Conjectured relation between even and odd central factorial numbers of the first kind

Prove that for all integers n ≥ 1 and k ≥ 1, the central factorial numbers of the first kind satisfy the identity 4^(−k) t(2n−1, 2k−1) = ∑_{q=k}^{n} 4^(−q) C(2q−1, 2k−1) t(2n, 2q), where t(·,·) denotes the central factorial numbers of the first kind and C(·,·) denotes the binomial coefficient.

Background

The paper studies generalized Dirichlet beta and Riemann zeta functions and develops identities involving central factorial numbers of the first and second kinds. These numbers t(n,ν) and T(n,ν) are defined via triangular recursions and reciprocity relations and play a key role in expressing integrals and series expansions.

After deriving several identities linking beta- and zeta-values with central factorial numbers, the author proposes a conjectural closed-form relation that connects odd-indexed central factorial numbers of the first kind t(2n−1,2k−1) with even-indexed ones t(2n,2q) through binomial coefficients and powers of 4. The author describes this as a possibly new identity.