Introductions to LucasL

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1 Itroductios to LucasL Itroductio to the Fiboacci ad Lucas ubers The sequece ow ow as Fiboacci ubers (sequece 0,,,, 3,, 8, 3...) first appeared i the wor of a aciet Idia atheaticia, Pigala (40 or 00 BC). Pigala's wor with the outai of cadece (ow ow as Pascal's triagle) ade hi the first ow perso to have looed ito Fiboacci ubers. Next, aother Idia atheaticia, Virahaa (6th cetury AD), too ote of the Fiboacci sequece through aalysis of a copletely differet proble. Virahaa cosidered the followig proble: assuig that lies of uits are coposed of syllables that ca be log or short a log syllable taes twice as log as a short syllable to articulate ad each lie of uits taes the sae tie to articulate o atter how it is coposed, how ay differet cobiatios of syllables are there for each lie of legth? Research of this questio was cotiued by the Idia scholar Heachadra ad the Idia atheaticia Gopala i the th cetury. Alost a half cetury later, the sequece was studied by the a whose ae is ost heavily lied to Fiboacci ubers, Leoardo of Pisa, a..a. Fiboacci (0). Fiboacci cosidered the faous growth of a idealized rabbit populatio proble. Later, Europea atheaticias bega to study various aspects of Fiboacci ubers. Researchers icluded J. Kepler (608), A. Girard (634), R. Sipso (73), É. Léger (837), É. Lucas (870, ), G. H. Hardy, ad E. M. Wright (938). Fro this group, it was Fracois Edouard Aatole Lucas (870, ) who gave Fiboacci ubers their ae. He also ivestigated a siilar sequece (sequece,, 3, 4, 7,, 8, 9, ), which was later coied Lucas ubers. I ay wors these sequeces are otated F ad L ( 0,,, ) to represet the first letters of the last aes Fiboacci ad Lucas. Evetually, it was established that both sequeces ca be aalytically exteded o coplex z-plaes ad that they satisfy the sae three-ter recurrece relatio, reflectig that the Fiboacci ad Lucas ubers are the sus of two eighborig ters: wz wz wz ; wz c F z c L z. For iteger arguets, Fiboacci ad Lucas ubers ca be elegatly represeted through the syetric relatios (icludig the golde ratio Φ ): F Φ Φ Φ Φ L Φ Φ Φ Φ ; ;. Defiitios of the Fiboacci ad Lucas ubers: For ay coplex Ν, the Fiboacci ubers ad Lucas ubers are defied by the forulas: ΦΝ cosν Π Φ Ν Φ Ν Φ Ν cosπ Ν,

2 where Φ is the golde ratio Φ. Coectios withi the group of the Fiboacci ad Lucas ubers ad with other fuctio groups Represetatios through ore geeral fuctios The Fiboacci ad Lucas ubers ad have the followig represetatios through ore geeral fuctios icludig soe hypergeoetric fuctios ad Meijer G fuctios: F U ;. Ν cos Π Ν F Ν, Ν ; 3 ; 4 si Π Ν F Ν, Ν ; ; 4 F Ν, Ν ; Ν; 4 F cosπ Ν F Ν, Ν ; Ν ; 4 ; Ν Π Ν siπ Ν Ν cosπ Ν F Ν, Ν ; 3 ; 4 siπ Ν F Ν, Ν ; ; 4 cos Π Ν F Ν, Ν ; ; 4 Ν si Π Ν F Ν, Ν ; 3 ; 4 F Ν, Ν ; Ν; 4 cosπ Ν F Ν, Ν ; Ν ; 4 ; Ν cos 3 Π Ν si3 Π Ν F Ν, Ν ; ; 4 Π Ν Ν siπ Ν F Ν, Ν ; 3 ; 4 F, ; 3 ; ; L F, ; ; ; F F, ; ; 4 ; L F, ; ; 4 ; siπ Ν Π, G 3,3 4 G, Ν, 4 Π Ν, Ν, Ν 0,, Ν Ν, Ν 0, Ν cosν Π Ν Π ; Ν G,, 4 Ν, Ν 0, Ν ; Ν Ν siπ Ν Π Ν siπ Ν Π, G 3,3 4 G, 3,3 4 Ν, Ν, Ν 0,, Ν Ν,, Ν, Ν, Ν ; Ν ; Ν Represetatios of Fiboacci ad Lucas ubers through each other ad through eleetary fuctios The Fiboacci ad Lucas ubers ad ca be represeted through each other by the followig forulas: L Ν L Ν Φ Ν si Π Ν L Ν F Ν Φ Ν si Π Ν F L L L L ; L L F F ; 0 F The Fiboacci ad Lucas ubers ad have the followig represetatios through eleetary fuctios:

3 3 Ν logw Π logw Ν Π logw Ν ΦΝ Ν ΠlogΦ Ν ΠlogΦ ; w cosπ Ν coshν csch cosπ Ν sih Ν cosπ Ν Ν Ν ΦΝ Ν ΠlogΦ Ν ΠlogΦ cos Ν csc cos 3 Π Ν si3 Π Ν expν csch exp Ν csch cosπ Ν Π Ν siπ Ν si Ν csc Φ Ν Φ Ν cosπ Ν Ν Φ Ν Π Ν siπ Νcos Ν si cosπ Ν siπ Ν si Ν si cosπ Ν coshν logφ cosπ Ν sihν logφ si Π Ν si Ν csc Π Ν sihν csch Π Ν cosπ Ν siπ Ν si Ν si siπ Νcos Ν si Π Ν siπ Νcos Ν si siπ Ν cosπ Ν si Ν si F si z siz ; z log Π. The best-ow properties ad forulas of the Fiboacci ad Lucas ubers Siple values at zero ad ifiity The Fiboacci ad Lucas ubers ad have the followig values at zero ad ifiity: F 0 0 L 0 F L Specific values for specialized variables The Fiboacci ad Lucas ubers F ad L with iteger arguet ca be represeted by the followig forulas:

4 4 F ; L ; F ; L F Φ Φ ; L Φ ; ; F Φ Φ ; L Φ Φ ; F F ; L L ; 3 F Φ Φ ΦΦ ; L Φ Φ ΦΦ ; For the cases of iteger arguets ; 0, the values of the Fiboacci ad Lucas ubers F ad L ca be described by the followig table: F L Aalyticity The Fiboacci ad Lucas ubers ad are etire aalytical fuctios of Ν that are defied over the whole coplex Ν-plae: Periodicity The Fiboacci ad Lucas ubers ad do ot have periodicity. Parity ad syetry The Fiboacci ad Lucas ubers ad geerically do ot have parity, but they have irror syetry: F F ; L L ; Poles ad essetial sigularities The Fiboacci ad Lucas ubers ad have oly the sigular poit Ν. It is a essetial sigular poit. Brach poits ad brach cuts The Fiboacci ad Lucas ubers ad do ot have brach poits ad brach cuts over the coplex Ν-plae. Series represetatios The Fiboacci ad Lucas ubers ad have the followig series expasios (which coverge i the whole coplex Ν-plae):

5 0 Ν 0 csch Ν 0 4 Ν 0 csch cosπ Ν 0 Π siπ Ν 0 Ν Ν 0 ; Ν Ν 0 F 0 Ν 0 csch Ν 0 Ν 0 Π Ν 0 csch Π csch Π Ν 0 0 Φ Ν 0 csch ΦΝ 0 Π Ν 0 Π csch Π Ν 0 Π csch Ν Ν 0 0 ΦΝ 0 csch ΦΝ 0 Π Ν 0 Π csch Π Ν 0 Π csch Ν Ν csch Π csch Ν Ν 0 0 OΝ Ν 0 0 OΝ Ν 0 logφ Ν Π Ν log3 Φ 3 Π logφ Ν3 ; Ν 0 log Φ Π Ν Π logφ Ν 3 ; Ν 0 logφ Ν OΝ ; Ν 0 csch Πcsch Πcsch Ν 0 csch Π csch Π csch OΝ ; Ν 0 Ν Asyptotic series expasios The asyptotic behavior of the Fiboacci ad Lucas ubers ad is described by the followig forulas: ΦΝ ; Ν Φ Ν ; Ν Φ Ν Ν ΠΝ csch csch Ν Π Ν Φ Ν cosν Π Φ Ν ΦΝ cosν Π Φ Ν ; Ν IΝ 0 ReΝ Π IΝ 0 IΝ 0 Π IΝ ReΝ 0 ; Ν IΝ 0 ReΝ Π IΝ 0 True Φ Ν IΝ 0 ReΝ Π IΝ 0 Ν ΠΝ csch IΝ 0 ReΝ Π IΝ 0 Π Νcsch Ν IΝ 0 ReΝ Π IΝ 0 Φ Ν cosν Π Φ Ν True Φ Ν cosν ΠΦ Ν ; Ν Other series represetatios The Fiboacci ad Lucas ubers F ad L for iteger oegative ca be represeted through the followig sus ivolvig bioials: F 0 ; F 0 ;

6 6 F 0 F F 0 0 ; ; ; F Π 4 0 ; F exp ta ; F 0 F 0 ; ; F 0 ; L 0 ;. Itegral represetatios The Fiboacci ad Lucas ubers F ad L have the followig itegral represetatios o the real axis: F 3 0 Π 3 cost sitt ; L Π 3 cost cost 3 sitt ;. Geeratig fuctios The Fiboacci ad Lucas ubers F ad L ca be represeted as the coefficiets of the series of the correspodig geeratig fuctios: t F t ; t t

7 7 t L t ;. t t Trasforatios: Additio forulas The Fiboacci ad Lucas ubers F ad L satisfy uerous additio forulas: F 0 F ; F 0 F ; F 3 0 F ; F F F ; F F F F F ; ; F F L F L ; ; F 0 F ; F F F F F ; ; F F L F L ; L F F L L ; L F F L L ; F Ν Trasforatios: Multiple arguets The Fiboacci ad Lucas ubers F ad L satisfy uerous idetities, for exaple the followig ultiple arguet forulas: F Ν si Π Ν Φ Ν

8 8 F Ν Φ Ν si Π Ν F Ν Φ Ν si Π Ν F L F ; ; F F p F p F p F p ; p F 0 F 3 0 F ; F ; F Ν 3 F Ν F Ν F Ν L F Ν F Ν ; F F 0 L ; L Ν 3 L Ν Φ Ν si L Ν L Ν Ν 4 Φ L F L ; L 0 L 3 0 L ; L ; L Ν 3 L Ν L Ν L Ν L L Ν L Ν ; L 0 L ; F 0 F L ; L L 0 F 0 F L 0 F 0 F F F ; F ; F ; F ; 0 L 0 F L ; L 0 L F F ; 0 Trasforatios: Products ad powers of the direct fuctio The Fiboacci ad Lucas ubers F ad L satisfy uerous idetities for products ad powers: cosν Π F F L L ; cosν Π F L ; F 3 3 F F 3 ; F 4 F F F F ; F 0 F F ; cosπ Ν L L L L ; cosπ Ν L L ; L 3 L 3 3 L ; L 4 L 4 4 L 6 ; L j 0 j j L j ; Idetities The Fiboacci ad Lucas ubers ad are solutios of the followig siple differece equatio with costat coefficiets:

9 9 wz wz wz ; wz c F z c L z. The Fiboacci ad Lucas ubers ad satisfy uerous recurrece idetities: ΦΝ Φ Φ Φ Ν Φ Φ Ν U 3 F Ν U 3 F Ν ; U 3 L Ν U 3 L Ν ; U 3 F Ν U 3 F Ν ; U 3 L Ν U 3 L Ν ; cosν Π Φ ΦΝ cosπ Ν Other idetities for Fiboacci ad Lucas ubers ad are just fuctioal idetities: F F l F l F F F l ; l F F F F F ; F F F F ; 4 F F F 3 F 4 F F 3 0 ; F a F b F c b c a ; F cb F ab F b F ac F bc F c F ba F cb F a a b c a b a c b c F Φ F ; F gcd, gcdf, F ; F F F F F F ; ta F ta F ta F ; F ; 0. L L F L L Coplex characteristics The Fiboacci ad Lucas ubers ad have the followig coplex characteristics for coplex arguets:

10 0 x y ; x y F x y Abs F x y 0 Φ x cosh Π y 4 Φ x cosπ x cos y logφ coshπ y Φ 4 x sih Π y cos Π x 4 Φ x siπ x si y logφ sihπ y Arg argf x y ta Φ x Φ x cosy logφ cosπ x coshπ y cosy logφ siπ x siy logφ sihπ y, Φ x Φ x siy logφ cosπ x coshπ y siy logφ cosy logφ siπ x sihπ y Re ReF x y Φx Φ x cosy logφ cosπ x coshπ y cosy logφ siπ x siy logφ sihπ y I IF x y Φx Φ x siy logφ cosπ x coshπ y siy logφ cosy logφ siπ x sihπ y Cojugate F x y Φx Φ x cosy logφ siy logφ siπ x sihπ ysiy logφ cosy logφ cosπ x coshπ y cosy logφ siy logφ Sig sgf x y Φ x cosπ x coshπ y siy logφ cosy logφ Φ x cosy logφ siy logφ siπ x cosy logφ siy logφ sihπ y Φ x cosh Π y 4 Φ x cosπ x cos y logφ coshπ y Φ 4 x sih Π y cos Π x 4 Φ x siπ x si y logφ sihπ y x y ; x y L x y Abs L x y x 4 x cos Π x 3 x y x y cosπ x y 4 y 4 y cosπ x y 4 x cosh Π y Arg Re I Cojugate argl x y ta Φ x cosy logφ Φ x cosπ x coshπ y cosy logφ Φ x siπ x siy logφ sihπ y, Φ x siy logφ Φ x cosπ x coshπ y siy logφ Φ x cosy logφ siπ x sihπ y ReL x y Φ x cosy logφ Φ x cosπ x coshπ y cosy logφ Φ x siπ x siy logφ sihπ y IL x y Φ x siy logφ Φ x cosπ x coshπ y siy logφ Φ x cosy logφ siπ x sihπ y L x y Φ x cosy logφ Φ x cosπ x coshπ y cosy logφ Φ x siπ x siy logφ sihπ y Φ x siy logφ Φ x cosπ x coshπ y siy logφ Φ x cosy logφ siπ x sihπ y Sig sgl x y Φ x cosy logφ x x siy csch x 4 x cosπ x y cosy csch siy csch x 4 x cos Π x 3 x y x y cosπ x y 4 y 4 y cosπ x y 4 x cosh Differetiatio The Fiboacci ad Lucas ubers ad have the followig represetatios for derivatives of the first ad th orders or the arbitrary fractioal order Α:

11 Ν Φ Ν Φ Ν logφcosπ Ν logφπ siπ Ν Ν ΦΝ logφ Φ Ν cosπ Ν logφ Φ Ν Π siπ Ν Ν Φ Ν log Φ Φ Ν Π Ν logφ Π Π Ν logφ Π ; Ν Φ Ν log Φ ΦΝ Π Ν Π logφ Π Ν Π logφ ; Α Ν Α ΝΑ Ν Π csch Α exp Π csch Ν QΑ, Π csch Ν exp Π csch Ν Ν Π csch Α QΑ, Π csch Ν Ν Α csch Α expν csch QΑ, Ν csch Α Ν Α ΝΑ QΑ, Π csch Ν Πcsch Ν Π csch Α QΑ, Π csch Πcsch Ν Ν Π csch Α QΑ Ν csch Ν Α csch Α Differetial equatios The Fiboacci ad Lucas ubers ad satisfy the followig third-order liear differetial equatio: w 3 Ν logφ w Ν Π log Φ w Ν logφ log Φ Π wν 0 ; wν c c c 3 Φ Ν siπ Ν, where c, c, ad c 3 are arbitrary costats. Idefiite itegratio Soe idefiite itegrals for Fiboacci ad Lucas ubers ad ca be evaluated as follows: Ν ΦΝ logφ cosπ ΝΠ siπ Ν Φ Ν log ΦΠ logφ Φ Ν log ΦΠ logφ Ν ΦΝ Π siπ ΝcosΠ Ν logφ Ν Α Ν ΝΑ Ν Α Α, Ν csch csch Α Ν Π csch Α Α, Ν Π csch Ν Π csch Α Α, Ν Π csch Ν Α Ν ΝΑ E Α Ν csch E Α Ν Π E Α Ν Π csch Laplace trasfors Laplace trasfors Ν f Νz of the Fiboacci ad Lucas ubers ad ca be represeted by the followig forulas: Ν z zcsch zcsch zcsch Π ; Rez logφ Ν z zcsch zcsch zcsch Π ; Rez logφ Suatio There exist ay forulas for fiite suatio of Fiboacci ad Lucas ubers, for exaple:

12 F ; 0 F F 0 0 F F F F 3 0 F z z z z F F z z 0 F pq z F qf pq z p F pq z p F qp z ; p z L p z p q 0 F F L F 0 F F F L ; 0 L L 0 0 L L L L 3 0 L z z L L z z z z 0 L pq z L qz L pq p z L pq p z L qp ; p z zl p p q 0 L L F L L L L Here are soe correspodig ifiite sus: z F z z z F 4 ϑ 0, 3 F F si Π F F F cos Π F F F 0 ; z F F F z z z z 4 z 0 z z L z. z z Ad here are soe ultiple sus: 0 0 0,j j j F j j j 0 j j ; p p j Liit operatio p j j F j F,p ; p F,p p F,p p F,p F, F. Soe forulas icludig liit operatios with Fiboacci ad Lucas ubers ad tae o syetrical fors:

13 3 li Ν F ΑΝ li Ν Φ Α L ΑΝ li Φ Α Ν 0 FΝ li Φ ; 0 li F Ν Ν LΝ L Ν Φ ; Other idetities The Fiboacci ubers F ca be obtaied fro the evaluatio of soe deteriates, for exaple: F if l if l. 0 else l Applicatios of the Fiboacci ad Lucas ubers Fiboacci ad Lucas ubers have uerous applicatios throughout algebraic codig theory, liear sequetial circuits, quasicrystals, phyllotaxies, bioatheatics, ad coputer sciece.

14 4 Copyright This docuet was dowloaded fro fuctios.wolfra.co, a coprehesive olie copediu of forulas ivolvig the special fuctios of atheatics. For a ey to the otatios used here, see Please cite this docuet by referrig to the fuctios.wolfra.co page fro which it was dowloaded, for exaple: To refer to a particular forula, cite fuctios.wolfra.co followed by the citatio uber. e.g.: This docuet is curretly i a preliiary for. If you have coets or suggestios, please eail coets@fuctios.wolfra.co , Wolfra Research, Ic.

SphericalHarmonicY. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

SphericalHarmonicY. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation SphericalHaroicY Notatios Traditioal ae Spherical haroic Traditioal otatio Y, φ Matheatica StadardFor otatio SphericalHaroicY,,, φ Priary defiitio 05.0.0.000.0 Y, φ φ P cos ; 05.0.0.000.0 Y 0 k, φ k k