Lommel Polynomials. Dr. Klaus Braun taken from [GWa] source. , defined by ( [GWa] 9-6)
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1 Loel Polyoials Dr Klaus Brau tae fro [GWa] source The Loel polyoials g (, efie by ( [GWa] 9-6 fulfill Puttig g / ( -! ( - g ( (,!( -!! ( ( g ( g (, g : g ( : h ( : g ( ( a relatio betwee the oifie Loel polyoials a the Bessel fuctio is give by Hurwitz s asyptotic forula ( [GWa] 9-65: Lea: Beig N resp N ( ( li h (! the zeros of ( resp ( puttig : the it hols ([TCh] 7, II, theor 6, [DDi] ie / N, i ( ( ( whereby is the oly iteger value for the ii L ( / L ( /, / / iii g( ( li li h ( li L (!!! g ( iv / / ( li li h ( li L (!!! v for large values of it hols O( vi (*, h ( h ( ( with h ( g (
2 Fro [GWa] 6-5, 3-6, 3-, we recall ( s s s ( ( s ( for Re( ( s s We suaries the above i Propositio : For the Loel polyoials the followig relatios hol true: i ( g ( li,! ˆ ( gˆ ( li! ii s s s ( li g( s! ( for Re( s ( iii ( (, g g (ote Lea (Fourier-Bessel epasio [GWa] 8: Let f ( be efie arbitrarily i the iterval (, a let f ( eist a (if it is a iproper itegral let it absolutely coverget a beig the zeros of ( Let a f ( ( ( : where / Let be ay iterval poit of a iterval ( a, b such that a b a such that f ( has liite total fluctuatio i ( a, b The the series a ( is coverget a its su is f ( f (, whereby a O( We recall Sheppar s result fro [GWa] 8-7, ie a ( ( ( f ( ( O( for
3 I [DDi] propositio, iii is prove builig a proper Riea-Stieltes itegral: The ter ( ( ( is aalytic outsie ay circle that cotais the fiite zeros of ( Hece it possesses a Lauret epasio about the origi that coverges uiforly o a i ay aulus whose isie bouary has the fiite zeros of ( i its iterior Let C be the cotour that ecircles the origi i a positive irectio a that lies withi the aulus The it hols [DDi] i C h ( ( ( Let ( the o-ecreasig step fuctio havig icrease of at the poit for,,, the it hols [DDi] / h ( /, h ( ( ( Rear: The Stieltes iverse forula gives the relatio to hyper-fuctios The Riea-Stieltes itegral represetatio the leas to propositio iii a The Loel polyoials buil a orthogoal polyoial syste of a Hilbert space The relatio to the Bagchi Forulatio of the Nya RH criterio is obvious [KBr] H with 3
4 We recall Euler s aalysis of the zeros of the Bessel fuctios With the otatios we follow [GWa] 5-, 5-5 We use the abbreviatio : to write the zeros of ( i the for Z / The zeros of ( are taig to be,, N, ie 3 N a it hols for ( ( ( I orer to eteriate the sallest zeros of ( Euler ifferetiate logarithically to coclue log ( provie that, a the last series is absolute coverget Puttig : a chage the orer of suatios results ito ( ( ( Base o this forula Euler obtaie a syste of equatios, which allow to calculate the a fro that to euce the sallest values of, ie Euler calculate to euce eg, /, 3 /3, / 8, 5 9/, 6 73/ 3, We suaries Euler s results above i the Lea Beig Z 5795, 76658, 3 87 the zeros of ( ( y a / N i log ( ( the zeros of ( it hols ii log ( for 5795 iii is the oly iteger value for the
5 With respect to [DDi] we efie the boue variatio fuctio ( : log (, which fulfills ( log ( a b ( ( ( b ( for, N whereby are the zeros of ( a a : a ( b ( : (! We ote the relatio ( ( ( ( ( 5
6 Herite Polyoials The Herite polyoials H ( fulfill the recursio forula H ( H ( ( b ( ( H ( Puttig a : (!! b (! / :, ( : e!, / / ( : e this gives the recursio forula ( : a ( ( b (, with ˆ ( : / / t e a H ( ( isg( ( trasfor the gives H t / / i t ( ( isg( e e ( ˆ Applyig the iverse Fourier Sice sg( / e is o we have Puttig f / ( it follows ( H ( ( t / / e si( t / / H ( ( e si( Substitutig the variables the leas to H ( f ( e si( Fro the Herite polyoials recursio forula the Hilbert trasfors ca be calculate by ˆ ( : ˆ ( ( ( ˆ a y y b ( ˆ ( / e si( The Herite polyoials a the Hilbert trasfore Herite polyoials buil a orthogoal syste of L (, 6
7 Refereces [KBr] Brau, K, A Note to the Bagchi Forulatio of the Nya RH criterio, wwwrieahypothesise [TCh] Chihara, TS, A Itrouctio to Orthogoal Polyoials, Matheatics a its Applicatios 3, Goro a Breach, New Yor, 978 [TCh] Chihara, TS, O co-recursive orthogoal polyoials, Proc Aer Mat Soc 8 (957, [DDi] Diciso, D, O Loel a Bessel Polyoials, Doctoral Dissertatio subitte to Uiversity of Michiga, 953 [SGr] IS Grashtey, IM Ryzhi, Table of Itegrals Series a Proucts, Fourth Eitio, Acaeic Press, New Yor, Sa Fracisco, Loo, 965 [GWa] G N Watso, A Treatise o the Theory of Bessel Fuctios, Cabrige Uiversity Press, Cabrige, Seco Eitio first publishe 9, reprite 996, 3,, 6 7
x !1! + 1!2!
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