Hypothesis for the Identity of the L2 (c, ρ, n) and (L)over-cap2((c)over-cap, (ρ)over-cap, (h)over-cap) Numbers and Its Application in the Theory of Waveguides

A guess is made that if for the limited arbitrary real numbers c and (c) over cap, for the less than unity real positive ones rho and (rho) over cap and for the natural n and the positive integer (n) over cap it holds: c = (c) over cap, rho = (rho) over cap and n = (n) over cap then the real positive numbers L-2(c, p, n) and (L) over cap (2)((c) over cap, (rho) over cap, (n) over cap), connected with the positive purely imaginary, resp. real zeros in x, resp. in (x) over cap of a special function, devised by two complex Phi(a, c; x) and Phi(a, c; px), resp. real (Phi) over cap((a) over cap, (a) over cap; (x) over cap) and (Phi) over cap((a) over cap, (a) over cap; (rho) over cap(x) over cap) Kummer, and two complex Psi(a, c; x) and Psi(a, c; rho x), resp. real (Psi) over cap((a) over cap, (a) over cap;(x) over cap) and (Phi) over cap((a) over cap, (a) over cap; (rho) over cap(x) over cap) Tricomi confluent hypergeometric ones of suitably selected complex a and real c, resp. real (a) over cap and (c) over cap parameters and a positive purely imaginary x or rho x, resp. real (x) over cap or (rho) over cap(x) over cap variable, coincide. This assumption, called Hypothesis for the identity of the L-2 (c, rho, n) and (L) over cap (2) ((c) over cap, (rho) over cap, (n) over cap) numbers, is substantiated numerically with a higher or lower accuracy, depending on the value of c ((c) over cap). The curve of equation, written through any of the quantities L-2 (c, rho, n) or (L) over cap (2) ((c) over cap, (rho) over cap, (n) over cap) which separates the domains of propagation of the normal TE0n and of the one of the two possible slow (TE) over cap (0 (n) over cap), modes - the (TE) over cap ((1))(0 (n) over cap), wave (n = (n) over cap) in the coaxial waveguide with ferrite, magnetized azimuthally in negative (clockwise) direction, is interpreted as a physical image of the result obtained in case c - (c) over cap - 3, postulating rho ((rho) over cap) as a relative thickness of the central conductor of the structure and n ((n) over cap) as an order of the mode.