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

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1 Multioial Notatios Traditioal ae Multioial coefficiet Traditioal otatio 1 2 ; 1, 2,, Matheatica StadardFor otatio Multioial 1, 2,, Priary defiitio ; 1, 2,, k k 1 ; 1 2 ; 1, 2,, 0 ; k ; 1, 2,, is the uber of ways of puttig 1 2 differet objects ito differet boxes with k i the kth box, k 1, 2,,. Specific values Specialized values ; ; 1, Values at fixed poits ; 0, 0,, 0 1

2 2 Geeral characteristics Doai ad aalyticity 1 2 ; 1, 2,, is a aalytical fuctio of 1, 2,, which is defied over ; 1, 2,, Syetries ad periodicities Mirror syetry ; 1, 2,, 1 2 ; 1, 2,, Perutatio syetry ; 1, ; 2, k j ; 1, 2,, k,, j,, 1 2 j k ; 1, 2,, j,, k,, ; k j k j Periodicity No periodicity Poles ad essetial sigularities With respect to k By variable k, 1 k, (with fixed other variables) the fuctio 1 2 ; 1, 2,, has a ifiite set of sigular poits: a) k N k j ; j, are the siple poles with residues 1 j1 ; N k k1 r 1 r r k1 r j ; r 1 j1 1jN k r 1 k1 r 1 r k1 b) k is the poit of covergece of poles, which is a essetial sigular poit ig k 1 2 ; 1, 2,, N k j, 1 ; j,, res k 1 2 ; 1, 2,, N k j r r j r 1 r k1 N k k1 1 j N k r 1 1 j1 ; k1 r 1 r k1 r 1 j 1 Brach poits

3 3 With respect to k The fuctio 1 2 ; 1, 2,, does ot have brach poits k 1 2 ; 1, 2,, ; 1 k Brach cuts With respect to k The fuctio 1 2 ; 1, 2,, does ot have brach cuts k 1 2 ; 1, 2,, ; 1 k Series represetatios Asyptotic series expasios ; 1, 2,, 1 a1 k 1 k 0 1 k 1 a k Bk, a, a 1 k k ; 1 a k 1 B, Α, z t Α t z t t 1 Α arga 1 Π ; 1, 2,, 1 a1 a 1 a 1 O 1 k ; 1 a k 1 arga 1 Π Other series represetatios ; 1, 2,, t k t 1 1, t 2 2,, t k ; Idetities Recurrece idetities Cosecutive eighbors ; 1, 2,, 1 2 1; 1, 2,, l1, l 1, l1,, ; 1, 2,, l 1 j l 1 2 1; 1, 2,, l1, l 1, l1,,

4 ; 1, 2,, l 1 Distat eighbors ; 1, 2,, j p 1 l p 1 p ; 1, 2,, j ; 1, 2,, l1, l 1, l1,, l 1 p 1 2 p; 1, 2,, l1, l p, l1,, 1 2 p; 1, 2,, l1, l p, l1,, j 1 p Geeralized Cauchy suatio k 1 k 1 k 1 p; j 1, j 2,, j q k h ; k 1 j 1, k 2 j 2,, k j j 1 0 j 0 j 0 h 1 k 1 k 2 k p q p q k h ; k 1, k 2,, k h 1 ; k 1 k 1 j 1 0 j 0 k 1 1 p h 1 j 2h ; j 1, j 2,, j 1 od 2 ; p j 0 Fuctioal idetities Relatios of special kid k j,0 k 1 k 2 k ; k 1, k 2,, k 1 k 1 2 k 2 k ; 1 k 1, 2 k 2,, k k 1 0 k 2 0 k 0 Differetiatio Low-order differetiatio ; 1, 2,, ; 1, 2,, ; 1, 2,, Ψ 1 Ψ 1 ; 1 2 ; 1, 2,, Ψ Ψ 3 1 Ψ 1 Ψ 1 2 Ψ 1 1 Ψ 1 1 ; Sybolic differetiatio k k

5 u 1 2 ; 1, 2,, u 1 u u s 1 u1 1 k 1 s u2 F u1a 1, a 2,, a u1, s 1; a 1 1, a 2 1,, a u1 1; 1 ; a 1 a 2 a u1 s 1 s k u s Suatio Fiite suatio o o k i,1 k i, ; k i,1,, k i, b 1 b ; b 1,, b ; k 1,1 0 k, 0 i 1 k i,j a i k i,j b i a i b j o Maxk 1,1,, k, i 1 i Σ 1 Σ 2 j 1 Σ 0 j 2 Σ 1 Σ 1 1 h 10k h h 1Σh Σ ; Σ k, k Σ 1, Σ 1 k 1, k 1 Σ 2,, Σ q2 k q2, k q2 Σ q1, Σ q1 k q, Σ q j Σ 1 Σ q ; Σ q k q, k q Σ q1, Σ q1 k q1, k q1 Σ q2,, Σ 2 k 2, k 2 Σ 1, Σ 1 k 0, Σ 0 Σ j1,σ j ; Σ 0 Σ 1 Σ Σ 0 Σ 1 Σ 1 Σ q 0 q Σ Σ 2 Σ 1 1 h 10k h h 1Σh Σ ; Σ k, k Σ 1, Σ 1 k 1, k 1 Σ 2,, Σ q2 k q2, k q2 Σ q1, j 1 Σ 0 j 2 Σ 1 j Σ 1 Σ q1 k q, Σ q Σ q ; Σ q k q, k q Σ q1, Σ q1 k q1, k q1 Σ q2,, Σ 2 k 2, k 2 Σ 1, Σ 1 k 0, Σ 0 2 Σ Σ 0 Σ ; Σ Σ 1, Σ 1 Σ 2,, Σ 2 Σ 1, Σ 1 Σ 0, Σ 0 ; Σ 0 Σ 1 Σ Σ 0 Σ 1 Σ 1 Σ q 0 q Represetatios through equivalet fuctios With related fuctios ; 1, 2,, k ; k

6 ; 1, 2,, 1 1 k 1 k ; Theores The ultioial expasio a 1 a 2 a, 1 2 ; 1, 2, a k k. 1, 2,, 0 The derivative of product z f k z 1, 2,, 0, 1 2 ; 1, 2,, f z k z k k The expected value of the uber of real roots of a syste of sparse polyoial equatios i variables The expected value of the uber of real roots of a syste of sparse polyoial equatios i variables ca be expressed i ultioials ad the volue of the correspodig Newto polytopes. Geeralized ultioial theore A x 1, x 2,, x, p 1, p 2,, p k k k x A kx 1 k, x 2,, x, p 1 1, p 2,, p ; k 0 A x 1, x 2,, x, p 1, p 2,, p k 1,k 2,, k 0, k1 k 2 k k 1 k 2 k ; k 1, k 2,, k x j k j k jp j ; p1, p 2,, p The volue of the d-diesioal regio d The volue V of the d-diesioal regio x k p k 1 is V 2 d d s p 1 k ; p 1 1, p 1 2,, p 1 d. History C. F. Hideburg (1779)

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