Real zeros of F hypergeometric polynomials 2 1 DDominici1 DepartmentofMathematics,StateUniversityofNewYorkatNewPaltz,1HawkDr.Suite9,NewPaltz,NY12561-2443,USA SJJohnston DepartmentofMathematicalSciences,UniversityofSouthAfrica,POBox392,UNISA0003,SouthAfrica KJordaan2 3 DepartmentofMathematicsandAppliedMathematics,UniversityofPretoria,LynnwoodRoad,Pretoria,0002,SouthAfrica 1 0 2 n a J Abstract 1 Weuseamethodbasedonthedivisionalgorithmtodetermineallthevaluesoftherealparametersbandcforwhichthe 2 hypergeometricpolynomials2F1( n,b;c;z)haven real, simplezeros. Furthermore,we usethe quasi-orthogonality − ] ofJacobipolynomialstodeterminetheintervalsonthereallinewherethezerosarelocated. A C Keywords: Orthogonalpolynomials,zeros,hypergeometricpolynomials. 2000MSC:33C05,33C45,42C05. . h t a m [ 1. Introduction 1 The F hypergeometricfunctionisdefinedby(cf. [1]) v 2 1 1 7 F (a,b;c;z)=1+ ∞ (a)k(b)k zk, z <1, 7 2 1 (c) k! | | 4 Xk=1 k 1. wherea, bandcarecomplexparameters, c0and | | − − whenz = 1providedthatRe(c a b+1) > 0. Whenoneofthenumeratorparametersisequaltoanonpositive ar integer,say−a= n, n N ,thes−eries−terminatesandthefunctionisapolynomialofdegreeninz. 0 − ∈ The problem of describing the zeros of the polynomials F ( n,b;c;z) when b and c are complex arbitrary 2 1 − parameters, has not been solved. Even when b and c are both real, the only cases that have been fully analyzed imposeadditionalrestrictionsonbandc. Recentpublications(cf.[4],[6],[7],[8],[11]and[13])consideredthezero locationofspecialclassesof F ( n,b;c;z)withrestrictionsonthe parameters bandc. Resultsonthe asymptotic 2 1 − zerodistributionofcertainclassesof F ( n,b;c;z)havealsoappeared(cf. [5],[10],[14],[15]and[27]). 2 1 − Emailaddresses:dominicd@newpaltz.edu(DDominici),johnssj@unisa.ac.za(SJJohnston),kerstin.jordaan@up.ac.za(K Jordaan) 1ResearchbythisauthorwassupportedbyaHumboldtResearchFellowshipforExperiencedResearchersfromtheAlexandervonHumboldt Foundation. 2ResearchbythisauthorwaspartiallysupportedbytheNationalResearchFoundationundergrantnumber2054423. PreprintsubmittedtoJournalofComputationalandAppliedMathematics January22,2013 Differenttypes of F ( n,b;c;z)have well-established connectionswith classical orthogonalpolynomials, no- 2 1 − tably the Jacobi polynomialsand the Gegenbauerorultrasphericalpolynomials(cf. [1]). Forthe rangesof the pa- rameterswherethesepolynomialsareorthogonal,informationaboutthezerosof F ( n,b;c;z)followsimmediately 2 1 − fromclassicalresults(cf. [1], [28]). Theasymptoticzero distributionof F ( n,b;c;z)whenbandc dependonn 2 1 − canbededucedfromrecentresultsbyKuijlaars,Mart´ınez-Finkelshtein,Mart´ınez-Gonza´lezandOrive(cf.[20],[21], [22], [23])onthe asymptoticzerodistributionofJacobipolynomialsP(α,β)(x) whentheparameters αandβdepend n onn. Conversely,ifthedistributionofthezerosof F ( n,b;c;z)isknown,thisleadstoinformationaboutthezero 2 1 − distribution of other special functions (cf. [6]). This makes knowledge of the zero distribution of F ( n,b;c;z) 2 1 − extremelyvaluable. Theorthogonalityofthepolynomials F ( n,b;c;z)giveninthenexttheoremfollowsfromtheorthogonalityof 2 1 − theJacobipolynomials(cf. [25, p. 257-261])andcanalso be proveddirectlyusingtheRodrigues’formulaforthe polynomials F ( n,b;c;z)(cf. [1,p. 99])aswasdonein[9]and[21]. 2 1 − Theorem1(cf.[9]). Letn N ,b, c Rand c0andb<1 n; −∞ − (ii) (0,1)forc>0andb>c+n 1; − (iii) (1, )forc+n 10. Then, ∈ ∈ (i) Forb>c+n,allzerosof F ( n,b;c;z)arerealandlieintheinterval(0,1). 2 1 − (ii) Forc0performthedivisionof f by f toobtain k k 1 k − f (x)=q (x)f (x) r (x). k 1 k 1 k k − − − Define r (x) ifr (x).0 f (x)= k k k+1 f (x) ifr (x) 0  k′ k ≡ andgeneratethesequenceofnumbersc1,c2,...where lc(f ) k+1 ifr (x).0 k ck =lc(fk 1) . 0 − ifrk(x) 0 ≡  3 When f isconstant,thealgorithmterminates. k Notethatthealgorithmmustterminate,sincethedegreesofthepolynomials f (x)decreaseoneachstep. k Thenwehavethefollowingtheoremwhichwewillapplyto F ( n,b;c;z). 2 1 − Theorem3(cf. [24],Theorem10.5.7,p.339). Let f beapolynomialofdegreenwithrealcoefficients. Then f has onlyrealzeros ifandonlyifthe abovealgorithmproducedn 1non-negativenumbersc ,...,c . Moreover, the 1 n 1 zerosof f areallrealandsimpleifandonlyifthenumbersc ,−...,c areallpositive. − 1 n 1 − 3. Mainresults We shallassumethroughoutourdiscussionthat b,c R with b,c , 0, 1,..., n+1. Theassumptiononb is ∈ − − madetoensurethat F ( n,b;c;z)isapolynomialofdegreen. 2 1 − Proposition4. Letb,c R. Then, ∈ 1. Thezerosof F ( 2,b;c;z)arerealandsimpleifandonlyifeither(seeFigure2): 2 1 − (i) c< 1andc0orb0 andb<0orc 1,c,0andb< 1orb>c+1. − − b b b=c+1 b=c -1 c -2 -1 c -1 Fhaigsuornely2:reVaallusiemspolfebzearnodscforwhich2F1(−2,b;c;z) Fhaigsuornely3:reVaallusiemspolfebzearnodscforwhich2F1(−3,b;c;z) Theorem5. Foranyintegern 4,thepolynomial F ( n,b;c;z)hasonlyrealandsimplezerosifandonlyif(c,b) 2 1 ≥ − belongstooneofthefourn-dependentregions ,..., definedby 1 4 R R = c+n 2 1, b<2 n , 2 R { − − } = c> 1, b>n 2, b>c+n 2 , 3 R { − − − } = 1c+n 1. Theremainingzeroisnegative. − − (ii) (1, )for1 n0andc+n 20and1 n0. 2 1 − − 2. Thediscriminantof 3b 3b(b+1) b(b+1)(b+2) F ( 3,b;c;z)=1 z+ z2 z3 2 1 − − c c(c+1) − c(c+1)(c+2) b2(b+1)(b c 1)(b c)2 isgivenby∆ =108 − − − (cf. [16])andtherefore F ( 3,b;c;z)hasrealsimpleroots 3 c4(c+1)3(c+2)2 2 1 − ifandonlyif∆ >0. 3 Thefollowingtwolemmaswillbeusedintheproofofourmainresult. 6 b c= 1 − 1 0 c − b=c+n 1 c=0 − b=c+n 2 − b=2 n − b=1 n − Figure6:ValuesofbandccorrespondingtothosedescribedinTheorem7 Lemma9. Let k n 2k n b 3 k b 1+c αk,l = (cid:16) −2 (cid:17)l(cid:16) −4− − (cid:17)l(cid:16) − −2 (cid:17)l k b 2 2k b 1 n k n 1 c −2− l −4− − l − −2 − l (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) andletthesequenceθ berecursivelydefinedby k θ =α θ , (1) k+1 k,1 k 1 − fork 2,...,n 2 ,with ∈{ − } nb (b c)(n 1) θ = , θ = − − . 1 − c 2 c(b+n 1) − Then, (b+1)(n+c) n 1 θ = α , k=1,2,... − , (2) 2k c(n+b+1) 1,k $ 2 % and nb n 2 θ2k+1 = α2,k , k=0,1,... − . (3) − c $ 2 % ProofofLemma9. Weprovetheresultbyinductiononk. Whenk =1,theright-handsideof(2)is (b+1)(n+c)(cid:16)1−2n(cid:17)(cid:16)−n+4b+1(cid:17)(cid:16)c−2b(cid:17) = (b−c)(n−1) c(n+b+1) b+1 n+b 1 n+c c(b+n 1) − 2 − 4− − 2 − (cid:16) (cid:17)(cid:16) (cid:17)(cid:16) (cid:17) whichisθ asrequired. 2 We now assume the result is true for k = t and prove the result true for k = t +1. If we let k = t+1 on the 7 b c= 1 − 1 0 c − b=c+n 1 c=0 − b=c+n 2 − b=2 n − b=1 n − Figure7:ValuesofbandccorrespondingtothosedescribedinTheorem8 right-handsideof(2),weobtain (b+1)(n+c) RHS = α c(n+b+1) 1,t+1 = (bc(+n+1)b(n++1c)) (cid:16)b1+−21n ++tt(cid:17)(cid:16)c−2nb+c++t(cid:17)t(cid:16)−n+nb4++b11++t(cid:17)t α1,t since (a)k+1 =(a+k)(a)k − 2 − 2 − 4− (1 n+2t(cid:16))(c b+(cid:17)2(cid:16)t)( n (cid:17)b(cid:16) 1+4t) (cid:17) =θ − − − − − bytheinductivehypothesis 2t( b 1+2t)( n c+2t)( n b+1+4t) − − − − − − =α θ 2t+1,1 2t =θ from(1) 2t+2 andtheresultfollowsbyinduction. Thesecondrelation(3)maybeprovedbyinductioninasimilarway. Lemma10. Letn 4. Then,forallk 2,...,n 1 , ≥ ∈{ − } (n k)(n+c k)(b+1 k)(b c+1 k) − − − − − (4) (n+b+2 2k)(n+b 2k)(n+b+1 2k)2 − − − and (n 1)(n+c 1)(b c) − − − (5) (n+b 2)(n+b 1)2 − − arepositiveifandonlyif(c,b) . 1 2 3 4 ∈R ∪R ∪R ∪R ProofofLemma10. Sincen 1>0and(n+b 1)2 >0foralln N,b R,weseethat(5)ispositiveifandonly − − ∈ ∈ 8 if (c,b) c<1 n, b>c, b<2 n = or 1 1 ∈{ − − } A ⊃R c>1 n, b1 n, b>c, b>2 n = ( )or 3 3 4 ∈{ − − } A ⊃ R ∪R c<1 n, b2 n = . ∈{ − − } ∅ (n k) Clearly − >0forallk 2,...,n 1 ,n Nandb R,b,,3 n,5 n,...,n 5,n 3. (n+b+1 2k)2 ∈{ − } ∈ ∈ − − − − − Furthermoreb>n 2ifandonlyifn+b+2 2k>0,n+b 2k>0andb+1 k>0forallk 2,...,n 1 . − − − − ∈{ − } Hence,whenb>n 2,(4)willbepositiveforallk 2,...,n 1 ifandonlyif − ∈{ − } (c,b) b>c+k 1, c+n k >0, k =2,...,n 1 = b>c+n 2, c> 1 = or 3 ∈{ − − − } { − − } R bc+k 1, c+n k >0, k =l+1,...,n 1 = or ∈{ − − }∩{ − − − } ∅ b>c+k 1, c+n k >0, k =2,...,l ∈{ − − }∩ bn 2 = . { − − − }∩{ − } ∅ Similarly,b<2 nifandonlyifb+1 k<0,n+b+2 2k<0andn+b 2k<0fork 2,...,n 1 .Hence, − − − − ∈{ − } whenb<2 n,(4)willbepositiveforallk 2,...,n 1 ifandonlyif − ∈{ − } (c,b) b>c+k 1, c+n k <0, k =2,...,n 1 = b>c+n 2, c<2 n = or 4 1 ∈{ − − − } { − − } A ⊃R b0, k =2,...,n 1 = b 1 = or 5 2 ∈{ − − − } { − } A ⊃R b>c+k 1, c+n k <0, k =2,...,l b0, k =l+1,...,n 1 = or ∈{ − − }∩{ − − − } ∅ b0, k =2,...,l b>c+k 1, c+n k <0, k =l+1,...,n 1 = . ∈{ − − }∩{ − − − } ∅ Fortheremainingcasewhere2 n 1,b>c+n 2 ∈{ − } − − − andn 4 1,b>c+n 2andb>n 3. − − ∈{ − } − − − Hence,when2 n0forallk 1,...,n 1 when(c,b) . Theresultnow k 1 2 3 4 ∈{ − } ∈R ∪R ∪R ∪R followsfromTheorem3. ProofofTheorem6. From[2],Corollary4(i),weknowthatfor 1<α<0and 1<β<0,theJacobipolynomials P(nα−1,β−1)(x)haverealsimplezerosand(n 2)ofthemareinthein−terval( 1,1).T−hesmallestzeroissmallerthan 1 andthelargestzeroislargerthan1. Equiv−alently,thesame istrueforthe−zerosofJacobipolynomialsP(α,β)(x)wh−en n 2<α< 1and 2<β< 1. − − − − (i) OneoftheconnectionsbetweenJacobipolynomialsandthepolynomials F ( n,b;c;z)isgivenby(cf. [25], 2 1 − p. 254,eq. 3) ( 1)n(1+β) x+1 P(α,β)(x)= − n F n,1+α+β+n;1+β; (6) n n! 2 1 − 2 ! whereα=b n candβ=c 1. Theconditions 2<β< 1and 2<α< 1areequivalentto 1

Real zeros of 2F1 hypergeometric polynomials

Similar Real zeros of 2F1 hypergeometric polynomials