Alternating Melham Sums and Divisibility in Lucas Polynomial Sequences

We introduce an alternating analogue of Melham’s sums for general (a,b)–type Lucas polynomial sequences. According to a previous result we restrict attention to the cases b=±1. Employing Ozeki–Prodinger-like identities, we establish that these alternating power (Melham) sums factor through explicit polynomials in the companion Lucas sequences, yielding sharp divisibility criteria. In particular, we prove that the sum(a2x2+4)W1(x)W3(x)⋯W2m+1(x)∑k=1n(−1)k W2k2m+1(x)is divisible by (w2n+1(x)−(−1)nax)2. Similar factorizations are obtained for the remaining three main theorems, and specialization to the classical Fibonacci, Lucas, Chebyshev and balancing polynomials is provided, extending earlier non-alternating results of Melham and others.