On the singularities of the continuous Jacobi transform when $\alpha +\beta =0$

Abstract:

Let $\alpha ,\beta > - 1$ and $\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x) = {(1 - x)^\alpha }{(1 + x)^\beta }P_\lambda ^{(\alpha ,\beta )}(x)$, where $P_\lambda ^{(\alpha ,\beta )}(x)$ is the Jacobi function of the first kind, $\lambda \geq - (\alpha + \beta + 1)/2$, and $- 1 < x \leq 1$. Let \[ {F^{(\alpha ,\beta )}}(\lambda ) = \frac {1} {{{2^{\alpha + \beta + 1}}}}\left \langle {f(x),\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x)} \right \rangle = \frac {1} {{{2^{\alpha + \beta + 1}}}}\int _{ - 1}^1 {f(x)\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x)dx} \] whenever the integral exists. It is known that for $\alpha + \beta = 0$, we have (*) \[ f(x) = \lim \limits _{n \to \infty } 4\int _0^n {{F^{(\alpha ,\beta )}}\left ( {\lambda - \frac {1}{2}} \right )} P_{\lambda - 1/2}^{(\beta ,\alpha )}( - x)\lambda \times \sin \pi \lambda \frac {{{\Gamma ^2}(\lambda + 1/2)}}{{\Gamma (\lambda + \alpha + 1/2)\Gamma (\lambda + \beta + 1/2)}}d\lambda \] almost everywhere in $[-1,1]$. In this paper, we devise a technique to continue $f(x)$ analytically to the complex $z$-plane and locate the singularities of $f(z)$ by relating them to the singularities of \[ g(t) = \int _0^\infty {{e^{ - \lambda t}}{F^{(\alpha ,\beta )}}(\lambda )} \frac {{d\lambda }}{{\Gamma (\lambda + \alpha + 1)}}.\] However, this will be done in the more general case where the limit in (*) exists in the sense of Schwartz distributions and defines a generalized function $f(x)$. In this case, we pass from $f(x)$ to its analytic representation \[ \hat f(z) = \frac {1} {{2\pi i}}\left \langle {f(x),\frac {1} {{x - z}}} \right \rangle ,\quad z \notin \operatorname {supp} f,\] and then relate the singularities of $\hat f(z)$ to those of $g(t)$.