The Z(c, n) numbers and their application in the theory of waveguides

The positive real numbers Z(c,n) are defined for c - arbitrary, real and n = 1, 2, 3, ... The cases c - diferent from zero or a negative integer, c - zero or an even negative integer and c - an odd negative integer, are considered. If a = c/2 - jk - complex, c - real, (c ≠ l, l = 0,-1,-2, ...), k - real, x = jz, z - real, positive and n = 1, 2, 3, ..., Z(c,n) numbers are called the positive purely imaginary zeros ζ^(c)_k,n in x of the complex Kummer confluent hypergeometric function Φ(a, c; x), on condition that k = 0. A theorem is formulated and proved numerically that extends the definition of numbers also for c = l in which Φ(a, c; x) has simple poles and reveals some of their properties. It is composed of three lemmas. Lemmas 1 and 2 substantiate the existence of quantities and determine them when l = 2p, p = 0,-1,-2, ... (l = 0,-2,-4, ..., c - zero or an even negative integer) and l = 2p - 1 (l = -1,-3,-5, ..., c - an odd negative integer), as the common limits of the couples of infinite sequences of positive real numbers {Z(l - ε,n)} and {Z(l+ε,n)}, and {Z(l-ε,n)} and {Z(l+ε,n+1)}), resp. for ε → 0 (ε - infinitesimal positive real number). Lemma 3 demonstrates the possibility to express the numbers Z(l,n), (l - an odd negative integer) through the ones Z(2-l,n), (the symmetry of both quantities with respect to the point c = 1) and presents the relation of Z(0,n) and Z(2,n) with the Ludolphian number π. Tables and graphs visualize the effect of parameters c and n on Z(c,n). The role of the latter is shown in the theory of azimuthally magnetized circular ferrite waveguides that support normal TE_0n modes.