Euler pentagonal number theorem
E697753
The Euler pentagonal number theorem is a fundamental result in number theory and combinatorics that gives a remarkable infinite product expansion for the generating function of partition numbers, involving exponents given by generalized pentagonal numbers.
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in combinatorics ⓘ result in number theory ⓘ |
| appliesTo |
analytic functions in the unit disk via q-series
ⓘ
formal power series in one variable ⓘ |
| characterizes | coefficients in the expansion of ∏_{n≥1}(1−x^n) ⓘ |
| connectsTo |
Euler function φ(q)=∏_{n≥1}(1−q^n)
ⓘ
modular forms ⓘ q-series ⓘ |
| describes | generating function of partition numbers ⓘ |
| expresses | alternating sum over generalized pentagonal exponents ⓘ |
| field |
combinatorics
ⓘ
number theory ⓘ partition theory ⓘ |
| gives | infinite product expansion for the partition generating function ⓘ |
| hasConsequence |
explicit formula for coefficients using generalized pentagonal numbers
ⓘ
sign pattern of coefficients in the product ∏_{n≥1}(1−x^n) ⓘ |
| hasDomain | complex variable q with |q|<1 in analytic context ⓘ |
| hasProofTechnique |
combinatorial arguments
ⓘ
generating functions ⓘ manipulation of infinite products ⓘ |
| historicalPeriod | 18th century mathematics ⓘ |
| implies | recurrence relations for the partition function p(n) ⓘ |
| introducedBy | Leonhard Euler NERFINISHED ⓘ |
| involves | generalized pentagonal numbers ⓘ |
| isCitedIn |
standard texts on combinatorial generating functions
ⓘ
standard texts on partition theory ⓘ |
| isFundamentalFor |
the study of integer partitions
ⓘ
the theory of q-series identities ⓘ |
| isPartOf | the theory of integer partitions ⓘ |
| isUsedIn |
asymptotic analysis of partition numbers
ⓘ
combinatorial proofs about partitions ⓘ proofs of identities in q-series ⓘ |
| isUsedTo | derive recurrence for p(n) involving p(n−k(3k−1)/2) ⓘ |
| namedAfter | Leonhard Euler NERFINISHED ⓘ |
| relatedTo |
Euler’s recurrence for the partition function
ⓘ
Jacobi triple product identity NERFINISHED ⓘ Rogers–Ramanujan identities NERFINISHED ⓘ |
| relates | infinite product expansions and power series expansions ⓘ |
| states | the infinite product ∏_{n≥1}(1−x^n) equals ∑_{k=−∞}^{∞}(−1)^k x^{k(3k−1)/2} ⓘ |
| topic | generalized pentagonal numbers sequence 1,2,5,7,12,15,… ⓘ |
| uses | generalized pentagonal numbers k(3k−1)/2 for integers k ⓘ |
Referenced by (1)
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