The theorem for existence and for the main properties of the L (c, h) numbers (c — real, h — restricted positive integer), is formulated with the help of three lemmas and proved numerically. Lemma 1 discloses the existence of quantities and determines them for c ^ I, (I = 0, —1, —2,...) as the common limits of some couples of infinite sequences of positive real numbers, constructed by means of the positive real zeros of a real Kummer confluent hypergeometric function of specially picked out parameters. Lemma 2 defines the same in case c = I (when the function in question has simple poles) as the common limit of the sequences of L(l — i, h) and L(l + i, h + 1) numbers in the sense of Lemma 1 attained, if the positive real number i becomes vanishingly small and shows also that under the circumstance referred to L(l + e, 1) approximates to zero. Lemma 3 states that for c = I and c = 1±/ it holds L(c, h) = L(2 — /, h) and L(l + l,h) = L(l — l,h), resp., and that L(0.5,n) and L(1.5,h) are related with the Ludolphian number n. The application of results obtained in the theory of waveguides is demonstrated.
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