Falling factorial

The falling factorial of order n is the product x(x−1)⋯(x−n+1), with the empty product equal to 1. It is a polynomial basis used in discrete mathematics and combinatorics.

Factorial: The product of all positive integers up to a given integer, with 0! defined as 1. For nonnegative integer x, the falling factorial of order x equals x!.
Forward difference: The operator that subtracts a function's value at x from its value at x+1. It maps the order-n falling factorial to n times the order-(n−1) one.
Permutations: Arrangements of objects in which order matters. The falling factorial counts ordered selections of n distinct objects from x choices.
Rising factorial: The product x(x+1)⋯(x+n−1), with the empty product equal to 1. It reverses the direction of the unit steps in the falling factorial.
Binomial coefficient: A number counting ways to choose a fixed-size subset from a finite set. Dividing a falling factorial by n! gives the binomial coefficient when n is a nonnegative integer.
Newton's forward formula: An interpolation formula expressing function values on an integer-spaced grid through forward differences. It expands functions using falling factorials as the natural discrete basis.
Sampling without replacement: A sampling process in which each selected item is removed from subsequent draws. Successive factors decrease because previously selected objects cannot be chosen again.
Pochhammer symbol: A notation for rising or falling products whose convention depends on the mathematical context. It is a standard compact notation for falling factorials, though conventions vary.
Polynomial: An expression formed from variables and coefficients using addition, subtraction, and multiplication with nonnegative integer exponents. Each fixed-order falling factorial is a polynomial in x.
Polynomial basis: A set of polynomials whose linear combinations represent every polynomial in a specified space. Falling factorials of successive orders form a basis for polynomials.

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