OI不得不知的那些数学定理

Binomial theorem

One can define$${r choose k}=frac{r,(r-1) cdots (r-k+1)}{k!} =frac{(r)_k}{k!}$$
Then, if (x) and (y) are real numbers with (|x| > |y|)( This is to guarantee convergence. Depending on (r), the series may also converge sometimes when (|x| = |y|).), and (r) is any complex number, one has

[(x+y)^r =sum_{k=0}^infty {r choose k} x^{r-k} y^k ]

Valid for (|x| < 1):$$(1+x)^{-1} = frac{1}{1+x} = 1 - x + x^2 - x^3 + x^4 - x^5 + cdots$$

Lagrange polynomial

[L(x) := sum_{j=0}^{k} y_j prod_{egin{smallmatrix}0le mle k\ m eq jend{smallmatrix}} frac{x-x_m}{x_j-x_m} ]

Lucas' theorem

For non-negative integers m and n and a prime p, the following congruence relation holds:$$inom{m}{n}equivprod_{i=0}^kinom{m_i}{n_i}pmod p,$$
where$$m=m_kp^k+m_{k-1}p{k-1}+cdots +m_1p+m_0,$$
and$$n=n_kp^k+n_{k-1}p{k-1}+cdots +n_1p+n_0$$
are the base (p) expansions of m and n respectively. This uses the convention that ( binom{m}{n} = 0) if (m < n).

A binomial coefficient ( binom{m}{n}) is divisible by a prime (p) if and only if at least one of the base (p) digits of (n) is greater than the corresponding digit of (m).

The p-th power mapping