Vandermonde's identity

Vandermonde's identity states that multiplying two binomial expansions gives a coefficient equal to a sum of products of binomial coefficients: \(\sum_k \binom{r}{k}\binom{s}{n-k}=\binom{r+s}{n}\).

Binomial theorem: The expansion of \((x+y)^n\) as a sum of terms weighted by binomial coefficients. Multiplying two binomial-theorem expansions produces Vandermonde's coefficient sum.
Hypergeometric distribution: The probability distribution for successes in draws without replacement from a finite population. Its normalization follows by summing products of binomial coefficients with Vandermonde's identity.
Binomial coefficient: The number of ways to choose a fixed-size subset from a finite set. Each term in the identity is a product of two such counts.
Vandermonde determinant: The determinant of a matrix whose rows or columns contain successive powers of variables, factoring into pairwise differences. It shares Vandermonde's name but concerns determinants rather than binomial-coefficient sums.
Cauchy product: The rule for multiplying two infinite series by summing products whose indices add to a fixed total. Its finite-sum form explains how coefficients combine when binomial polynomials are multiplied.
Binomial convolution: A sum combining two binomial-coefficient sequences according to indices with a fixed total. Vandermonde's identity evaluates this convolution in closed form.
Generating function: A formal power series whose coefficients encode a sequence and whose algebra reflects sequence operations. Binomial expansions act as generating functions for the coefficients being convolved.
Vandermonde's identity for negative integers: An extension of the binomial-coefficient identity using generalized binomial coefficients for negative integer parameters. It changes the coefficient convention and typically turns the finite sum into an infinite series.
Coefficient extraction: A method for finding the multiplier of a specified power in a polynomial or power series. Extracting the coefficient of a fixed power gives both sides of the identity.
Pascal's triangle: A triangular arrangement in which each entry is the sum of the two entries above it. Repeated use of Pascal's recurrence yields related binomial-sum identities.

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Abel's binomial theoremCompared with: It rearranges binomial coefficients, whereas Abel’s theorem expands powers with shifted factors.