The (Bessel, Jacobi, Laguerre, Legendre)-type fourth-order linear ordinary differential equations: remarks on history and special function solutions

26 May 2006, 3 pm

 

Speaker: Prof W N Everitt (Birmingham)

The Hermite, Jacobi, Laguerre, Legendre second-order ordinary linear differential equations, and their sets of orthogonal polynomials, have a long history of discovery.
In 1929 Bochner proved that these four sets of orthogonal polynomials are unique in the sense that they are generated by second-order symmetric ordinary linear differential expressions, i.e. Sturm-Liouville differential expressions.
The search for equivalent results using fourth-order symmetric differential equations, in the 1930s, ended with the discoveries by H.L. Krall, in 1938 and 1940, of the (Jacobi, Laguerre, Legendre)-type fourth-order differential equations and their corresponding sets of orthogonal polynomials. There is no Hermite-type fourth-order differential equation.
Krall also proved the equivalent of the second-order Bochner result for the fourth-order type differential equations.
Likewise the second-order linear ordinary Bessel differential equation has a long and indeed remarkable history. The structured, higher-order Bessel ordinary linear differential equations were discovered by Everitt and Markett in 1994. In particular the fourth-order Bessel-type differential equation has now been extensively studied.
In both the second-order and fourth-order cases there are remarkable confluent limit operations between the solutions of the Jacobi et al and the (Jacobi et al)-type equations, and the solutions of the of the second-order Bessel and the fourth-order Bessel-type equation, respectively.
The lecture will outline these results and properties.