Pith Number

pith:CDLPVDPM

pith:2022:CDLPVDPMF47IEWU3YMJOFJVCG5

not attested not anchored not stored refs pending

An invitation to formal power series

Benjamin Sambale

Formal power series ring operations prove Newton's binomial theorem, Jacobi's triple product, and Rogers-Ramanujan identities without analysis.

arxiv:2205.00879 v7 · 2022-04-28 · math.HO · math.CO · math.NT

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\pithnumber{CDLPVDPMF47IEWU3YMJOFJVCG5}

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4 Citations open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

Combining ideas from various authors we are able to prove Newton's binomial theorem, Jacobi's triple product, the Rogers--Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan's partition congruences, generating functions of Stirling numbers and Jacobi's four-square theorem.

C2weakest assumption

That the ring operations and formal manipulations on power series (addition, multiplication, substitution, differentiation) suffice to establish the listed identities without any appeal to analytic properties or convergence.

C3one line summary

An expository lecture proving Newton's binomial theorem, Jacobi's triple product, Rogers-Ramanujan identities, Ramanujan partition congruences and related results using formal power series.

Formal links

2 machine-checked theorem links

Cited by

1 paper in Pith

An Introduction to Algebraic Combinatorics

Receipt and verification

First computed	2026-08-04T00:35:55.121069Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

10d6fa8dec2f3e825a9bc312e2a6a2375eaa74552677c3bc29f5476caddfa114

Aliases

arxiv: 2205.00879 · arxiv_version: 2205.00879v7 · doi: 10.48550/arxiv.2205.00879 · pith_short_12: CDLPVDPMF47I · pith_short_16: CDLPVDPMF47IEWU3 · pith_short_8: CDLPVDPM

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/CDLPVDPMF47IEWU3YMJOFJVCG5 \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 10d6fa8dec2f3e825a9bc312e2a6a2375eaa74552677c3bc29f5476caddfa114

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "70caaeb355115856e75242cccaf2e38979789973b6c699a3ac4823ebae24232b",
    "cross_cats_sorted": [
      "math.CO",
      "math.NT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.HO",
    "submitted_at": "2022-04-28T15:07:34Z",
    "title_canon_sha256": "70fcfd33517e56bef10f1333cddf010679f30236d730fc4272e3114fa5964753"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2205.00879",
    "kind": "arxiv",
    "version": 7
  }
}