Hankel determinant identities for convolution powers not divisible by three

Prove the conjectured periodic identities for the Hankel determinants H_n(M(x)^r) when r is congruent to 1 or 2 modulo 3, including the stated equalities, zero values, paired-sum identities, and parameter condition involving a with |a|=1 for r=4 and r=7.

Background

For convolution powers r congruent to 1 or 2 modulo 3, the paper reports more complicated determinant patterns and gives explicit theorems for r=4, 5, and 7. It then formulates a general conjecture covering both residue classes.

The identities are supported by calculations for r<27 through the authors’ software package, but Conjecture 7 is stated in general rather than proved for all admissible r. The source text’s displayed formulas contain OCR-corrupted exponents and subscripts; the statement here preserves the explicitly stated structure while identifying the relevant determinant family and parameter conditions.

References

Conjecture 7. For r = 1 or 2 (mod 3), we have H3rn(F(x,r)) = H3rn+1(F(x,r)) = H3rn+r(F(x,r)) = H3rn+r+1(F(x,r)) =Q. H3rn+2r(F(x,r)) = H3rn+2r+1(F(x,r)) =0. H3rn+2r-1(F(x,r)) + H3rn+2r+2(F(x,r))= ((2r)(n+1))"-2. H3rn+2(F(x,r)) + H3rn-1(F(x,r)) =yr(r -3). where | a| = |3| = |7| = 1.

— Hankel determinants for convolution powers of Motzkin numbers  (2502.21050 - Wang et al., 28 Feb 2025) in Conjecture 7, Section 1