q-series

A power series in a variable q, often with coefficients or factors that encode arithmetic, combinatorial, or partition information.

Formal power series: An algebraic expression with infinitely many terms whose coefficients are manipulated without requiring numerical convergence. Many q-series identities are first interpreted algebraically, without assigning a value to q.
q-binomial theorem: An identity expanding a q-shifted product or quotient as a power series with q-binomial coefficients. It supplies basic product-to-series expansions used throughout q-series.
Partition of an integer: A representation of a positive integer as a sum of positive integers, with order disregarded. The generating function for partitions is the reciprocal of the Euler product.
Leonhard Euler: An eighteenth-century mathematician whose work established foundational results across analysis, number theory, and combinatorics. Euler derived foundational product and partition identities that became central q-series examples.
Theta function: A special function defined by a sum over an integer lattice, often with strong transformation properties. Theta functions are bilateral q-series and central examples of modular behavior.
Infinite product: A product with infinitely many factors, defined through limits or formal algebra. Products such as ∏(1−qⁿ) are central q-series representations.
Basic hypergeometric series: A series whose successive terms have a ratio that is a rational function of qⁿ. This is the standard structured class containing many named q-series.
Partition generating function: The series ∏ₙ≥1(1−qⁿ)⁻¹, whose coefficient of qⁿ counts partitions of n. It is a canonical q-series whose coefficients have direct arithmetic meaning.
Euler's partition theorem: The theorem that partitions into distinct parts are equinumerous with partitions into odd parts. Its generating-function proof uses a basic equality of infinite q-products.
q-analogue: A deformation of a mathematical object that depends on q and recovers the original when q approaches 1. Many q-series identities arise by replacing ordinary numbers or functions with q-dependent analogues.

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