Binomial

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

For fixed n

Values at fixed points

General characteristics

Domain and analyticity

(({{n}, {k}})) is an analytical function of n and k which is defined over (ℂ^2).

Symmetries and periodicities

Mirror symmetry

Periodicity

No periodicity

Poles and essential singularities

With respect to k

For fixed n, the function (({{n}, {k}})) has only one singular point at k =(OverTilde[∞]).  It is an essential singular point.

With respect to n

For fixed k, the function (({{n}, {k}})) has an infinite set of singular points:
a)  n==-j/;j∈(SuperPlus[ℕ]), are the simple poles with residues ((((-1))^j)/(j!k!(-j-k)!))/;k∉ℤ;
b) n==(OverTilde[∞]) is the point of convergence of poles, which is an essential singular point.

Branch points

With respect to k

The function (({{n}, {k}})) does not have branch points with respect to k.

With respect to n

The function (({{n}, {k}})) does not have branch points with respect to n.

Branch cuts

With respect to k

The function (({{n}, {k}})) does not have branch cuts with respect to k.

With respect to n

The function (({{n}, {k}})) does not have branch cuts with respect to n.

Series representations

Generalized power series

Expansions at generic point n==(Subscript[n, 0])

Expansions at generic point k==(Subscript[k, 0])

Asymptotic series expansions

Expansions at n==∞

Expansions at k==∞

Expansions at n==2k==∞

Residue representations

Other series representations

Integral representations

On the real axis

Contour integral representations

Identities

Recurrence identities

Consecutive neighbors

Distant neighbors

Functional identities

Consecutive neighbors

Distant neighbors

Additional relations between contiguous functions

Relations of special kind

Identities involving determinants

Differentiation

Low-order differentiation

With respect to n

With respect to k

Symbolic differentiation

With respect to n

With respect to k

Summation

Finite summation

Infinite summation

G.Huvent (2006)

Infinite summation

Multiple sums

R. Lyons, P. Paule, A. Riese: A Computer Proof of a Series Evaluation in Terms of Harmonic Numbers Applicable Algebra in Engineering, Communication and Computing  13, 327-333 (2002)

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

Representations through equivalent functions

With related functions

Inequalities

Zeros

Theorems

The binomial expansions

(((a+b))^n)==(∑_(k=0)^n)((n/k))(a^k)(b^(n-k));

(((a+b))^n)==(∑_(k=0)^n)(({{n}, {k}})) a (((a-k z))^(k-1)) (((b+k z))^(n-k))/;n∈(SuperPlus[ℕ]),a≠0

The derivative of products

(((∂^n)(f(z) g(z)))/(∂(z^n)))==(∑_(k=0)^n)(({{n}, {k}})) (((∂^k)f(z))/(∂(z^k))) (((∂^(n-k))g(z))/(∂(z^(n-k)))).

Representations of polynomials in Bezier form

Every polynomial (Subscript[p, n])(x) can be represented in Bezier form, that is, as (Subscript[p, n])(x)==(∑_(k=0)^n)(Subscript[β, k]) (Subscript[B, k]^n)(x), 
where the Bernstein polynomials (Subscript[B, k]^n)(x) are given as (Subscript[B, k]^n)(x)==(({{n}, {k}}))(((x^k)(1-x))^(n-k)).

Inversion of the Hilbert matrix

The  inverse  (A^(-1)) of the Hilbert matrix A=(Subscript[{(Subscript[a, i j])}, 1≤i,j≤n]) with entries (Subscript[a, i j])=(1/(i+j-1)) has the entries

(((-1))^(i+j))(i+j-1)(({{n+i-1}, {n-j}}))(({{n+j-1}, {n-i}}))(((({{i+j-2}, {i-1}})))^2).

Generalized binomial theorem

(Subscript[A, n])(x,y,p,q)==(∑_(k=0)^n)(({{n}, {k}})) k! (k+x) (Subscript[A, n-k])(k+x,y,p-1,q)/;(Subscript[A, n])(x,y,p,q)==(∑_(k=0)^n)(({{n}, {k}})) (((k+x))^(k+p)) (((n-k+y))^(n-k+q))∧n∈ℕ∧p,q∈ℤ

Weyl ordering

The Weyl ordering of the operator product (Subscript[(((OverHat[q])^n)((OverHat[p])^m)), W]) of the operators (OverHat[q]) and (OverHat[p]) is given by

(Subscript[(((OverHat[q])^n)((OverHat[p])^m)), W])=(2^(-n))(∑_(k=0)^n)(({{n}, {k}}))((OverHat[q])^(n-k))((OverHat[p])^m)((OverHat[q])^k).

History

– Chia Hsien(1050), al-Karaji (about 1100), Omar al-Khayyami (1080), Bhaskara Acharya (1150), al-Samaw'al (1175), Yang Hui (1261), Tshu shi Kih (1303), Shih-Chieh Chu (1303), M. Stifel (1544), Cardano (1545), Scheubel (1545), Peletier (1549), Tartaglia (1556), Cardan (1570), Stevin (1585), Faulhaber (1615), Girard (1629), Oughtred (1631), Briggs (1633), Mersenne (1636), Fermat (1636), Wallis (1656), Montmort (1708), De Moivre (1730)
-  B. Pascal (1653) gave a recursion relation
– I. Newton (1676) studied cases with fractional arguments
– G.W. Leibniz (1695)
–  L. Euler (1774, 1781) used notations with parentheses
– C. F. Gauss (1812)
– A. von Ettinghausen (1826) introduced the binomial symbol
– Förstemann (1835) gave combinatorical interpretation of binomial coefficient

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