Discovery Archive

Explore Ramanujan's Discoveries

Search, filter and compare identities, theorems, conjectures, formulas, and notebook contribution families.

Complete Catalogue

Browse 177 discovery and contribution-family records grouped by field, source, result type and difficulty.

Source-Aware Notes

Each entry distinguishes published papers, notebooks, later names, collaborations and records still needing exact citation review.

Study Path

Use search, category filters, related discoveries and formula permalinks to move from readable summaries into the mathematics.

32 entries shown from 177 detailed discovery and contribution-family records.

SourceResult type

Sort

Bernoulli Numbers

Bernoulli-Number Congruences

Contribution family

Bernoulli-Number Congruences is catalogued as a identity in Bernoulli Numbers.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Bernoulli Numbers

Bernoulli-Number Identities

\frac{x}{e ^{x} - 1} = n = 0 \sum \infty B_{n} \frac{x ^{n}}{n !}

Bernoulli-Number Identities is catalogued as a identity in Bernoulli Numbers.

Plain language: A central formula or notation associated with Bernoulli-Number Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Bernoulli Numbers

Bernoulli-Polynomial Identities

Contribution family

Bernoulli-Polynomial Identities is catalogued as a identity in Bernoulli Numbers.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Combinatorial Identities

Contribution family

Combinatorial Identities is catalogued as a identity in Combinatorial Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Combinatorial Identities

Combinatorial Sums

Contribution family

Combinatorial Sums is catalogued as a identity in Combinatorial Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Combinatorial Identities

Finite Combinatorial Identities

Contribution family

Finite Combinatorial Identities is catalogued as a identity in Combinatorial Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Divisor Functions

Divisor-Function Congruences

Contribution family

Divisor-Function Congruences is catalogued as a identity in Divisor Functions.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Divisor Functions

Divisor-Sum Convolution Identities

Contribution family

Divisor-Sum Convolution Identities is catalogued as a identity in Divisor Functions.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Divisor Functions

Sigma-Function Identities

σ_{k} (n) = d ∣ n \sum d^{k}

Sigma-Function Identities is catalogued as a identity in Divisor Functions.

Plain language: A central formula or notation associated with Sigma-Function Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Divisor Functions

Sums of Powers and Divisor Sums

Contribution family

Sums of Powers and Divisor Sums is catalogued as a identity in Divisor Functions.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Eisenstein and Modular Series

Eisenstein-Series Congruences

Contribution family

Eisenstein-Series Congruences is catalogued as a identity in Eisenstein and Modular Series.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Gamma-Function Identities

Gamma and Beta Integral Identities

Contribution family

Gamma and Beta Integral Identities is catalogued as a identity in Gamma-Function Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Gamma-Function Identities

Γ (z + 1) = z Γ (z)

Gamma-Function Identities is catalogued as a identity in Gamma-Function Identities.

Plain language: A central formula or notation associated with Gamma-Function Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Gamma-Function Identities

Gamma-Ratio Identities

Contribution family

Gamma-Ratio Identities is catalogued as a identity in Gamma-Function Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Infinite Products

Eta-Product Identities

Contribution family

Eta-Product Identities is catalogued as a identity in Infinite Products.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Infinite Products

Infinite-Product Transformations

(a; q)_{\infty} = k = 0 \prod \infty (1 - a q^{k})

Infinite-Product Transformations is catalogued as a identity in Infinite Products.

Plain language: A central formula or notation associated with Infinite-Product Transformations.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Modular Equations and Elliptic Functions

Modular Equations of Degree 3

Contribution family

Modular Equations of Degree 3 is catalogued as a identity in Modular Equations and Elliptic Functions.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Modular Equations and Elliptic Functions

Modular Equations of Degree 5

Contribution family

Modular Equations of Degree 5 is catalogued as a identity in Modular Equations and Elliptic Functions.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Modular Equations and Elliptic Functions

Modular Equations of Degree 7

Contribution family

Modular Equations of Degree 7 is catalogued as a identity in Modular Equations and Elliptic Functions.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Partition Theory

Restricted Partition Identities

Contribution family

Restricted Partition Identities is catalogued as a identity in Partition Theory.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

q-Series and Basic Hypergeometric Series

False Theta Identities

Contribution family

False Theta Identities is catalogued as a identity in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

q-Series and Basic Hypergeometric Series

Partial Theta Identities

Contribution family

Partial Theta Identities is catalogued as a identity in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

q-Series and Basic Hypergeometric Series

q-Binomial Identities

\frac{1}{( z ; q ) _{\infty}} = n = 0 \sum \infty \frac{z ^{n}}{( q ; q ) _{n}}

q-Binomial Identities is catalogued as a identity in q-Series and Basic Hypergeometric Series.

Plain language: A central formula or notation associated with q-Binomial Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

q-Series and Basic Hypergeometric Series

Root-of-Unity Identities

Contribution family

Root-of-Unity Identities is catalogued as a identity in q-Series and Basic Hypergeometric Series.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Ramanujan Polynomials

Ramanujan Polynomial Identities

Contribution family

Ramanujan Polynomial Identities is catalogued as a identity in Ramanujan Polynomials.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Ramanujan Theta Functions

Ramanujan Theta Identities

Contribution family

Ramanujan Theta Identities is catalogued as a identity in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Ramanujan Theta Functions

Theta Addition and Subtraction Formulas

Contribution family

Theta Addition and Subtraction Formulas is catalogued as a identity in Ramanujan Theta Functions.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Ramanujan Theta Functions

Theta Product Representations

f (a, b) = (- a; ab)_{\infty} (- b; ab)_{\infty} (ab; ab)_{\infty}

Theta Product Representations is catalogued as a identity in Ramanujan Theta Functions.

Plain language: A central formula or notation associated with Theta Product Representations.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Rogers-Ramanujan Identities

First Rogers-Ramanujan Identity

n = 0 \sum \infty \frac{q ^{n^{2}}}{( q ; q ) _{n}} = \frac{1}{( q ; q ^{5} ) _{\infty} ( q ^{4} ; q ^{5} ) _{\infty}}

First Rogers-Ramanujan Identity is catalogued as a identity in Rogers-Ramanujan Identities.

Plain language: Rogers discovered the identities earlier; Ramanujan independently rediscovered them and deepened their significance.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Rogers-Ramanujan Identities

Rogers-Ramanujan Modular Relations

Contribution family

Rogers-Ramanujan Modular Relations is catalogued as a identity in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Rogers-Ramanujan Identities

Rogers-Ramanujan Product Identities

Contribution family

Rogers-Ramanujan Product Identities is catalogued as a identity in Rogers-Ramanujan Identities.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery

Rogers-Ramanujan Identities

Second Rogers-Ramanujan Identity

n = 0 \sum \infty \frac{q ^{n (n + 1)}}{( q ; q ) _{n}} = \frac{1}{( q ^{2} ; q ^{5} ) _{\infty} ( q ^{3} ; q ^{5} ) _{\infty}}

Second Rogers-Ramanujan Identity is catalogued as a identity in Rogers-Ramanujan Identities.

Plain language: Rogers discovered the identities earlier; Ramanujan independently rediscovered them.

Level: Advanced
Source: Source catalogue
Status: Needs Review

Open the entry for context, source notes, related formulas, and later mathematical developments.

Read Discovery