The N numbers: definition and application

It is proved numerically that the limits lim_k→-∞ 2|k|ζ_k,n^(c)/ζ_0,n^(c) and lim_k→-∞ 2|a|ζ_k,n^(c)/ζ_0,n^(c), where ζ_k,n^(c) is the nth positive purely imaginary zero of the complex Kummer confluent hypergeometric function Φ(a,c;x) in x (n = 1,2,3....), with a = c/2 - jk - complex (k - real), c = 2Rea - restricted positive integer and x = jz - positive purely imaginary ( z - real, positive), exist and coincide. Their common value is a finite positive real number, depending on c and n to which is given a special name: N(c,n) number. The definition of the latter is extended to the case c - real and z - real, positive or negative, too. Analyzing similar expressions that take in the purely imaginary roots ξ_k,n^(c) of a certain transcendental equation, involving beside the above function the real derivative difference Bessel ones, finite limits for k → -∞ and k → +∞ are found. Accordingly, the classes of real N_- and N₊ numbers are introduced. Since the zeros of Φ and the roots of the equation mentioned specify the eigenvalue spectrum of normal TE_0n modes in the azimuthally magnetized circular ferrite and ferrite-dielectric waveguides, the numbers advanced are of importance in the theory of these structures. It is shown that they allow us to formulate the mathematical conditions and to deduce the material ones for operation of the latter as digital phase shifters for normal TE_0n mode.