Multiindices

Multiindices are a convenient bookkeeping tool used in analysis, algebra, and combinatorics to handle many variables. A multiindex α in N^n is an n-tuple α = (α1, ..., αn) of nonnegative integers. The length or order of α is |α| = α1 + ... + αn. For a function f: R^n → R, the differential operator D^α is defined by D^α f = ∂^{|α|} f / ∂x1^{α1} ... ∂xn^{αn}. The factorial associated to α is α! = α1! ... αn!, and for x ∈ R^n one writes x^α = x1^{α1} ... xn^{αn}.

Multiindices are used to encode sums over derivatives and monomials in a compact form. If β ≥ α componentwise, the multinomial coefficient is β choose α = β! / (α! (β−α)!). In Taylor expansions, f(x) = ∑_{|α|≥0} D^α f(a) / α! · (x − a)^α, where (x − a)^α is the monomial x1 − a1 to the α1 power times ... times xn − an to the αn power. D^α f(a) denotes the α-th partial derivative at a.

Common special cases include α with many zeros, yielding ordinary partial derivatives, and α with all components nonzero, giving mixed and higher-order derivatives. In practice, multiindices simplify notations across several domains, including partial differential equations, differential geometry, numerical analysis, and probability, where higher-order derivatives and moments are organized efficiently.

Example: in R^3, α = (2, 0, 1) corresponds to ∂^3 f/∂x1^2 ∂x3, with α! = 2!, and x^α = x1^2 x3.