For $\alpha,\beta \in \mathbb{C},\, \alpha$ a non-negative integer, we define $$A_{\alpha,\beta}(t)=(\sinh t)^{2\alpha+1}(\cosh t)^{2\beta+1} $$ and $$ \mathcal{L}_{\alpha,\beta}=\frac{d^2}{dt^2}+\frac{A'_{\alpha,\beta}(t)}{A_{\alpha,\beta}(t)}\frac{d}{dt}.$$ Then the Jacobi function $\phi_\lambda^{\alpha,\beta}(r)$ is the unique solution on $(0,\infty)$ of the initial value problem $$\mathcal{L}_{\alpha,\beta}f=-(\lambda^2+(\alpha+\beta+1)^2)f,\, f(0)=1,f'(0)=0. $$ We have the following asymptotic expansion $\phi_\lambda^{\alpha,\beta}$ (see [1, p. 219]) for $\alpha,\beta $ half odd integers with $\alpha> -\frac{1}{2},$ $$ \sqrt{A_{\alpha,\beta}(t)}\phi_\lambda^{\alpha,\beta}(t)= \sum_{m=0}^M a_m(t)\frac{\mathcal{J}_{\alpha+m}(\lambda t)}{\lambda^{m+\alpha+1/2}}+R_M(λ, t),$$ where $a_m$ are holomorphic and $M\geq 0.$ $\mathcal{J}_{\alpha}(x)=\sqrt{x}J_\alpha(x),$ where $J_\alpha$ is the Bessel function of first kind.
My question: Is the above asymptotic expansion also valid when $\alpha> -1$ and $\beta$ is any integer such that $\alpha \pm\beta>-1$? It will be immensely helpful if someone suggests some papers or books regarding this. Thanks in advance.
[1] Brandolini, Luca, Gigante, Giacomo: Equiconvergence theorems for chebli-trimeche hypergroups. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(2), 211–265 (2009)