Fundamental solutions of singular differential equations with a~Bessel $D_B$ operator

L. N. Lyakhov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 278 (2012), pp. 148-160

Résumé

We prove a theorem on the fundamental solution of an ordinary differential equation in which the role of even-order derivatives is played by powers of the Bessel operator and the role of odd-order derivatives is played by the derivatives of integer powers of the Bessel operator. The result obtained has allowed us to derive formulas for the fundamental solutions of classical singular equations with the Bessel operator when the index of the Bessel operator can take negative values greater than $-1$; in this case the dimension $N$ of the Euclidean space and the total sum $|\gamma|$ of the indices of the Bessel operators that appear in the equation should satisfy the condition $N+|\gamma|-1>0$.