Rogers–Ramanujan identities

The Rogers–Ramanujan identities equate two infinite products with q-series whose coefficients count partitions whose parts differ by at least two.

Integer partition: An integer partition expresses a nonnegative integer as a sum of positive integers, with order disregarded. The identities’ series coefficients count partitions subject to restrictions on successive parts.
Leonhard Euler: Leonhard Euler was an eighteenth-century mathematician whose work shaped number theory, analysis, and combinatorics. His partition theorem is an early model for identities linking distinct partition conditions.
q-series: A q-series is a power series whose coefficients or terms depend on a parameter q. Both identities are equations between q-series and infinite products.
Partition congruence: A partition congruence is a divisibility relation among values of the integer partition function. The identities’ product structure helps expose arithmetic patterns in partition counts.
Schur's partition theorem: Schur's partition theorem equates partitions into distinct parts not divisible by three with partitions whose parts differ by at least three. It replaces the identities’ gap of two with a stronger gap and a different product restriction.
Partition with difference two: A partition with difference two has successive parts differing by at least two. This is the precise restriction counted by the identities’ coefficients.
Srinivasa Ramanujan: Srinivasa Ramanujan was an Indian mathematician whose work included deep results in number theory and infinite series. His rediscovery and extension of the identities brought them to mathematical attention.
Partition into parts congruent to a: A partition into parts congruent to a restricts every part to a specified residue class modulo a fixed integer. The product sides select part sizes from particular residue classes modulo five.
Modular form: A modular form is a highly symmetric holomorphic function on the upper half-plane with prescribed transformation laws. The identities belong to a broader landscape where q-series acquire modular transformation properties.
Gordon's theorem: Gordon's theorem generalizes the Rogers–Ramanujan identities to families of restricted partitions and product formulas. The identities occur as the first nontrivial cases of its broader parameterized pattern.

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Rogers–Ramanujan continued fractionRelated: They share the fraction’s name and are linked to it through related product and modular formulas.