{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-analytic-ordinary-generating-function",
  "semantic_family_id": "sf-analytic-ordinary-generating-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "解析普通生成函數"
  },
  "summary": {
    "zh-Hant": "數列(a_n)在明示收斂域內以ordinary power series A(x)=sum_{n≥0}a_nx^n解析編碼。"
  },
  "statement": {
    "zh-Hant": "coefficient extraction [x^n]A=a_n是編碼責任。有限partial sum只使用明示prefix。對a_n=r^n，A(x)=1/(1-rx)僅在|rx|<1成立。對a_n=a_{n-1}+a_{n-2}且初值a_0,a_1明示，A(x)=(a_0+(a_1-a_0)x)/(1-x-x^2)，boundary terms不可遺漏。formal power-series identity不自動是analytic equality；不得在radius外代值，有限coefficient prefix不決定完整OGF或radius，名稱中的analytic也不授權未證的逐項微分。"
  },
  "formula": {
    "tex": "A(x)=G(a,x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mrow><mi>A</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><mi>G</mi><mo>(</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo>)</mo></mrow></mrow></math>",
    "semantic_ast": {
      "type": "equation",
      "lhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "A",
          "glyph": "A"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      },
      "rhs": {
        "type": "function_call",
        "callee": {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        },
        "arguments": [
          {
            "type": "symbol",
            "name": "a",
            "glyph": "a"
          },
          {
            "type": "symbol",
            "name": "x",
            "glyph": "x"
          }
        ]
      }
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "429472cdfac2dc7b4c3dbad02a1fe6e3d9f9c6ad3b1bd56bebd238b60cf4d0ee"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "exact families使用有理coefficients與有理evaluation point，並明示足以保證收斂的domain。"
    },
    {
      "id": "a2",
      "zh-Hant": "recurrence family固定二階Fibonacci型homogeneous recurrence及兩個初值。"
    }
  ],
  "symbols": [
    {
      "token": "A",
      "role": "analytic_ordinary_generating_function",
      "role_zh": "解析普通生成函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "evaluation_variable",
      "role_zh": "變數",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "ordinary_generating_operator",
      "role_zh": "普通生成函數算子",
      "scope": "representative_formula"
    },
    {
      "token": "a",
      "role": "coefficient_sequence",
      "role_zh": "係數數列",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-recurrence-relation",
      "title_zh": "遞迴關係",
      "reason_zh": "recurrence-to-OGF。"
    },
    {
      "id": "mko-power-series-radius-of-convergence",
      "title_zh": "冪級數收斂半徑",
      "reason_zh": "analytic domain。"
    },
    {
      "id": "mko-sequence",
      "title_zh": "數列",
      "reason_zh": "coefficient data。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "sequence與coefficient extraction一一對位。",
      "geometric closed form必須附收斂域。",
      "recurrence乘上x^n求和時初值產生明示boundary terms。"
    ],
    "common_misconception_zh": "把formal identity在任意x直接當數值等式，或由有限prefix斷言完整生成函數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本節點限ordinary且analytic的明示slice；不自動涵蓋exponential generating functions或formal-only algebra。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入生成函數運算與漸近係數分析。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-analytic-ordinary-generating-function.md 整理；2026-10-15 脈衝 R03（使用者手動接續 2026-11-13 工作序列；實際日期仍為 2026-10-15），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-15"
  }
}