Binomial Notations Traditional name Traditional notation Mathematica StandardForm notation Primary definition

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1 Bioial Notatios Traditioal ae Bioial coefficiet Traditioal otatio Matheatica StadardFor otatio Bioial, Priary defiitio ; 0 For Ν, Κ egative itegers with, the bioial coefficiet Ν caot be uiquely defied by a liitig proce- Κ dure based o the above defiitio because the two variables Ν, Κ ca approach egative itegers, with at differet speeds. For egative itegers with we defie: ; 0 Specific values Specialized values For fixed ;

2 ; siπ ; Π 1 Values at fixed poits Geeral characteristics Doai ad aalyticity is a aalytical fuctio of ad which is defied over Syetries ad periodicities Mirror syetry

3 Periodicity No periodicity Poles ad essetial sigularities With respect to For fixed, the fuctio has oly oe sigular poit at. It is a essetial sigular poit. ig , With respect to For fixed, the fuctio has a ifiite set of sigular poits: a) ;, are the siple poles with residues 1 ; ; b) is the poit of covergece of poles, which is a essetial sigular poit. ig res , 1 ;,, ; Brach poits With respect to The fuctio does ot have brach poits with respect to With respect to The fuctio does ot have brach poits with respect to

4 4 Brach cuts With respect to The fuctio does ot have brach cuts with respect to With respect to The fuctio does ot have brach cuts with respect to Series represetatios Geeralized power series Expasios at geeric poit 0 For the fuctio itself H 0 H H 0 H 0 Ψ Ψ ; i S i 0 1 i i 1 1 i 0 i 1 i 0 ; H 0 H H 0 H 0 Ψ Ψ O 0 3 ; i S i 0 1 i i 1 1 i 0 i 1 i 0 O 0 3 ; siπ F 1a 1, a,, a 1, 1; a 1 1, a 1,, a 1 1; 1 0 ; Π 0 a 1 a a 1 0 1

5 i S i i i 0 ; 0 i O 0 ; Expasios at geeric poit 0 For the fuctio itself cosπ H 0 H Π cotπ 0 0 Π 4 0 H 0 H 0 1 Ψ Ψ ; cosπ H 0 H Π cotπ 0 0 Π 4 0 H 0 H 0 1 Ψ Ψ O Π 1 0 si Π 0 F 1a 1, a,, a 1, ; a 1 1, a 1,, a 1 1; 1 0 Π 0 ; a 1 a a O 0 Asyptotic series expasios Expasios at B, 1, t Α t ; B, Α, 1 t t 1 Α arg 1 Π O 1 ; arg 1 Π Expasios at

6 siπ B,, Π 0 ; B, Α, z t Α t z t t 1 Α arg Π siπ O 1 Π ; arg Π Expasios at Π O 1 5 ; Residue represetatios res z z 1 z 1 0 ; res z z 1 z 1 0 ; Other series represetatios cos Π cos Π ; Itegral represetatios O the real axis Π Π t 1 t t ; 1 Π Cotour itegral represetatios Π z1 z 1 z 1 z Idetities Recurrece idetities

7 7 Cosecutive eighbors Distat eighbors Fuctioal idetities Cosecutive eighbors

8 8 Distat eighbors ; ; ; ; Additioal relatios betwee cotiguous fuctios Relatios of special id z 1 z z 4 z z 1 z 3 z 4 z 1 z z 3 z 4 z 3 z 1 z z 4 z 1 z 3 z 4 z 3 z 1 z z 3 z 4 z p p p 1 ; p p ; ; z 1 z z w z w 4 0 z ; Idetities ivolvig deteriats

9 a l x a l y 0 0 l a 1 0 a x l 1 l 1 a x y 1 a y x y a l x a l y 0 0 l a 1 0 x y a y l 1 l 1 a x y 1 a x Differetiatio Low-order differetiatio With respect to Ψ 1 Ψ Ψ 1 Ψ1 Ψ 1 1 Ψ 1 1 With respect to Ψ1 Ψ Ψ 1 Ψ1 Ψ 1 1 Ψ 1 1 Sybolic differetiatio With respect to siπ Π 0 1 ; 1 1

10 siπ Π a 1 a a F 1a 1, a,, a 1, 1; a 1 1, a 1,, a 1 1; 1 ; S 1 1 ; With respect to Π 1 0 si Π a 1 a a 1 F 1a 1, a,, a 1, ; a 1 1, a 1,, a 1 1; 1 Π ; Suatio Fiite suatio ; ; ; , ; ; ;

11 ; r r r 1 r ; r r Ifiite suatio Π Π Π Π Π Π Π Π 81 3

12 Π Π Π Π Π cot cot cot cot cot

13 cot cot cot cot cot cot cot cot Π 6 log Π 18 log

14 Π 4 log Π 774 log G.Huvet (006) 3 Π 3 Ifiite suatio cscπ siπ ; si Π si Π csc Π csc Π ; l 0 p l 0 p od p l 0 od p ; p p p ax , 1,, l l ; l

15 15 Multiple sus ; ; ; H H ; 1 1 l 0 l l l l, ; 1 l 0 l l l l 1, l 1, ; 1 l 0 l Α Β Α l Β l 1 Α 1, Β 1, ; Α Β 1 1, Α Β 1 l Α 1, l Β l l 1 0 l 0 l l, ; H H ; R. Lyos, P. Paule, A. Riese: A Coputer Proof of a Series Evaluatio i Ters of Haroic Nubers Applicable Algebra i Egieerig, Couicatio ad Coputig 13, (00) Represetatios through ore geeral fuctios Through other fuctios

16 16 Ivolvig soe hypergeoetric-type fuctios ;, Represetatios through equivalet fuctios With related fuctios z z ; , z 1 C z Iequalities ; ; 1

17 d Λ d Α Β Λ 0 d Α Β a b 0 d Β Λ a d Α 0 d Α Λ b ; d Β 0 d Α Α 1 Α Β Λ Λ 0 axa 1, a,, a 0 axb 1, b,, b 0 Geeralized Cauchy Schwarz iequality. Λ 0 gives the Cauchy Schwarz iequality d Λ d Γ Λ Μ Ν d Γ Μ Ν 0 u v 0 x y 0 d Λ Ν d Γ Μ u 0 d Α Α 1 d Λ Μ d Γ Ν y 0 d Γ Λ Ν d Γ Λ Μ d Μ x d Ν Ν ; 0 Μ Ν 0 Γ Γ 0 Λ Λ 0 axu 1, u,, u 0 axv 1, v,, v 0 axx 1, x,, x 0 axy 1, y,, y d Α Β Γ d Α d Α d Γ d Α Β d Α Γ d Β Γ ; d Α Α 1 Α Β Γ 0 Zeros ; ; Theores The bioial expasios a b 0 a b ; a b a a z1 b z ;, a 0 0 The derivative of products f z gz z f z z 0 gz z.

18 18 Represetatios of polyoials i Bezier for Every polyoial p x ca be represeted i Bezier for, that is, as p x 0 where the Berstei polyoials B x are give as B x x 1 x. Β B x, Iversio of the Hilbert atrix The iverse A 1 of the Hilbert atrix A a i 1 i, with etries a i 1 has the etries i1 1 i i 1 i 1 1 i i i 1. Geeralized bioial theore A x, y, p, q x A x, y, p 1, q ; 0 A x, y, p, q xp y q p, q 0 Weyl orderig The Weyl orderig of the operator product q p W of the operators q ad p is give by q p W q p q. 0 History Chia Hsie(1050), al-karai (about 1100), Oar al-khayyai (1080), Bhasara Acharya (1150), al-saaw'al (1175), Yag Hui (161), Tshu shi Kih (1303), Shih-Chieh Chu (1303), M. Stifel (1544), Cardao (1545), Scheubel (1545), Peletier (1549), Tartaglia (1556), Carda (1570), Stevi (1585), Faulhaber (1615), Girard (169), Oughtred (1631), Briggs (1633), Mersee (1636), Ferat (1636), Wallis (1656), Motort (1708), De Moivre (1730) - B. Pascal (1653) gave a recursio relatio I. Newto (1676) studied cases with fractioal arguets G.W. Leibiz (1695) L. Euler (1774, 1781) used otatios with paretheses C. F. Gauss (181) A. vo Ettighause (186) itroduced the bioial sybol Förstea (1835) gave cobiatorical iterpretatio of bioial coefficiet

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