Abel's binomial theorem

Abel's binomial theorem expresses a power of a sum as a binomial sum whose terms contain powers of one summand shifted linearly by the other.

Binomial coefficient: A number counting selections of a fixed size from a finite set, appearing as a coefficient in binomial expansions. These coefficients weight the terms in Abel’s sum.
Niels Henrik Abel: A Norwegian mathematician whose work shaped the theory of algebraic equations, elliptic functions, and series. The theorem bears Abel’s name and belongs to his algebraic legacy.
Power of a sum: An expression formed by raising the sum of two or more quantities to an integer power. Abel’s theorem expands this familiar expression in a nonstandard form.
Lagrange inversion theorem: A theorem that extracts coefficients of a compositional inverse of a formal power series. Abel-type identities arise in coefficient calculations related to inversion formulas.
Multinomial theorem: The expansion of an integer power of a sum of several terms using multinomial coefficients. It generalizes ordinary binomial expansion by adding summands, not shifted powers.
Falling factorial: The product of consecutive descending integers, often written as a polynomial in its argument. A falling-factorial expansion gives a standard route to deriving Abel’s identity.
Paolo Ruffini: An Italian mathematician who gave an early proof that the general quintic equation cannot be solved by radicals. Ruffini’s work formed part of the algebraic setting of Abel’s early career.
Polynomial: An expression built from variables and coefficients using addition, multiplication, and nonnegative integer powers. The theorem is a finite equality between polynomials.
Combinatorial enumeration: The counting of finite structures according to specified properties. Binomially weighted polynomial sums can encode counts of decomposable structures.
Vandermonde's identity: An identity expressing a binomial coefficient of a sum as a convolution of binomial coefficients. It rearranges binomial coefficients, whereas Abel’s theorem expands powers with shifted factors.

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Abel’s theoremCompared with: This separate result by Abel has no divisor or period-lattice criterion.