Supercongruences modulo p^(3k) for [x^(rn)]((1+x)^α(1−x)^(−β))^n with arbitrary integer exponents, and proofs of conjectures of P. Bala on twenty sequences and four infinite families
Year: 2026
Type: preprint
Abstract: For integers r ≥ 1 and α, β let u(N) = [x^(rN)] ((1+x)^α (1−x)^(−β))^N. We prove that for every prime p ≥ 5, all integers α, β (of either sign) and all n, k ≥ 1, u(n p^k) ≡ u(n p^(k−1)) (mod p^(3k+v)), where v = min(v_p(α), v_p(β)). For α, β ≥ 1 and without the factor p^v this was proved in our earlier note on the Schröder-type sequences; the extension covers negative exponents, the polynomial case (1+x)^α (1−x)^|β|, and the primes dividing both exponents. The proof splits u(n p^k) according to the residue of the summation index modulo p: the terms with p | i are compared with the previous level by the Jacobsthal–Kazandzidis congruence, extended here to binomial coefficients with negative upper argument; the remaining terms are written exactly as T(pa)·p²·(−1)^(b−1)·ce/((pa+b)(pd−b)) times a p-adic unit; the sum over b is expanded about the centre b = p/2, where the even centred alternating sums vanish; and the surviving odd moments are controlled by a congruence Σ_a w(a)(2a+1)^(2j+1) ≡ 0 (mod N), proved for generalized binomial weights of either sign by an identity for derivatives of (1+z)^a (1−z)^(−b) modulo N. As consequences we prove the supercongruences a(n p^k) ≡ a(n p^(k−1)) (mod p^(3k)), p ≥ 5, conjectured by P. Bala in the OEIS for twenty sequences — A091527, A119259, A123164, A156894, A241023, A262732, A333481, A333482, A333561, A333562, A333563, A333715, A348410, A351856, A351857, A352373, A370097 (for p ≥ 5), A370100, A370101, A370102 (eighteen of them were open) —, the same towers for A002003 and A103885 (whose entries record only the case k = 1), and four of his conjectures on infinite families: [x^n]((1+x)^r(1−x)^s)^n for all integers r, s (A352373); [x^(rn)]((1+x)/(1−x))^(sn) (A333715); [x^(an)] S(x)^(bn) for the large Schröder function S (A333481, A333482); and [x^(rn)] G(x)^(sn) for the generating function G of A079489 (A333563). Each sequence is identified with a member of the family from its definition in the OEIS — by Lagrange inversion, changes of variables in formal residues, a count of matchings of sunlet graphs and the duplication formula — and the identifications are checked against all the terms in the OEIS. The case p = 3 of the conjecture in A370097 remains open. The conjectures were found with the author's SyntheticMind engine, which scans the open congruence conjectures of the OEIS, fits each sequence against the families whose towers it can prove and checks the fits on all available terms; the method of this note has been added to the engine. The proofs and the text were prepared with the assistance of an AI system under the author's direction (including a separate AI review pass over the proofs); the author checked every step and takes full responsibility for the content. Files: Blanco_Gomez_2026_supercongruences_integer_exponents_EN.pdf is the note (8 pages); Blanco_Gomez_2026_supercongruencias_exponentes_enteros_ES.pdf is the Spanish version (same content); verify_familia_general.py is an independent verification script (Python 3, exact arithmetic, standard library only) that checks the identifications against the OEIS data and from the definitions (including a brute-force count of matchings of sunlet graphs), Bala's conjectures numerically with the exponent 3k attained, the theorem on a grid of exponents of all signs, the refinement modulo p^(3k+v), every lemma of the proof (also for negative exponents, v ≥ 1 and the degenerate cases), the four families including primes dividing a + b or s + 2r, and controls (failure for p = 3 and for (1+x+x^2)^n); verify_familia_general.log is its output (ALL CHECKS PASSED); datos_oeis.json contains the OEIS terms used.
Author: Blanco Gómez, Roberto
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222497294
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