ON A GENERALIZATION OF BESSEL-FUNCTIONS SATISFYING HIGHER-ORDER DIFFERENTIAL-EQUATIONS

[摘要] Bessel-type functions {J(lambda)(alpha,M)(x)}(lambda greater than or equal to 0) with two parameters alpha greater than or equal to -1/2 and M greater than or equal to 0, which include the classical normalized Bessel function for M = 0, are introduced as a certain confluent limit of Koornwinder's Laguerre-type polynomials {L(n)(alpha,N)(x)}(n is an element of N0). For any alpha is an element of N-0, they arise as solutions of a spectral lambda-dependent differential equation of order 2 alpha + 4. A necessary and sufficient condition to transform the differential equation into symmetric form is given in terms of an overdetermined system of linear equations. It is shown that for alpha = 0, 1, 2, a solution of this problem exists and leads to a (symmetric) fourth-, sixth- and eighth-order differential equation, respectively. For any alpha greater than or equal to -1/2, we also derive a second-order differential equation, but with coefficients depending nonlinearly on the eigenvalue parameter lambda. Finally, two differential expressions of order 2 alpha + 2, alpha is an element of N-0, are constructed which map the classical Bessel functions onto their nonclassical counterparts, and vice versa. This result may be used to establish an orthogonality relation for the Bessel-type functions (in a distributional sense) with respect to the distributional weight function (2/Gamma(alpha + 1))x(2 alpha+1) + M delta(x), delta denoting the point mass centered at the origin.

[发布日期] 1994-10-20