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1 ON TURÁN S INEQUALITY FOR LEGENDRE POLYNOMIALS HORST ALZER a, STEFAN GERHOLD b, MANUEL KAUERS c2, ALEXANDRU LUPAŞ d a Morsbacher Str. 0, 5545 Waldbröl, Germay alzerhorst@freeet.de b Christia Doppler Laboratory for Portfolio Risk Maagemet, Viea Uiversity of Techology, Viea, Austria sgerhold@fam.tuwie.ac.at c Research Istitute for Symbolic Computatio, J. Kepler Uiversity, Liz, Austria mauel.kauers@risc.ui-liz.ac.at d Departmet of Mathematics, Uiversity of Sibiu, 2400 Sibiu, Romaia alexadru.lupas@ulsibiu.ro Abstract. Let (x) = P (x) 2 P (x)p + (x), where P is the Legedre polyomial of degree. A classical result of Turá states that (x) 0 for x [, ] ad =, 2, 3,... Recetly, Costatiescu improved this result. He established h ( + ) ( x2 ) (x) ( x ; =,2,3,...), where h deotes the -th harmoic umber. We preset the followig refiemet. Let be a iteger. The we have for all x [,]: with the best possible factor α ( x 2 ) (x) α = µ [/2] µ [(+)/2]. Here, µ = 2 2( 2 ) is the ormalized biomial mid-coefficiet. Keywords. Legedre polyomials, Turá s iequality, ormalized biomial mid-coefficiet Mathematics Subject Classificatio. 26D07, 33C45. S. Gerhold was supported by FWF grat SFB F305, the Christia Doppler Associatio (CDG), the Austria Federal Fiacig Agecy ad Bak Austria. 2 M. Kauers was supported by FWF grat SFB F305 ad P663-N2.

2 2 ON TURÁN S INEQUALITY FOR LEGENDRE POLYNOMIALS. Itroductio The Legedre polyomial of degree ca be defied by P (x) = d!2 dx (x2 ) ( = 0,,2,...), which leads to the explicit represetatio P (x) = [/2] 2 ( ) ν ν=0 (2 2ν)! ν!( ν)!( 2ν)! x 2ν. (As usual, [x] deotes the greatest iteger ot greater tha x.) The most importat properties of P (x) are collected, for example, i [] ad [6]. Legedre polyomials belog to the class of Jacobi polyomials, which are studied i detail i [3] ad [3]. These fuctios have various iterestig applicatios. For istace, they play a importat role i umerical itegratio; see [2]. The followig beautiful iequality for Legedre polyomials is due to P. Turá [5]: (.) (x) = P (x) 2 P (x)p + (x) 0 for x ad. 3 This iequality has foud much attetio ad several mathematicias provided ew proofs, farreachig geeralizatios, ad refiemets of (.). We refer to [8,, 9, 4] ad the refereces give therei. I this paper we are cocered with a remarkable result published by E. Costatiescu [7] i He offered a ew refiemet ad a coverse of Turá s iequality. More precisely, he proved that the double-iequality (.2) h ( + ) ( x2 ) (x) 2 ( x2 ) is valid for x [,] ad. Here, h = +/2+ +/ deotes the -th harmoic umber. It is atural to ask whether the bouds give i (.2) ca be improved. I the ext sectio, we determie the largest umber α ad the smallest umber β such that we have for all x [,]: α ( x 2 ) (x) β ( x 2 ). We show that the right-had side of (.2) is sharp, but the left-had side ca be improved. It turs out that the best possible factor α ca be expressed i terms of the ormalized biomial mid-coefficiet ( ) 2 µ = (2 ) = ( = 0,,2,... ) (2) We remark that µ has bee the subject of recet umber theoretic research; see [2] ad [5]. I our proof we reduce the desired refiemet of Turá s iequality to aother iequality, which also depeds polyomially o Legedre polyomials. This latter iequality is ameable to a recet computer algebra procedure [0, ]. The procedure sets up a formula that ecodes the iductio step of a iductive proof of the iequality ad, replacig the quatities P (x),p + (x),... by real variables Y,Y 2,..., trasforms the iductio step formula ito a polyomial formula i fiitely may variables. The recurrece relatio of the Legedre polyomials traslates ito polyomial equatios i the Y k, which are added to the iductio step formula. The truth of the resultig formula for all real Y,Y 2,... ca be decided algorithmically ad is a sufficiet (i geeral ot ecessary!) coditio for the truth of the iitial iequality, if we assume that sufficietly may iitial values have bee checked. 3 A ice aecdote about Turá reveals that he used (.) as his visitig card ; see [4].

3 ON TURÁN S INEQUALITY FOR LEGENDRE POLYNOMIALS 3 2. Mai result The followig refiemet of (.2) is valid. Theorem. Let be a atural umber. For all real umbers x [,] we have (2.) α ( x 2 ) P (x) 2 P (x)p + (x) β ( x 2 ) with the best possible factors (2.2) α = µ [/2] µ [(+)/2] ad β = 2. Proof. We defie for x (, ) ad : f (x) = (x) x 2. We have f (x) α = β = /2. First, we prove that f is strictly icreasig o (0,) for 2. Differetiatio yields f (x) = 2x (x) + ( x 2 ) (x) ( x 2 ) 2. Usig the well-kow formulas P (x) = + x 2(xP (x) P + (x)) ad ( + )P + (x) = (2 + )xp (x) P (x) we obtai the represetatio (2.3) ( x 2 ) 2 f (x) = ( )xp (x) 2 (2x 2 + x 2 )P (x)p + (x) + ( + )xp + (x) 2. We prove the positivity of the right-had side of (2.3) o (0,) by typig I[]:= << SumCracker.m SumCracker Package by Mauel Kauers c RISC Liz V I[2]:= ProveIequality[ (( ) xlegedrep[, x] 2 (2x 2 + x 2 )LegedreP[, x]legedrep[ +, x] + ( + ) x LegedreP[ +, x] 2 ) > 0, From 2,Usig {0 < x < },Variable ] ito Mathematica, obtaiig, after a couple of secods, the output Out[2]= True It follows from this that f is strictly icreasig o (0,) for 2. Sice we coclude that f is eve. Thus, we obtai P (x) = ( ) P ( x), (2.4) f (0) < f (x) < f () for < x <, x 0. We have Therefore, P () = ad P () = ( + ). 2 () = 0 ad () =.

4 4 ON TURÁN S INEQUALITY FOR LEGENDRE POLYNOMIALS Applyig l Hospital s rule gives (2.5) f () = lim x (x) x 2 = 2 () = 2. Sice P 2k (0) = 0 ad P 2k (0) = ( ) k µ k, we get (2.6) f 2k (0) = µ k µ k ad f 2k (0) = µ k 2. Combiig (2.4) (2.6) we coclude that (2.) holds with the best possible factors α ad β give i (2.2). Remarks. () The proof of the Theorem reveals that for 2 the sig of equality holds o the left-had side of (2.) if ad oly if x =,0, ad o the right-had side if ad oly if x =,. (2) The umbers µ p µ q (p,q = 0,,2,... ; p q) are the eigevalues of Liouville s itegral operator for the case of a plaar circular disc of radius lyig i R 3 ; see [6]. (3) The automated provig procedure ca be applied to (2.) directly. However, owig to the computatioal complexity of the method, we did ot obtai ay output after a reasoable amout of computatio time. (4) The Mathematica package SumCracker used i the proof of the Theorem cotais a implemetatio of the provig procedure described i [0]. It is available olie at (5) The ormalized Jacobi polyomial of degree is defied for α,β > by R (α,β) (x) = 2 F (, + α + β + ;α + ;( x)/2). The special case α = β leads to the ormalized ultraspherical polyomial R (α,α) (x) = 2 F (, + 2α + ;α + ;( x)/2) = ( ) 2 (α + ) ( x 2 ) α d dx ( x2 ) +α, where (a) deotes the Pochhammer symbol. Obviously, we have R (0,0) (x) = P (x). We cojecture that the followig extesio of our Theorem holds. Cojecture. Let α > /2 ad. For all x [,] we have a (α) with the best possible factors ( x 2 ) R (α,α) (x) 2 R (α,α) a (α) = µ (α) [/2] µ(α) [(+)/2] (x)r(α,α) + ad b (α) = (x) b(α) ( x 2 ) 2(α + ). Here, µ (α) = µ / ( +α). (6) Gasper [9] has show that the ormalized Jacobi polyomials satisfy R (α,β) (x) 2 R (α,β) (x)r(α,β) + (x) 0 ( x ) if ad oly if β α >. More geeral criteria for a family of orthogoal polyomials to satisfy a Turá-type iequality are give by Szwarc [4]. Ackowledgemet. We thak the referee for brigig refereces [9] ad [4] to our attetio.

5 ON TURÁN S INEQUALITY FOR LEGENDRE POLYNOMIALS 5 Refereces [] M. Abramowitz ad I. Stegu (eds.), Hadbook of Mathematical Fuctios with Formulas ad Mathematical Tables, Dover, New York, 965. [2] H. Alzer ad B. Fuglede, Normalized biomial mid-coefficiets ad power meas, J. Number Th. 5 (2005), [3] G.E. Adrews, R. Askey ad R. Roy, Special Fuctios, Cambridge Uiv. Press, Cambridge, 999. [4] R. Askey, Rememberig Paul Turá, J. Approx. Th. 86 (996), [5] T. Bag ad B. Fuglede, No two quotiets of ormalized biomial mid-coefficiets are equal, J. Number Th. 35 (990), [6] C. Berg ad B. Fuglede, Liouville s operator for a disc i space, Mauscr. Math. 67 (990), [7] E. Costatiescu, O the iequality of P. Turá for Legedre polyomials, J. Iequal. Pure Appl. Math. 6(2) (2005), Art. 28. [ [8] D.K. Dimitrov, Higher order Turá iequalities, Proc. Amer. Math. Soc. 26 (998), [9] G. Gasper, A iequality of Turá type for Jacobi polyomials, Proc. Amer. Math. Soc. 32 (972), [0] S. Gerhold ad M. Kauers, A procedure for provig special fuctio iequalities ivolvig a discrete parameter, Proceedigs of ISSAC 05, ACM Press (2005), [] S. Gerhold ad M. Kauers, A computer proof of Turá s iequality, J. Iequal. Pure Appl. Math. 7(2) (2006), Art. 42. [ [2] Y.L. Luke, The Special Fuctios ad Their Approximatios, Vol. 2, Academic Press, New York, 969. [3] G. Szegö, Orthogoal Polyomials, 4th ed., Colloquium Publicatios, vol. 23, Amer. Math. Soc., Rhode Islad, 975. [4] R. Szwarc, Positivity of Turá determiats for orthogoal polyomials, I: Harmoic Aalysis ad Hypergroups, (K.A. Ross et al., eds.), Birkhäuser, Bosto-Basel-Berli, 998, [5] P. Turá, O the zeros of the polyomials of Legedre, Časopis Pest. Mat. Fys. 75 (950), [6] E.T. Whittaker ad G.N. Watso, A Course of Moder Aalysis, Cambridge Uiv. Press, Cambridge, 952.