SPECTRAL PROPERTIES OF SOLUTIONS OF HYPERGEOMETRIC-TYPE DIFFERENTIAL-EQUATIONS

[摘要] The second-order differential equation sigma(X)y'' + tau(X)gamma' + lambday = 0 is usually called equation of hypergeometric type, provided that sigma, tau are polynomials of degree not higher than two and one, respectively, and lambda is a constant. Their solutions are commonly known as hypergeometric-type functions (HTFs). In this work, a study of the spectrum of zeros of those HTFs for which lambda = -nutau' - 1/2nu(nu - 1)sigma'', nu is-an-element-of R, and sigma, tau are independent of nu, is done within the so-called semiclassical (or WKB) approximation. Specifically, the semiclassical or WKB density of zeros of the HTFs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy. Applications to the Gaussian and confluent hypergeometric functions as well as to Hermite functions are shown.

[发布日期] 1994-05-20

[关键词] DIFFERENTIAL EQUATIONS;ZEROS;SPECIAL FUNCTIONS;SEMICLASSICAL APPROXIMATION