Table Of Content

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, c<N0 = 0,1,2,... and − { } 0 3 α(α+1)...(α+k 1) , k N, 1 (α)k =1 − , k =∈ 0, α,0 v:  Xi isPochhammer’ssymbol.Thisseriesconvergeswhen z <1andalsowhenz=1providedthatRe(c a b)>0and | | − − 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:[email protected](DDominici),[email protected](SJJohnston),[email protected](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 c<N . Then F ( n,b;c;z)isthenthdegreeorthogonalpolynomial 0 0 2 1 ∈ ∈ − − forthen-dependentpositiveweightfunction zc 1(1 z)b c n ontheintervals − − − | − | (i) ( ,0)forc>0andb<1 n; −∞ − (ii) (0,1)forc>0andb>c+n 1; − (iii) (1, )forc+n 1<b<1 n. ∞ − − Asaconsequenceoforthogonality,weknowthatforeachn,thenzerosof F ( n,b;c;z)arereal,simpleandlie 2 1 − intheintervaloforthogonalityforthecorrespondingrangesoftheparameters(see,forexample,[12],Theorem4)as illustratedinFigure1. b G 1 b=c+n 1 − c b=1 n − G 3 G 2 Figure1:Valuesofbandcforwhich2F1( n,b;c;z)isorthogonalandhasnrealsimplezerosintheintervals(0,1),( ,0)and(1, )areindicated − −∞ ∞ byregions 1, 2and 3respectively. G G G In his classical paper (cf. [19]), Felix Klein obtained results on the precise number of zeros of F (a,b;c;z) 2 1 thatlie in eachof the intervals( ,0),(0,1)and (1, )by generalizingearlierresultsofHilbert(cf. [17]). These −∞ ∞ Hilbert-Kleinformulasarevalidforhypergeometricfunctionsandnotonlyforpolynomials. Szego¨ recapturedthese results for the special case of Jacobi polynomials P(α,β)(x), which have a representation as F ( n,b;c;z), in the n 2 1 − intervals( , 1),( 1,1)and(1, )(cf. [28],p.145,Theorem6.72). Thenumberandlocationof therealzerosof −∞ − − ∞ F ( n,b;c;z)forbandcrealcanbededucedasfollows. 2 1 − 2 Theorem2(cf. [11],Theorem3.2). Letn N,b, c Randc>0. Then, ∈ ∈ (i) Forb>c+n,allzerosof F ( n,b;c;z)arerealandlieintheinterval(0,1). 2 1 − (ii) Forc<b<c+n,c+ j 1<b<c+ j, j=1,2,...,n, F ( n,b;c;z)has jrealzerosin(0,1).Theremaining 2 1 − − (n j) zeros of F ( n,b;c;z) are all non-realif (n j) is even, while if (n j) is odd, F ( n,b;c;z) has 2 1 2 1 − − − − − (n j 1)non-realzerosandoneadditionalrealzeroin(1, ). − − ∞ (iii) For0 < b < c, allthezerosof F ( n,b;c;z)arenon-realifniseven,while ifnisodd, F ( n,b;c;z)has 2 1 2 1 − − onerealzeroin(1, )andtheother(n 1)zerosarenon-real. ∞ − (iv) For n < b < 0, j < b < j+1, j = 1,2,...,n, F ( n,b;c;z)has j realnegativezeros. The remaining 2 1 − − − − (n j) zeros of F ( n,b;c;z) are all non-realif (n j) is even, while if (n j) is odd, F ( n,b;c;z) has 2 1 2 1 − − − − − (n j 1)non-realzerosandoneadditionalrealzeroin(1, ). − − ∞ (v) Forb< n,allzerosof F ( n,b;c;z)arerealandnegative. 2 1 − − Thevaluesoftheparametersbandcforwhich F ( n,b;c;z)hasexactlynrealsimplezerosin(0,1)givenin 2 1 − Theorem2 (i)and(ii) correspondto thosein Theorem1(ii) whilethe parametervaluesinTheorem1(i) thatensure thatallthezerosof F ( n,b;c;z)arereal,simpleandnegativearethesameasthoseinTheorem2(iv)and(v).The 2 1 − valuesofbandcinTheorem1(iii)forwhichnzerosarein(1, )canalsobeobtainedfromTheorem2(iv)and(v) ∞ usingthetransformation(cf. [1,p. 79,(2.3.14)]) (c b) F ( n,b;c;z)= − n F ( n,b;1 n+b c;1 z) 2 1 2 1 − (c) − − − − n duetoPfaff. AnaturalquestiontoaskiswhethertheparameterrangesinTheorems1and2aretheonlyvaluesofb,c Rfor ∈ which F ( n,b;c;z)havenrealsimplezeros. Inthispaper,weuseamethodthatdoesnotrelyonorthogonalityto 2 1 − determinealltherealvaluesoftheparametersbandcforwhich F ( n,b;c;z)havenrealsimplezeros. Weapply 2 1 − analgorithmwhichcountsthezerosofpolynomialswithrealcoefficientsandtheirmultiplicities. Wealsodetermine theintervalswheretherealzerosarelocatedforthesevaluesofbandc. 2. Thealgorithm Recallthatgiventwopolynomials f(x)andg(x),withdeg(f) deg(g),thereexistuniquepolynomialsq(x)and ≥ r(x) suchthat f(x) = q(x)g(x)+r(x) with deg(r) < deg(g). We willdenotethe leadingcoefficientofa polynomial f(x)=a xn+a xn 1+ +a bylc(f)=a . n n 1 − 0 n − ··· Weusethefollowingalgorithm(cf. [24]). Let f(x)bearealpolynomialwithdeg(f)=n 2. Define ≥ f (x):= f(x) and f (x):= f (x) 0 1 ′ andproceedfork Nasfollows: ∈ Ifdeg(f )>0performthedivisionof 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< 1andc<b<0. − (ii) 1<c<0andb>0orb<c. − (iii) c>0 andb<0orc<b. 2. Thezerosof F ( 3,b;c;z)arerealandsimpleifandonlyifeither(seeFigure3): 2 1 − (i) c< 2 and1+c<b< 1. − − (ii) 2<c< 1 and 1<b<1+c. − − − (iii) c> 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<b<2 n , 1 R { − − } = c> 1, b<2 n , 2 R { − − } = c> 1, b>n 2, b>c+n 2 , 3 R { − − − } = 1<c<0, c+n 2<b<n 2 . 4 R {− − − } 4 b c= 1 − b=n 2 − 1 0 c − b=c+n 1 c=0 − b=c+n 2 − b=2 n − b=1 n − Figure4:Valuesofbandcforwhich2F1( n,b;c;z),n=4,5,... hasnrealsimplezeros − Theparametervalues(c,b) describedinTheorem5forwhich F ( n,b;c;z),n = 4,5,... 1 2 3 4 2 1 ∈ R ∪R ∪R ∪R − has n real simple zeros are illustrated by the grey and diagonally shaded regionsin Figure 4 with the grey regions indicatingthoseparametervaluesthatextendtheresultsinTheorems1and2. Next, we turn ourattention to the location of the zeros of F ( n,b;c;z)for those parametervalueslocated in 2 1 − thegreyshadedregionsinFigure4wherethepolynomialsarenolongerorthogonalandthelocationoftherealzeros cannotbeobtainedusingTheorems1and2. Forthesevaluesof bandc,thepolynomials F ( n,b;c;z)arequasi- 2 1 − orthogonalof order1 and, in some cases, order2 (cf. [3] and [2]). Theorem3 in [2] and Theorem6 in [18] yield informationonthezerolocationofquasi-orthogonalpolynomialswithnon-varyingweightfunctions.However,these resultscannotbeappliedto F ( n,b;c;z)sincetheirweightfunctiondependsonn. Weuseinformationaboutthe 2 1 − zerosofJacobipolynomialstoobtainthefollowingthreeresults. Theorem6. Letn Nandb,c R. Then, F ( n,b;c;z)hasallitszerosrealandsimpleand(n 2)ofthemliein 2 1 ∈ ∈ − − (i) (0,1)for 1<c<0andc+n 2<b<c+n 1. Oneoftheremainingzerosliesin(1, )andtheotherone − − − ∞ in( ,0). −∞ (ii) (1, )for1 n< b< 2 nandb n+1<c <b n+2. Oneoftheremainingzerosliesin( ,0)andthe ∞ − − − − −∞ otheronein(0,1). (iii) ( ,0)for 1 < c < 0and1 n < b < 2 n. Oneoftheremainingzerosliesin(1, )andtheotheronein −∞ − − − ∞ (0,1). Theorem6appliestotheparametervaluesillustratedinFigure5. Theorem7. Letn Nandb,c R. Then, F ( n,b;c;z)hasallitszerosrealandsimpleand(n 1)ofthemliein 2 1 ∈ ∈ − − (i) (0,1)for 1<c<0andb>c+n 1. Theremainingzeroisnegative. − − (ii) (1, )for1 n<b<2 nandc<b n 1. Theremainingzeroisnegative. ∞ − − − − (iii) ( ,0)for 1<c<0andb<1 n. Theremainingzeroliesin(0,1). −∞ − − TheparametervaluesdescribedinTheorem7areillustratedinFigure6. Theorem8. Letn Nandb,c R. Then, F ( n,b;c;z)hasallitszerosrealandsimpleand(n 1)ofthemliein 2 1 ∈ ∈ − − (i) (0,1)forc>0andc+n 2<b<c+n 1. Theremainingzeroisintheinterval(1, ). − − ∞ 5 b c= 1 − b=n 2 − 1 0 c − b=c+n 1 c=0 − b=c+n 2 − b=2 n − b=1 n − Figure5:ValuesofbandccorrespondingtothosedescribedinTheorem6 (ii) (1, )forb<1 nandc+n 2<b<c+n 1. Theremainingzeroliesin(0,1). ∞ − − − (iii) ( ,0)forc>0and1 n<b<2 n. Theremainingzeroliesin(1, ). −∞ − − ∞ Figure7illustratestherangeoftheparametersbandcreferredtoinTheorem8. 4. Proofs ProofofProposition4. 1. Since 2b b(b+1) F ( 2,b;c;z)=1 z+ z2, 2 1 − − c c(c+1) weseethat F ( 2,b;c;z)=0ifandonlyif 2 1 − b(c+1) √b(c+1)(b c) z= ± − . b(b+1) Hence,thezerosof F ( 2,b;c;z)arerealandsimpleifandonlyifb(c+1)(b c)>0. 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, b<c, b<2 n = or 2 2 ∈{ − − } A ⊃R c>1 n, b>c, b>2 n = ( )or 3 3 4 ∈{ − − } A ⊃ R ∪R c<1 n, b<c, b>2 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 b<c+k 1, c+n k <0, k =2,...,n 1 = b<c+1, c<2 n = or ∈{ − − − } { − } ∅ b<c+k 1, c+n k <0, k =2,...,l b>c+k 1, c+n k >0, k =l+1,...,n 1 = or ∈{ − − }∩{ − − − } ∅ b>c+k 1, c+n k >0, k =2,...,l ∈{ − − }∩ b<c+k 1, c+n k<0, k=l+1,...,n 1 b>n 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 b<c+k 1, c+n k >0, k =2,...,n 1 = b<c+1, c> 1 = or 5 2 ∈{ − − − } { − } A ⊃R b>c+k 1, c+n k <0, k =2,...,l b<c+k 1, c+n k >0, k =l+1,...,n 1 = or ∈{ − − }∩{ − − − } ∅ b<c+k 1, c+n k >0, k =2,...,l b>c+k 1, c+n k <0, k =l+1,...,n 1 = . ∈{ − − }∩{ − − − } ∅ Fortheremainingcasewhere2 n<b<n 2or,morespecifically, 2<b+n 2k<0withb k+1<0forall − − − − − k 2,...,n 1 ,theonlynon-emptypossibilityisthat(4)ispositivefork =n 1ifandonlyifc> 1,b>c+n 2 ∈{ − } − − − andn 4<b<n 2whereas(4)ispositivefork 2,3,...,n 2 ifandonlyifc> 1,b>c+n 2andb>n 3. − − ∈{ − } − − − Hence,when2 n<b<n 2,(4)ispositiveforallk 2,...,n 1 ifandonlyif(c,b) . 4 − − ∈{ − } ∈R Since = and = ,theresultfollows. 1 4 1 2 5 2 A ∩A R A ∩A R ProofofTheorem5. Weapplythealgorithmtothepolynomial f(z)= F ( n,b;c;z). 2 1 − Wehave(cf.[25],p.69,ex.1) nb f (z)= f (z)= F ( n+1,b+1;c+1;z). 1 ′ 2 1 − c − UsingRaimundasViduñas’Maplepackageforcontiguousrelationsof F hypergeometricseries(cf. [29],[30]),we 2 1 obtain 1 c+n 1 (b c)(n 1) f (z)= z − f (z) − − F ( n+2,b;c+1;z). 0 n − b+n 1! 1 − c(b+n 1) 2 1 − − − Thisrelationcaneasilybeverifiedbycomparingcoefficients.Thus, (b c)(n 1) f (z)=r (z)= − − F ( n+2,b;c+1;z). 2 1 c(b+n 1) 2 1 − − Inthenextstep(k=2),weget f (z)=q (z)f (z) r (z), 1 1 2 2 − 9 with n(b+n 1)2(b+n 2) (n 2)(c+n 2) (n 1)(c+n 1) q (z)= − − z+ − − − − 1 (n 1)(c+n 1)(b c)" b+n 3 − b+n 1 # − − − − − and (b+n 1)(b 1 c)n(n 2) r (z)= − − − − F ( n+3,b 1;c+1;z). 2 c(c+n 1)(b+n 3) 2 1 − − − − Setting f (z)=θ F ( n+k,b+2 k;c+1;z), k 1,...,n 1 , k k 2 1 − − ∈{ − } weseethatingeneralweneedacontiguousrelationoftheform F ( n+k 1,b+3 k;c+1;z) 2 1 − − − θ θ =q (z) k F ( n+k,b+2 k;c+1;z) k+1 F ( n+k+1,b+1 k;c+1;z), k−1 θk 1 2 1 − − − θk 1 2 1 − − − − fork 2,...,n 2 ,with ∈{ − } nb (b c)(n 1) θ = , θ = − − . 1 − c 2 c(b+n 1) − Using Viduñas’package,we obtain(1) fork = 2,3,... andfromLemma9we concludethat θ iswell defined k andnon-zeroforallk 1,...,n 1 whenc,0and(c,b) .Thus, 1 2 3 4 ∈{ − } ∈R ∪R ∪R ∪R lc(f ) c = k+1 , k N, k lc(f ) ∈ k 1 − whichimpliesthat (b c)(n 1)( n+2) (b) (c) n! (n 1)(c+n 1)(b c) c1 = c(−b+n−−1) (c−+1)n−n2−(2n−n2−)2!(−n)nn(b)n = (b−+n−2)(b−+n−1−)2 and,fork 2,...,n 1 , ∈{ − } θ ( n+k+1) (b+1 k) (n k+1)!(c+1) ck = θk+1 − (c+1) n−k−1(n k −1)!n−k−1( n+k− 1) (b+3n−k+k1) (kn−1 k)(n k+cn)−(kb−1 k−+1)−( c+1+b− k) − n−k+1 − n−k+1 = − − − − − . (n 2k+b+2)(n 2k+b)(b 2k+1+n)2 − − − FromLemma10,weknowthatc >0forallk 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<c<0 − − − − − − − − and c+n 2 < b < c+n 1. Furthermore, the intervals ( 1,1), (1, ) and ( , 1) are transformed to − − − ∞ −∞ − (0,1), (1, ) and ( , 1) respectively under the linear mapping x = 2z 1. Thus, when c ( 1,0) and ∞ −∞ − − ∈ − b (c+n 2,c+n 1), F ( n,b;c;z)hasn 2real,simplezerosintheinterval(0,1),onezeroin(1, ) 2 1 an∈donezer−oin( ,−0)foreach−n N. − ∞ −∞ ∈ 10