Differential equations for symmetric generalized ultraspherical polynomials

Abstract:

We look for differential equations satisfied by the generalized Jacobi polynomials $\{ P_n^{\alpha ,\beta ,M,N}(x)\} _{n = 0}^\infty$ which are orthogonal on the interval $[- 1,1]$ with respect to the weight function \[ \frac {{\Gamma (\alpha + \beta + 2)}}{{{2^{\alpha + \beta + 1}}\Gamma (\alpha + 1)\Gamma (\beta + 1)}}{(1 - x)^\alpha }{(1 + x)^\beta } + M\delta (x + 1) + N\delta (x - 1),\] where $\alpha > - 1$, $\beta > - 1$, $M \geq 0$, and $N \geq 0$. In the special case that $\beta = \alpha$ and $N = M$ we find all differential equations of the form \[ \sum \limits _{i = 0}^\infty {{c_i}(x){y^{(i)}}(x) = 0,\quad y(x) = P_n^{\alpha ,\alpha ,M,M}(x),} \] where the coefficients $\{ {c_i}(x)\} _{i = 1}^\infty$ are independent of the degree n. We show that if $M > 0$ only for nonnegative integer values of $\alpha$ there exists exactly one differential equation which is of finite order $2\alpha + 4$. By using quadratic transformations we also obtain differential equations for the polynomials $\{ P_n^{\alpha , \pm 1/2,0,N}(x)\} _{n = 0}^\infty$ for all $\alpha > - 1$ and $N \geq 0$.