Generalized Theorem for the G1(c, n) Numbers

Abstract

The Generalized theorem for existence and for the main properties of the G 1 (c,n) numbers [finite real positive numbers, related with the zeros $\kappa_{z,n}^{(c)}$ in the imaginary part $\kappa$ of the complex first parameter $a$ of the complex Kummer confluent hypergeometric function $\Phi(a, c;x)$ with $a=c/2+j\kappa$ - complex, $c=2Rea$ - real $(c\neq l=0, -1, -2, \ldots), \kappa$ - real (positive, negative or zero, $-\infty ), $x=jz$ - positive purely imaginary, $z$ - real, positive and $n=1,2,3\ldots, c, n$ - restricted], is laid down by means of three lemmas and proved numerically. Lemma 1 reveals their existence and defines them for $c$ - positive or negative real numbers, different from zero or negative integers $(c\neq l)$ as the limits of the infinite sequences of real numbers $\{D_{1}(c, n, z)\}(D_{1}(c, n, z)=\kappa_{z,n}^{(c)}z)$ for $z\rightarrow 0$ . Lemma 2 deals with the special case $c=l$ [in which $\Phi(a, c;x)$ has no sense] and determines $G_{1}(c, n)$ as the common limit of the sequences $\{G_{1}(l-\varepsilon, n)\}$ and $\{G_{1}(l+\varepsilon, n+1)\}$ . ( $\varepsilon$ - an infinitesimal real positive number) at $\varepsilon\rightarrow 0$ . It demonstrates also that under the same condition $\{G_{1}(l+\varepsilon, 1)\}$ gravitates to zero. Lemma 3 is a compendium of the basic features of the numbers [identity of $G_{1}(l, n)$ in the meaning of Lemma 2 with G 1 (2 - l, n) in that of Lemma 1, symmetry of $G_{1}(1+l, n)$ and $G_{1}(1-l, n)$ with respectto the point $c=1$ , bond of G 1 (0.5, n) and G 1 (1.5, n) with the Ludolphian number and intervals of monotony]. An example for application of the newly advanced quantities in the theory of the circular waveguide, entirely filled with azimuthally magnetized ferrite which sustains normal $TE_{0n}$ modes, is given.