Open AccessOn a Third Kind of Characteristic Numbers of the Spheroidal Functions
Author(s) -
Donald R. Rhodes
Publication year - 1970
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.67.1.351
Subject(s) - orthogonality , infinity , mathematics , eigenvalues and eigenvectors , mathematical analysis , range (aeronautics) , alpha (finance) , real number , order (exchange) , integral equation , physics , quantum mechanics , geometry , statistics , materials science , finance , economics , composite material , construct validity , psychometrics
A set of real positive numbers γαn (c ) was found recently to be associated with the spheroidal functions of real order α > -1 as a consequence of their double orthogonality on (-1, 1) and (- ∞, ∞). In the range -1 < α < 0 these numbers are shown to be determined by the eigenvalues of a new integral equation for the spheroidal functions. Thus they represent a third kind of characteristic numbers. The new equation ceases to be an integral equation above the range -1 < α < 0 but it leads to computational formulas for γαn (c ) that appear from numerical results to be valid for all real α > -1.