Closed formula for the odd lacunary coefficients in the regularized Hermite case

Derive a closed formula for the odd-indexed coefficients \(\beta_{2n-1}\) in the recurrence defining the lacunary orthogonal polynomials \(R_{2n+1}(x)=H_{2n+2}(x)+\beta_{2n+1}H_{2n}(x)\) associated with the regularized Hermite functional \(w_h\).

Background

In Example 3.8, the paper studies lacunary orthogonal polynomials generated from the monic Hermite polynomials by the relation $R_n(x)=H_{n+1}(x)+\beta_n H_{n-1}(x)$, where the coefficients $\beta_n$ are determined by the regularized Hermite functional $w_h$. The coefficients satisfy a two-step recurrence obtained from the factorization relations involving $\alpha_n$ and $\beta_n$.

The even subsequence admits the explicit formula $\beta_{2n}=n+\tfrac12$, but the authors state that they cannot obtain a closed formula for the odd subsequence $\beta_{2n-1}$. Determining such a formula would complete the explicit description of the lacunary orthogonal polynomials in this example.