Cui–Sun Conjecture 2.1: integral Legendre telescopers

Li-Li Cui and Zhi-Hong Sun, Curious identities involving Legendre polynomials and Apéry-like numbers, arXiv:2607.12330v1, page 10, state:

Suppose that m, p ∈ Z^+. Then there are integral polynomials f_i(t) with degree i (i = 0, 1, …, m − 1) such that (1 − x)^{m+1} Σ_{n=0}^{p−1} (2n + 1)^{2m+1} P_n(x) = pL_m(p, 1 − x)P_{p−1}(x) − pL_m(−p, 1 − x)P_p(x), where L_m(p, t) = (2p + 1)^{2m} t^m + f_{m−1}((2p + 1)^2) t^{m−1} + · · · + f_1((2p + 1)^2) t + f_0.

Equation (1.1), page 1, specifies:

The famous Legendre polynomials {P_n(x)} are given by P_0(x) = 1, P_1(x) = x and (n + 1)P_{n+1}(x) = (2n + 1)xP_n(x) − nP_{n−1}(x) (n ≥ 1).

The coefficient family is chosen once for each positive m and works for every positive p. The model is an identity in ℚ[X], with integral coefficient polynomials in ℤ[s]; the endpoint parameter of L is an integer.

The frozen declaration D5/S1/Recurrence/Sun/LegendreOddPowerTelescoper.result proves the strong, p-independent reading of Conjecture 2.1. Equality as polynomials also includes x = 1 and entails every rational specialization without division by 1 − x.

Issue #12574 preregisters the Tier 1 numbered conjecture, its verbatim source, the full quantifiers, literature check and proof route. The bounded literature search recorded no settlement in the searched scope. The paper supplies lower cases, rather than the general all-m assertion; its known cases are comparisons, not new results.

For a monomial in ℤ[s], define

$$ A(s^d)=\sum_{j=1}^{d}\left(4^j\binom{2d}{2j}+2^{2j-1}\binom{2d}{2j-1}\right)s^{d-j}, $$

and extend linearly. Its symmetric-difference identity is

$$ 2r(Af)(r^2)=(r+1)f((r+2)^2)+(r-1)f((r-2)^2)-2rf(r^2). $$

The operator lowers positive degree by one and annihilates constants. Thus its iterates on s^m terminate, and

$$ \ell_m(s,t)=\sum_{j=0}^{m}(-1)^j t^{m-j}A^j(s^m) $$

satisfies $(t+A)\ell_m=t^{m+1}s^m$. Set $f_i=(-1)^{m-i}A^{m-i}(s^m)$ and evaluate s at $(2p+1)^2$ to obtain the source-shaped L. The Legendre recurrence turns the inverse identity into a boundary difference; summing it over n < p gives the claimed formula.

A counterexample would be a positive m for which no p-independent integral family of the specified exact degrees satisfies the polynomial identity for every positive p. The route would fail if the operator identity, exact degree lowering, termination or Legendre boundary difference failed.

The public declarations are P, L, claim and result. The three private content lemmas are A_iter_degree, c_telescoping and g_sum. The binomial monomial operator identity and its finite reindexing are bind-only local have terms inside op_identity, not escape witnesses; structural rewrites and source bridges are also local proof terms. Only pinned Mathlib modules are imported. The Library note Library/Analytic/cuisun2026legendre.md records the source, verbatim statement and DOI locator.

Tier 1 external named open problem. Resolution: Proved. Admission basis: open-problem-resolution (#12574; Proved). Utility: none; the theorem constructs a family uniformly for unbounded m and p, rather than certifying a finite instance or enumerating bounded cases.

| theorem | proof_shape | decisive content | | --- | --- | --- | | A_iter_degree (private) | content | Leading coefficients establish nonzero iterates of exact successive degrees and their termination. | | c_telescoping (private) | content | The terminating inverse cancels all interior coefficients and supplies the local L_bracket. | | g_sum (private) | content | The local g_succ_sub combines the finite inverse with the recurrence to give the Legendre boundary sum. | | result | content | Constructs the common integral family and discharges the entire source-shaped claim. |

Information-escape registration is paused under CLAUDE.md §3.9.

What the settlement shows

Proved in this module. Exact degree lowering and termination (A_iter_degree), the terminating inverse (c_telescoping / L_bracket) and the Legendre boundary sum (g_succ_sub / g_sum) are the escape witnesses used by result. Binomial expansion and finite reindexing establish the operator identity as local normalization, rather than as a content declaration. Degree lowering and annihilation of constants make A nilpotent on degree at most m; the proof uses this for the iterates of s^m. The local operator extension and bracket identity in g_sum identify the boundary difference with $(2n+1)^{2m+1}(1-X)^{m+1}P_n$. The coefficient family is p-independent, integral, nonzero and of exact degree i for every i < m; these properties occur in claim and are discharged by result.

Derived algebraically from the coefficient argument; not separately kernel-checked. The local equality A_coeff_top gives the top coefficient multiplier

$$ 4\binom{2d}{2}+2\binom{2d}{1}=8d^2. $$

Iterating this equality gives

$$ \operatorname{lc}(f_i)=(-1)^{m-i}\prod_{d=i+1}^{m}8d^2 =(-1)^{m-i}8^{m-i}\left(\frac{m!}{i!}\right)^2. $$

This closed form is an algebraic consequence of the coefficient argument, not a separately exported or separately compiled Lean theorem. The kernel-checked public conclusion asserts exact degree and nonvanishing. Reflection $L_m(-p-1,t)=L_m(p,t)$ follows directly from the even power and the common square in the definition of L; it does not need an extra public declaration.

Computed, m ≤ 3. The integer operator reproduces

$$ \ell_1=st-8, $$ $$ \ell_2=s^2t^2-(32s+48)t+256, $$ $$ \ell_3=s^3t^3-(72s^2+400s+256)t^2+(2304s+6656)t-18432. $$

With s = (2p+1)^2 these agree term by term with Corollary 2.1 (equation (2.4)), Corollary 2.5 and Theorem 2.4 of the source. The following exact symbolic command checks the coefficient comparison:

python3 - <<'PY'
import sympy as S
s, t = S.symbols('s t')
def A(f):
    return S.expand(sum(c * sum((4**j * S.binomial(2*d, 2*j) +
        2**(2*j-1) * S.binomial(2*d, 2*j-1)) * s**(d-j)
        for j in range(1, d+1)) for (d,), c in S.Poly(f, s).terms()))
expected = [s*t-8, s**2*t**2-(32*s+48)*t+256,
    s**3*t**3-(72*s**2+400*s+256)*t**2+(2304*s+6656)*t-18432]
for m in range(1, 4):
    q, value = s**m, 0
    for j in range(m+1):
        value += (-1)**j * t**(m-j) * q
        q = A(q)
    assert q == 0 and S.expand(value-expected[m-1]) == 0
print('COEFFICIENT_COMPARISON_OK m=1..3')
PY

Open. Uniqueness of the family is not proved in this module. Nor is a corresponding telescoper for other orthogonal-polynomial families, including v_n, proved here; their recurrences require their own operator correspondence.

Proved / open consequence boundary. The source-shaped identity is available for every m, so the identity input previously provided by the paper's cases m ≤ 4 has no m cutoff. No all-m mod-p or higher-power congruence is a formal conclusion of this module. Such consequences still require the source's specialization formulas, denominator controls and congruence estimates; their all-m analogues remain open here. The paper's bilinear identities, squared-polynomial sums and Apéry-like-number results are separate statements and receive no automatic all-m extension from this result.

The preregistration's citing-work and literature readings are orchestrator-reported; the bounded search does not establish worldwide novelty or priority. Semantic Scholar returned HTTP 429 in that search. No analytic extension, uniqueness theorem, other-family result or all-m supercongruence is claimed as kernel-checked by this module.