Computer Algebra Algorithms for Orthogonal Polynomials and Special Functions

Transcription

1 Computer Alger Algorithms for Orthogol Polyomils d Specil Fuctios Prof. Dr. Wolfrm Koepf Deprtmet of Mthemtics Uiversity of Kssel oepf@mthemti.ui-ssel.de

2 Olie Demostrtios with Computer Alger I will use the computer lger system Mple to demostrte d progrm the lgorithms preseted. Of course we could lso esily use y other system lie Mthemtic or MuPAD. We first give short itroductio out the cpilities of Mple.

3 Computtio of Power Series Assume give epressio f depedig of the vrile we would lie to compute formul for the coefficiet of the power series f represetig f. 0

4 Algorithm Iput: epressio f Determie holoomic differetil eutio DE homogeeous d lier with polyomil coefficiets y computig the derivtives of f itertively. Covert DE to holoomic recurrece eutio RE for. Solve RE for. Output: resp. È

5 Computtio of Holoomic Differetil Eutios Iput: epressio f Compute c 0 f c f c J f J with still udetermied coefficiets c j. Sort this w. r. t. lierly idepedet fuctios ± d determie their coefficiets. Set these zero d solve the correspodig lier system for the uows c 0 c c J. Output: DE:c 0 f c f c J f J 0.

6 Alger of Holoomic Fuctios We cll fuctio tht stisfies holoomic differetil eutio holoomic fuctio. Sum d product of holoomic fuctios tur out to e holoomic. We cll seuece tht stisfies holoomic recurrece eutio holoomic seuece. Sum d product of holoomic seueces re holoomic. A fuctio is holoomic iff it is the geertig fuctio of holoomic seuece.

7 Hypergeometric Fuctios The power series whose coefficiets A hve rtiol term rtio is clled the geerlized hypergeometric fuctio. 0 p p A F A A p

8 Coefficiets of Hypergeometric Fuctios For the coefficiets of the hypergeometric fuctio we get the formul where - is clled the Pochhmmer symol or shifted fctoril.! 0 p p p F

9 Emples of Hypergeometric Fuctios si e 0 F0 0 F 3/ Further emples: cos rcsi rct l erf L α... ut for emple ot t... 4

10 Idetifictio of Hypergeometric Fuctios Assume we hve s 0. How do we fid out which p F this is?

11 Idetifictio Algorithm Iput: Compute r : d chec whether the term rtio r is rtiol. Fctorize r. Output: red off the upper d lower prmeters d iitil vlue.

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13 Recurrece Eutios for Hypergeometric Fuctios Give seuece s s hypergeometric sum s F how do we fid recurrece eutio for s?

14 Celie Fsemyer s Algorithm Iput: summd F Compute F j i stz : F i 0... I j 0... J Brig this ito rtiol orml form d set the umertor coefficiet list w.r.t. zero. Output: Sum the resultig recurrece eutio for F w.r.t..

15 Drwcs of Fsemyer s Algorithm I esy cses this lgorithm succeeds ut: I my cses the lgorithm geertes recurrece eutio of too high order. The lgorithm is slow. If e.g. I d J the lredy 9 lier eutios hve to e solved. Therefore the lgorithm might fil.

16 Idefiite Summtio Give seuece fid seuece s which stisfies s s s. Hvig foud s mes defiite summtio esy sice y telescopig for ritrry m m s s. Idefiite summtio is the iverse of. m

17 Gosper s Algorithm Iput: hypergeometric term. Compute p r ± [] with p d gcd r j for ll p r j Fid polyomil solutio f of the recurrece eutio f - r f - p. Output: the hyperg. term s r p f. 0.

18 Defiite Summtio: Zeilerger s Algorithm Zeilerger hd the rillit ide to use modified versio of Gosper s lgorithm to compute defiite hypergeometric sums s F. Note however tht wheever s is itself hypergeometric term the Gosper s lgorithm pplied to F fils!

19 Zeilerger s Algorithm Iput: summd F For suitle J ± set : F σ F σ J FJ. Apply the followig modified versio of Gosper s lgorithm to : I the lst step solve t the sme time for the coefficiets of f d the uows σ j ±. Output y summtio: The recurrece eutio RE : s σ s σ J s J 0.

20 The output of Zeilerger s Algorithm We pply Zeilerger s lgorithm itertively for J util it succeeds. If J is successful the the resultig recurrece eutio for s is of first order hece s is hypergeometric term. If J > the the result is holoomic recurrece eutio for s. Oe c prove tht Zeilerger s lgorithm termites for suitle iput. Zeilerger s lgorithm is much fster th Fsemyer s.

21 Represettios of Legedre Polyomils 0 F P 0 F / 0 / / / F / / F

22 Dougll s Idetity Dougll 907 foud the followig idetity 6 7 d c d c d c d c F. d c d c d c d c

23 Cluse s Formul Cluse s formul gives the cses whe Cluse 3 F fuctio is the sure of Guss F fuctio: The right hd side c e detected from the left hd side y Zeilerger s lgorithm.. 3 F F

24 A Geertig Fuctio Prolem Recetly Folmr Borem showed me ewly developed geertig fuctio of the Legedre polyomils d sed me to geerte it utomticlly. Here is the uestio: Write G z α : 0 α P s hypergeometric fuctio! z

25 Geertig Fuctio s Doule Sum We c te y of the four give hypergeometric represettios of the Legedre polyomils to write GzD s doule sum. The the tric is to chge the order of summtio z p z p α α

26 Automtic Computtio of Ifiite Sums Wheres Zeilerger s lgorithm fids Chu- Vdermode s formul for ± 0 the uestio rises to detect Guss idetity for c ± i cse of covergece. c c c F c c c c c F Γ Γ Γ Γ

27 Solutio The ide is to detect utomticlly d the to cosider the limit s m ˆ. Usig pproprite limits for the Γ fuctio this d similr uestios c e hdled utomticlly y Mple pcge of Vidus d Koorwider. c F c c c c m c F m m m m

28 The WZ Method Assume we wt to prove idetity f ~ s with hypergeometric terms f d s~. Dividig y s~ we my put the idetity ito the form s : F.

29 Rtiol Certificte If Gosper s lgorithm pplied to F-F is successful the it geertes rtiol multiple G of F i.e. G R F such tht F - F G - G By telescopig this proves s s 0 hece the idetity. Secod proof: Dividig y F we my prove F F R R F F purely rtiol idetity.

30 Differetil Eutios Zeilerger s lgorithm c esily e dpted to geerte holoomic differetil eutios for hyperepoetil sums s F. For this purpose the summd F must e hyperepoetil term i.e. F F.

31 Petovse s Algorithm Petovse s lgorithm is dptio of Gosper s. Give holoomic recurrece eutio it determies ll hypergeometric term solutios. Petovse s lgorithm is slow especilly if the ledig d trilig terms hve my fctors. Mple 9 will coti much more efficiet lgorithm due to Mr v Hoeij.

32 Comiig Zeilerger s d Petovse s Algorithm Zeilerger s lgorithm my ot give recurrece of first order eve if the sum is hypergeometric term. This rrely hppes though. Therefore the comitio of Zeilerger s lgorithm with Petovse s gurtees to fid out whether give sum c e writte s hypergeometric term. Eercise 9.3 of my oo gives 9 emples for this situtio ll from p. 556 of Prudiov Brychov Mrichev: Itegrls d Series Vol. 3: More Specil Fuctios. Gordo Brech 990.

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34 Emples As emple we te d Eercise 9.3 resp. PBM : c c c. 3 4 F

35 Idefiite Itegrtio To fid recurrece d differetil eutios for hypergeometric d hyperepoetil itegrls oe eeds cotiuous versio of Gosper s lgorithm. Almvist d Zeilerger gve such lgorithm. It fids hyperepoetil tiderivtives if those eist.

36 Recurrece d Differetil Eutios for Itegrls Applyig the cotiuous Gosper lgorithm oe c esily dpt the discrete versios of Zeilerger s lgorithm to the cotiuous cse. The resultig lgorithms fid holoomic recurrece d differetil eutios for hypergeometric d hyperepoetil itegrls.

37 Emple As emple we would lie to fid holoomic eutios for t S y : t dt Resultig recurrece eutios: 0 y S y S y S y y S y y 0 0

38 Emple ctd. Solvig oth recurrece eutios shows tht Sy must e multiple of S y ~ Computig the iitil vlue Γ Γ y Γ y S 00 : dt proves tht the ove is idetity. 0

39 Emple The itegrl stisfies the differetil eutio from which it c e derived tht 0 4 dt t t I I I I. I π

40 Rodrigues Formuls Usig Cuchy s itegrl formul h! h t π i t γ for the th derivtive mes the itegrtio lgorithm ccessile for Rodrigues type epressios f dt d g h. d

41 Orthogol Polyomils Hece we c esily show tht the fuctios re the Legedre polyomils d re the geerlized Lguerre polyomils. d d P!! e d d e L α α α

42 Geertig Fuctios If Fz is the geertig fuctio of the seuece f the y Cuchy s formul d Tylor s theorem 0 z f z F.! 0 dt t t F i F f γ π

43 Lguerre Polyomils Hece we c esily prove the followig geertig fuctio idetity z z 0 α z e L z α for the geerlized Lguerre polyomils.

44 Bsic Hypergeometric Series Isted of cosiderig series whose coefficiets A hve rtiol term rtio A /A ± we c lso cosider such series whose coefficiets A hve term rtio A /A ±. This leds to the -hypergeometric series. ; 0 s r s r A ϕ

45 Coefficiets of the Bsic Hypergeometric Series Here the coefficiets re give y where deotes the -Pochhmmer symol. ; ; ; ; ; r s s r A 0 ; j j

46 Further -Epressios -Pochhmmer symol: -fctoril: -Gmm fuctio: -iomil coefficiet: -rcets: ; lim ; [ ] ;! z z z Γ ; ; ; ; ; [ ].

47 -Chu-Vdermode Theorem For ll clssicl hypergeometric theorems correspodig -versios eist. For emple the - Chu-Vdermode theorem sttes tht c ϕ ; c c / ; c; d c e proved y -versio of Zeilerger s lgorithm.

48 -Hypergeometric Orthogol Polyomils All clssicl orthogol systems hve severl -hypergeometric euivlets. The Little d the Big -Legedre Polyomils respectively re give y ; p ϕ. ; ; ; 3 c c P ϕ

49 Opertor Eutios -orthogol polyomils stisfy - holoomic recurrece eutios with respect to d i the clssicl Hh cse holoomic -differece eutios. For the ltter oe uses Hh s -differece opertor D f f f.

50 Sclr Products Give: sclr product f g : f g dµ with o-egtive mesure µ supported i the itervl []. Prticulr cses: solutely cotiuous mesure dµ ρd discrete mesure ρ supported y À discrete mesure ρ supported y.

51 Orthogol Polyomils A fmily P of polyomils is orthogol w. r. t. the mesure µ if 0 P. if 0 if 0 m d m P P m

52 Clssicl Fmilies The clssicl orthogol polyomils c e ltertively defied s the polyomil solutios of the differetil eutio σ P τ P λp 0. Coclusios: implies τ d e d 0 implies σ c coefficiet of implies λ -d

53 Clssifictio The clssicl systems c e clssified ccordig to the scheme σ 0 powers σ Hermite polyomils σ Lguerre polyomils σ Bessel polyomils σ Jcoi polyomils

54 Weight fuctio The weight fuctio ρ correspodig to the differetil eutio stisfies Perso s differetil eutio d σ ρ τ ρ d Hece it is give s ρ C e σ τ d σ.

55 Clssicl Discrete Fmilies The clssicl discrete orthogol polyomils c e defied s the polyomil solutios of the differece eutio σ P τ P λp Coclusios: implies τ d e d 0 implies σ c coefficiet of implies λ -d 0.

56 Clssifictio The clssicl discrete systems c e clssified ccordig to the scheme σ trslted Chrlier pols. σ fllig fctorils σ Chrlier Meier Krwtchou pols. σ Nα- Hh polyomils

57 Weight fuctio The weight fuctio ρ correspodig to the differece eutio stisfies Perso s differece eutio ρ τ ρ σ Hece it is give s ρ σ ρ σ τ.

58 Clssicl -Fmilies The -orthogol polyomils of the Hh clss c e defied s the polyomil solutios of the -differece eutio σ DD / P τ DP λp 0. Coclusios: implies τ d e d 0 implies σ c coefficiet of implies λ [ ] / [ ] d[ ]

59 Clssifictio The clssicl -systems c e clssified ccordig to the scheme σ 0 powers d -Pochhmmers σ discrete -Hermite polyomils II σ -Chrlier -Lguerre pols. σ - - Big - Jcoi pols.

60 Weight fuctio The weight fuctio ρ correspodig to the -differece eutio stisfies the - Perso differetil eutio D σ ρ τ ρ Hece it is give s ρ σ ρ τ σ.

61 Computig Differetil Eutio from Recurrece Eutio From the differetil or -differece eutio oe c determie the three-term recurrece eutio for P i terms of the coefficiets of σ d τ. Usig this iformtio i the opposite directio oe c fid the correspodig differetil or -differece eutio from give three-term recurrece eutio.

62 Emple Give the recurrece eutio P P α P 0 oe fids tht for α ¼ trslted Lguerre polyomils d for α < ¼ Meier d Krwtchou polyomils re solutios.

63 Emple Give the recurrece eutio P P α P 0 oe fids tht for every α there re - orthogol polyomil solutios.

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