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

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1 Fiboacci Notatios Traditioal ame Fiboacci umber Traditioal otatio F Ν Mathematica StadardForm otatio FiboacciΝ Primary defiitio F Ν ΦΝ cosν Π Φ Ν Specific values Specialized values F ; F ; F Φ Φ ; F Φ Φ ; F F ;

2 F Φ Φ ; Φ Φ Values at fied poits F 0 0 F F F 3 F 4 3 F F F F F F 0 Values at ifiities F Geeral characteristics Domai ad aalyticity F Ν is a etire aalytical fuctio of Ν which is defied over the whole comple Ν-plae ΝF Ν Symmetries ad periodicities Parity

3 F F ; Mirror symmetry F Ν F Ν Periodicity No periodicity Poles ad essetial sigularities The fuctio F Ν has oly oe sigular poit at Ν. It is a essetial sigular poit ig Ν F Ν, Brach poits The fuctio F Ν does ot have brach poits Ν F Ν Brach cuts The fuctio F Ν does ot have brach cuts Ν F Ν Series represetatios Geeralized power series Epasios at geeric poit Ν Ν 0 For the fuctio itself F Ν F Ν0 Ν 0 csch Ν 0 4 Ν 0 csch cosπ Ν 0 Π siπ Ν 0 Ν Ν 0 0 Π Π cosπ Ν 0 csch siπ Ν 0 Ν 0 csch F Ν0 Ν Ν 0 ; Ν Ν 0

4 F Ν F Ν0 Ν 0 csch Ν 0 4 Ν 0 csch cosπ Ν 0 Π siπ Ν 0 Ν Ν 0 0 Π Π cosπ Ν 0 csch siπ Ν 0 Ν 0 csch F Ν0 Ν Ν 0 OΝ Ν F Ν F Ν 0 csch Ν 0 Ν 0 Π Ν 0 csch Π csch Π Ν 0 csch Π csch Ν Ν F Ν F Ν0 OΝ Ν 0 Epasios at Ν F Ν logφ Ν Π Ν log 3 Φ 3 Π logφ Ν3 ; Ν F Ν logφ Ν Π Ν log 3 Φ 3 Π logφ Ν3 OΝ F Ν csch Π csch Π csch Ν F Ν logφ Ν OΝ ; Ν 0 Asymptotic series epasios F Ν ΦΝ cosν Π Φ Ν ; Ν Φ Ν ImΝ 0 ReΝ Π ImΝ 0 F Ν Ν ΠΝ csch csch Ν Π Ν ImΝ 0 Π ImΝ ReΝ 0 ImΝ 0 ReΝ Π ImΝ 0 ; Ν Φ Ν cosν Π Φ Ν True

5 F Ν ΦΝ ; Ν Other series represetatios F F F ; ; ; F ; F ; F Π 4 ; F ep ta ; F F ; ; F ; Itegral represetatios

6 6 O the real ais Of the direct fuctio F Π 3 cost sitt ; Limit represetatios F lim ma loglogd Μ m loglogμ Μ,,m ; d m d Σ 0 m d 0 m Σ 0 m Geeratig fuctios t F t ; t t Differetial equatios Ordiary liear differetial equatios ad wrosias w 3 Ν logφ w Ν Π log Φ w Ν logφ log Φ Π wν 0 ; wν c F Ν c L Ν c 3 Φ Ν siπ Ν Trasformatios Additio formulas F m F F m F m F ; m F m F L m F m L ; m F m F m ; m F m F m F F F m ; m F m F m L F L m ; m

7 F m F m ; m F m F m ; m F m3 F m ; m F Ν F Ν L Ν F F F ; F F F ; 0 Cassii's formula Multiple argumets F Ν L Ν F Ν si Π Ν Φ Ν F Ν F Ν F Ν F Ν F Ν Φ Ν si Π Ν F Ν F Ν F Ν F Ν F Ν Φ Ν si Π Ν F F p F p F p F p ; p F L F ; F F ; F Ν 3 F Ν F Ν F m Ν L m F m Ν m F m Ν ; m

8 m m F m F F m F ; m m F m m m m F L m ; m m m F m F F m L m m L m ; m F m m ; m m m F m m m F m m ; m Products, sums, ad powers of the direct fuctio Products of the direct fuctio F Ν F Ν F Ν cosν Π F F m L m L m ; m Powers of the direct fuctio F Ν F Ν F Ν cosν Π F L ; F 3 3 F F 3 ; F 4 4 F F 4 8 F F 4 6 ; F 4 F F F F ;

9 F m m m m m F m m F m ; m Related trasformatios F Ν L Ν L Ν Idetities Recurrece idetities Cosecutive eighbors F Ν F Ν F Ν F Ν F Ν F Ν F Ν F Ν Φ Ν Φ F Ν Φ F Ν Φ Ν Distat eighbors F Ν m U m 3 F mν m U m 3 F mν ; m F Ν m U m 3 F Νm m U m 3 F Νm ; m Fuctioal idetities Fuctioal equatios wz wz wz ; wz c F z c L z Relatios of special id F Ν F Ν F Ν cosν Π F F l F l F F F l ; l F F F F F ;

10 F F m F m m F m ; m 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 gcdm, gcdf m, F ; m F m m F F m F F F ; m ta F ta F ta F ; Comple characteristics Real part ReF y Φ Φ cosy logφ cosπ coshπ y cosy logφ siπ siy logφ sihπ y ReF y siπ siy csch sihπ y cosπ cosy csch coshπ y cosy csch Imagiary part ImF y Φ Φ siy logφ cosπ coshπ y siy logφ cosy logφ siπ sihπ y ImF y cosπ coshπ y siy csch cosy csch siπ sihπ y siy csch

11 Absolute value F y 0 Φ cosh Π y 4 Φ cosπ cos y logφ coshπ y Φ 4 sih Π y cos Π 4 Φ siπ si y logφ sihπ y F y cos Π cosh Π y 4 cosπ cos y csch coshπ y 4 siπ si y csch sihπ y Argumet argf y ta Φ Φ cosy logφ cosπ coshπ y cosy logφ siπ siy logφ sihπ y, Φ Φ siy logφ cosπ coshπ y siy logφ cosy logφ siπ sihπ y argf y ta siπ siy csch sihπ y cosπ cosy csch coshπ y cosy csch, cosπ coshπ y siy csch cosy csch siπ sihπ y siy csch Cojugate value F y Φ Φ cosy logφ siy logφ siπ sihπ ysiy logφ cosy logφ cosπ coshπ y cosy logφ siy logφ F y cosπ coshπ y cosy csch siy csch siπ siy csch cosy csch sihπ y cosy csch siy csch Sigum value

12 sgf y Φ cosπ coshπ y siy logφ cosy logφ Φ cosy logφ siy logφ siπ cosy logφ siy logφ sihπ y Φ cosh Π y 4 Φ cosπ cos y logφ coshπ y Φ 4 sih Π y cos Π 4 Φ siπ si y logφ sihπ y sgf y siπ cosy csch siy csch sihπ y cosy csch siy csch cosπ coshπ y cosy csch siy csch 3 4 cos Π cosh Π y 4 cosπ cos y csch coshπ y 4 siπ si y csch sihπ y Differetiatio Low-order differetiatio F Ν Ν ΦΝ Φ Ν logφ cosπ Ν logφ Π siπ Ν F Ν Ν Φ Ν cosπ Ν Π log Φ logφ Φ Ν logφ Π siπ Ν Symbolic differetiatio F Ν Ν Φ Ν log Φ Φ Ν Π Ν logφ Π Π Ν logφ Π ; F Ν Ν F Ν log Φ ΦΝ cosπ Ν log Φ Π cos Π Ν log Φ ; Fractioal itegro-differetiatio Α F Ν ΝΑ Ν Π csch Α ep Π csch ΝQΑ, Π csch Ν ep Π csch Ν Α Ν Ν Π csch Α QΑ, Π csch Ν Ν Α csch Α epν csch QΑ, Ν csch

13 3 Itegratio Idefiite itegratio Ivolvig oly oe direct fuctio F a Ν Ν a Φ a Ν logφ cosπ a Ν Π siπ a Ν log Φ Π logφ F Ν Ν Φ Ν logφ cosπ Ν Π siπ Ν log Φ Π logφ Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig power fuctio Φ Ν Φ a Ν Ν Α F a Ν Ν Ν Α a Ν Α Α, a Ν csch csch Α a Ν Π csch Α Α, a Ν Π csch a Ν Π csch Α Α, a Ν Π csch ΝΑ Ν Α F Ν Ν Ν Α Α, Ν csch csch Α Ν Π csch Α Α, Ν Π csch Ν Π csch Α Α, Ν Π csch Itegral trasforms Laplace trasforms t F t z z csch z csch z csch Π ; Rez logφ Summatio Fiite summatio F F

14 F F ; F F 3 ; F z z z z F F ; z z F q F pq z p F pq z p F qp z F pq z ; p q p z L p z F F L F ; F F F Ifiite summatio z F z z z F ϑ 0, F F si Π F cos Π F 0 ; F F F F z F F F z z z z 4 z as a formal power series

15 F Φ F Φ Multiple sums m 0 m 0 m 0,j m j j F m j j j 0 j j ; p p j p j j F j F,p ; p F,p p F,p p F,p F, F Operatios Limit operatio F Ν lim Ν L Ν F ΑΝ lim Φ Α Ν F Ν m F Ν lim Φ ; m Ν F mν F Ν Represetatios through more geeral fuctios Through hypergeometric fuctios Ivolvig F F Ν Ν Π Ν cos F Ν, Ν ; 3 ; 4 si Π Ν F Ν, Ν ; ; F Ν ΘΝ siν Π F Ν, Ν ; Ν; 4 ΘΝ siν Π cosν Π Ν F, Ν ; Ν ; Ν F Ν F, Ν ; Ν; 4 cosπ Ν Ν F, Ν ; Ν ; 4 ; Ν

16 6 F Ν Π Ν Ν siπ Ν cosπ Ν F Ν, Ν ; 3 ; 4 siπ Ν F Ν, Ν ; ; F Ν 0 Π Ν Ν cosπ Ν siπ Ν F Ν, Ν; 3 ; 4 siπ Ν F Ν, Ν; ; 4 F Ν Π Ν siπ Ν F Ν, Ν ; ; 4 Ν siπ Ν cosπ Ν F Ν, Ν ; 3 ; F F, ; 3 ; ; F F, ; ; 4 ; F F, ; 3 ; 4 ; F F, ; ; ; F 4 F, ; ; ; F F, ; ; 4 ; F F, ; ; 4 ; F F, ; ; 4 ; F 3 F F 3 8 F 7 8, 3 8, 3 8 ; ; 9 ; ; 9 ; ;

17 F F, ; 3 ; 4 ; Ivolvig p F q F 3F,, ; 3, ; 4 ; F 4 3F,, ; 3, ; 4 ; Through Meijer G Classical cases for the direct fuctio itself siπ Ν F Ν Π , G 3,3 4 Ν, Ν, Ν 0,, Ν ; Ν F Ν Ν Π G,, 4 Ν, Ν 0, Ν cosν Π Ν Π G,, 4 Ν, Ν 0, Ν ; Ν Geeralized cases for the direct fuctio itself siπ Ν F Ν Π , G 3,3, Ν, Ν, Ν 0,, Ν ; Ν Through other fuctios Ivolvig some hypergeometric-type fuctios F Ν F Ν F U ; Represetatios through equivalet fuctios With elemetary fuctios F Ν Ν logw Π logw Ν Π logw Ν ; w

18 F Ν ΦΝ Ν ΠlogΦ Ν ΠlogΦ F Ν ΦΝ Ν ΠlogΦ Ν ΠlogΦ F Ν epν csch ep Ν csch cosπ Ν F Ν Π Ν siπ Νcos Ν si cosπ Ν siπ Ν si Ν si F Ν cosπ Ν coshν logφ cosπ Ν sihν logφ F Ν si Π Ν si Ν csc Π Ν sihν csch F Ν Π Ν cosπ Ν siπ Ν si Ν si siπ Νcos Ν si F Ν Π Ν siπ Νcos Ν si siπ Ν cosπ Ν si Ν si F si z siz ; z log Π With Lucas umbers F Ν L Ν L Ν F m L m L m L m L m L m ; m F Ν L Ν L Ν Φ Ν si Π Ν L Ν L Ν

19 9 Other idetities Idetities ivolvig determiats F if l if l 0 else l Theorems Zecedorf theorem Every positive iteger ca be decomposed i a uique way as a sum of Fiboacci umbers, such that o two of these umbers are cosecutive i the Fiboacci sequece. Fiboacci substitutio After actig o A times with the Fiboacci substitutio A A B, B A the resultig sequece cotais F As ad F Bs. A trascedetal umber F is a trascedetal umber. The umbers of primary ad secodary spirals i the positios of leaves The umbers of primary ad secodary spirals i the positios of leaves or scales alog a plat stem are early always two cosecutive Fiboacci umbers. Hirmer's cojecture The umber of the largest set of oitersectig circles arraged alog the circumferece of a give circle ad agle Π GoldeRatio betwee cosecutive midpoits is give by the Fiboacci umbers F. History J. Kepler (608) A. Girard (634); R. Simpso (73) É. Léger (837) É. Lucas (870, ) G.H. Hardy ad E.M. Wright (938)

20 0 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a ey to the otatios used here, see Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for eample: To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: This documet is curretly i a prelimiary form. If you have commets or suggestios, please commets@fuctios.wolfram.com , Wolfram Research, Ic.

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