates this sectio, we record a umber of estimates for orthogoal olyomials ad related quatities, ad also establish some simle cosequeces of them. Throughout, we assume that W = ex ( Q) F C + ad that () holds. Moreover, we deote the zeros of the th orthoormal olyomial by Note that for large eough, x < x ; < x ; < ::: < x < x : as follows from Lemma 3.(ii) below. First some estimates ivolvig Q : Lemma 3. (i) For ; ad for some > ; a < x < x < a ; s ja j Q a = Q (a ) C; () T (a ) (ii) From () follows that for each " > ; (iii) Uiformly for ad s [ ; ] ; (iv) For ; s Q T (a ) (a ) ; () ja j = O : () T (a ) = O ( " ) ;! : (3) Q L [ ] (s) s C: a = a (v) Give " >, there exists C > such that J ; = a T (a ) : (4) Q (x) a (a x) (x a ) dx C " : Proof (i) The rst two estimates are (3.7) ad (3.8) of Lemma 3.4 i [8,. 69] ad art of (3.8) of Lemma 3.5 i [8,. 7]. The third follows from Lemma 3.5(c)

12 i [8,. 7]. (ii) We have from () ad (i) above, T (a ) = O (Q (a ) " ) = O ( " ) ;! : (iii) After the substitutio s = L (x), the desired iequality becomes jq (x)j (a x) (x a ) C; x [a ; a ] : This is a weaker form of (3.4) of Lemma 3.8 i [8,. 77]. (iv) This is (3.5) of Lemma 3. i [8,. 8]. (v) Now as T is quasi-icreasig i (; d) ad quasi-decreasig i (c; ) ; a Q (x) a (a x) (x a ) dx a xq (x) = a T (x) (a x) (x a ) dx C a xq (x) max ft (a ) ; T (a )g a (a x) (x a ) dx = C max ft (a ) ; T (a )g ; by de itio of a. The (ii) gives the result. Next, we record estimates ivolvig the orthoormal olyomials ad their asymtotics. Some of this was already discussed i the revious sectio, though here we shall give more details. Recall that there is a equilibrium measure with total mass ad suort o [a ; a ] ad satisfyig (9). Now is absolutely cotiuous ad has desity, so that for E [a ; a ] ; (E) = (t) dt where > i (a ; a ) ad a E a (t) dt = : t is ofte referable to work o the xed iterval [ [a ; a ], which varies with. Accordigly, we de e so that > i ( (x) = ; ) ad L [ ] (x) ; x ( ; ) ; (x) dx = : ; ], rather tha o

13 As i the roof of Theorem, we let be the fuctio de ed by (5). Moreover, we use the same J otatio for some itegrals as i the roof of Theorem. Lemma 3. (i) For each " > ; T k W k L() =6 =3 (a ) max ; T (a =6 ) = O =6+" : (5) a ja j (ii) There exists such that for ; x a ad x a ; (6) where 8 s 9 < = : T (a ja j = ) ; =3 : (7) (iii) Let < < 4. Without assumig (), we have for ; k W k L() : (iv) Uiformly for ad [; ] ; = ( W ) L [ ] (cos ) si r = cos 4 + () + " () ; (8) where the error fuctio " satis es su su [;] ad for some (; ), ad some C > ; (v) For some > ; j" ()j < (9) sufj" ()j : [ ; ]g C : () J ;4 = j" ()j Q L [ ] (cos ) d C : Proof (i) The relatio is Theorem.8 i [8,. ]. The uer boud follows because of our boud o T (a ) i the revious lemma. (ii) See [8, Theorem.9(f),. 3]. 3

14 (iii) This is art of Theorem 3.6 i [8,. 36]. (iv) The asymtotic (8) with the estimate () o the error term " is Theorem.4 i [8,. 6]. The uiform boud (9) follows from the estimate [8, Theorem.7,. ] su j W j (x) j(x a ) (x a )j =4 C: x The the substitutio x = L (cos ) gives uiformly for ad [; ] ; = j W j L [ ] (cos ) si C: Thus the left-had side of (8) is bouded uiformly i ad [; ], ad the rst term o the right-had side of (8) is also uiformly bouded. The the same is true of ". (v) From () ad (), j" ()j Q L [ ] (cos ) d C max fq (a ) ; Q (a )g C : Next, the boud (9) o " ad the boud () o Q (a ) give! + j" ()j Q L [ ] (cos ) d C : Now add these last two estimates. 4 Estimatio of the tegrals this sectio, we ish the estimatio of the J itegrals de ed i the roof of Theorem (). We use the same otatio as there. We rst boud some tail itegrals. Lemma 4. There exists > such that the followig hold: (a) a a ( W ) = O : (b) J = ( W ) Q = O : [a ;a ] Proof 4

15 (a) This is really cotaied i Theorem 8.4(b) of [8,. 38]. Now = ( W ) a a ( W ) if deg(s)< a = if deg(s)< = + O ; a a (x S (x)) W (x) dx a (x S (x)) W (x) dx= if deg(s)< (x S (x)) W (x) dx by (8.3) i Theorem 8.4(b) [8,. 38]. Here we are usig the extremal roerties of leadig coe ciets of orthogoal olyomials, ad the fact that our class of weights is cotaied i the class F s = cosidered there. (b) Now a3 a ( W ) Q a3 Q (a 3 ) ( W ) a C ; () by () ad (a) of this lemma. To hadle the itegral over [a 3 ; d), we use a restricted rage iequality from [8, (4.), Lemma 4.4,. 99]. Choosig = ad = t = there, gives j W j (x) ex (U (x)) k W k L() where U is a exlicitly give fuctio. We shall ot eed this exlicit form; istead we eed the estimate (4.8) from Lemma 4.5 there. t yields where C; C > are ideedet of ad r. The d a 3 ( W ) Q = U (a r ) C r C ; r 3; () 3 ( W ) (a r ) Q (a 3r ) a r dr C k W k L () 3 ex C r C r a r dr: Here we have used our boud () o Q (a r ). Next, we ote the relatio [8, (3.47), Theorem 3.(b),. 79] a r a r rt (a r ) C ; r > : (3) 5

16 This ad our boud from Lemma 3.(i) o k W k L() give d a 3 ( W ) Q C 3 ex( C 4 C ): summary, this iequality, ad () give d a ( W ) Q C ; ad a similar relatio holds over (c; a ). the estimatio of J ;, we shall eed smoothess roerties of the fuctio Lemma 4. (a) For ad [; ] ; (b) For ad ; (; ), () = (t) dt; [; ] : cos C () C (si ) : (4) j () ()j C! =4 j j jsi j : (5) (c) For ad u; v [; ] ; C ju vj [ ] (u) [ ] (v) C ju vj =3 : (6) (d) Let >. For ad u; v [ ; ] ; (u) (v) C ju vj= 5 : (7) [ ] [ ] Proof (a) Theorem 6.(b) i [8,. 46] asserts that for ad u [ ; ] ; C u h (u) (u) C h (u) where ad h (u) = ( = u + ) + u + ja j T (a ) : 6

17 The we deduce that C u (u) C u : The () = (cos ) si satis es (4). (b) By Theorem 6.3(a) i [8,. 48], with (s) = s =, ad by this last idetity, jcos j () ()j C j C cos j h (cos ) =4 : j si (c) t follows from (a) that for ad [; ] ; C () C mi f; g : O itegratig this iequality, we see that for ; [; ] ; =4 C j j j () ()j C j j 3 : Settig = [ ] (u) ad = [ ] (v) gives for all ad u; v [; ] ; [ ] C (u) [ ] [ ] (v) ju vj C (u) [ ] (v) 3 : (d) Now the left-had side of (7) equals [ ] (v) [ ] (u) [ ] (u) [ ] (v) 3 6 [ ] (v) [ ] =4 (u) 7 C 4 5 si [ ] (v) si [ ] (u) by (4) ad (5). We cotiue this usig (6) as jv si [ ] 5 uj = mi si [ ] (u) ; si (v)o [ ] : ; (v) Now assume that u; v [ ; ]. The as [ ] () = ad [ ] () =, we see from (6) that [ ] (u) ; [ ] (v) C 3 ; C 4 7

18 ad so mi si [ ] (u) ; si [ ] The (7) follows. Now we tur to estimatio of J ; : Lemma 4.3 For some > ; J ; = o (v) C 5 : Q L [ ] (cos ) si cos () d = O : Proof As i the roof of Theorem, Jackso tye theorems yield the estimate jj ; j C su juj= jg ( + u) g ()j d; where g () = Q L [ ] cos [ ] () si [ ] () = [ ] () : (We take the di erece g ( + u) =. From (), for juj =; g () as if + u lies outside [; ]). Let jg ( + u) g ()j d C = C [ ] () d= h [ ] () [ ] i C =3 ; (8) by (6). A similar estimate holds for the itegral over [; ]. Now cosider [ ; ] ad juj =. For some betwee + u ad ; Q L [ ] cos [ ] ( + u) Q L [ ] cos () [ ] = Q L [ ] cos [ ] () si [ ] () = () [ ] juj C= si [ () ] ; by (4) ad Lemma 3.(iii). Here the lower boud i (6) ad the fact that () = ; () = show that [ ] () C ; C 8

19 ad hece, Q L [ ] cos [ ] ( + u) Q L [ ] cos [ ] () C : Also, from (7), [ ] ( + u) [ ] () C =+5 ; ad from the uer boud i (6), si [ ] ( + u) si [ ] () C =3 : Combiig these estimates i the obvious way ad usig our boud () for Q gives jg ( + u) g ()j C = [ ] ( + u) + C =3 = [ ] ( + u) + C =+5 : tegratig ad makig the obvious substitutio gives jg ( + u) g ()j d C maxf;=+5g = C =+5 ; recall that = =. Together with (8), this shows that for some > ; jj ; j C : Fially, we ote that a similar estimate holds for J ;3. Refereces [] A.. Atekarev, V. S. Buyarov, J.S. Dehesa, Asymtotic Behaviour of L Norms ad Etroy for Geeral Orthogoal Polyomials, Russia Acad. Sci. Sb. Math., 8(995), [] V.S. Buyarov, J.S. Dehesa, A. Martiez-Fikelshtei, E.B. Sa, Asymtotics of the formatio Etroy for Jacobi ad Laguerre Polyomials with Varyig Weights, J. Arox. Theory, 99(999), [3] V.S. Buyarov ad E.A. Rakhmaov, Families of Equilibrium Measures i a Exteral Field o the Real Axis, Math. Sborik., 9(999), [4] R. DeVore ad G.G. Loretz, Costructive Aroximatio, Sriger, Berli,

20 [5] P. Deift, Orthogoal Polyomials ad Radom Matrices: A Riema Hilbert Aroach, Courat Lecture Notes, 999. [6] J.S. Dehesa, A. Martiez-Fikelshtei, J. Sachez-Ruiz, Quatum formatio Etroies ad Orthogoal Polyomials, J. Com. Al. Math., 33(), [7] T. Kriecherbauer ad K. McLaughli, Strog Asymtotics of Polyomials Orthogoal with Resect to Freud Weights, teratioal Mathematics Research Notices, 6(999), [8] A.L. Levi ad D.S. Lubisky, Orthogoal Polyomials for Exoetial Weights, Sriger, New York,. [9] G.G. Loretz, M. vo Golitschek ad Y. Makovoz, Costructive Aroximatio: Advaced Problems, Sriger, Berli, 996. [] H.N. Mhaskar, troductio to the Theory of Weighted Polyomial Aroximatio, World Scieti c, 996. [] H.N. Mhaskar ad E.B. Sa, Where Does The Su Norm of a Weighted Polyomial Live?, Costr. Arox, (985), 7-9. [] H.N. Mhaskar ad E.B. Sa, Where Does The L Norm of a Weighted Polyomial Live?, Tras. Amer. Math. Soc., 33(987), 9-4. [3] P. Nevai, Geza Freud, Orthogoal Polyomials ad Christo el Fuctios: A Case Study, J. Arox. Theory, 48(986), [4] E.A. Rakhmaov, Strog Asymtotics for Orthogoal Polyomials Associated with Exoetial Weights o R, (i) Methods of Aroximatio Theory i Comlex Aalysis ad Mathematical Physics, (eds. A.A. Gochar ad E.B. Sa ), Nauka, Moscow, 99, [5] E.B. Sa ad V. Totik, Logarithmic Potetials with Exteral Fields, Sriger, Berli, 997. [6] J. Sachez-Ruiz, J. S. Dehesa, Etroic tegrals of Orthogoal Hyergeometric Polyomials with Geeral Suorts, J. Comut. Al. Math., 8(), 3-3. [7] W. Va Assche, Asymtotics for Orthogoal Polyomial s, Sriger Lecture Notes i Maths., Vol. 65. Sriger, Berli, 987. [8] W. Va Assche, R.J. Yaez, J.S. Dehesa, Etroy of Orthogoal Polyomials with Freud Weights ad formatio Etroies of the Harmoic Oscillator Potetial, J. Math. Phys., 36(995),