Theorem for the Identity of the $G_{1}(c, n)$ and $L_{1}(c,\ n)$ Numbers and Its Application to the Theory of Waveguides

It is proved numerically that for the same restricted positive integers $c$ and natural numbers $n$ the real positive numbers $G_{1}(c, n)$ and $L_{1}(c, n)$, connected with certain kinds of zeros of the complex Kummer confluent hypergeometric function $\Phi(a, c;x)$ of a specially selected complex first parameter $a$, second parameter $c$-acquiring values, as pointed out above and a positive purely imaginary variable $x$, coincide, provided $n$ designates the number of the zeros mentioned. This statement is advanced as a Theorem for the identity of the $G_{1}(c, n)$ and $L_{1}(c, n)$ numbers. The physical interpretation of the truthfulness of the result established made, reflects the possibility to substantiate in two different ways the existence of specific envelope curves in (to compute by means of two algorithms) the phase diagrams of the azimuthally magnetized circular ferrite waveguide for normal $TE_{0n}$ modes in case of negative magnetization which bound the wave propagation from the side of higher frequencies.