The core of the theory of L_2(c, p, n) numbers (c, p - real, 0 < p < 1, n - restricted positive integer) - the theorem for existence and for the basic characteristics of these new real positive quantities, is formulated by means of three lemmas and proved numerically. The first discovers the existence of the same and ascertains them provided c ≠ l, (l = 0,-1, -2,...) as the common limits of definite couples of infinite sequences of real positive numbers, composed through the real positive zeros of a special function, devised, using real Kummer and Tricomi confluent hypergeometric ones of selected in an appropriate way parameters. The second determines them for c = l (when the Kummer function is not defined) with the help of the equality L_2(c, p, n) = L_2(2 - l, p, n). The third lemma says that for any admissible c, p and n, it is fulfilled: L_2(c, p, n) = L_2(2 - c, p, n) and L_2(1 + h, p, n) = L_2(1 - h, p, n), (h = ±(1 - c)). A problem for slow guided wave transmission is pointed out in whose solution the outcomes obtained could be employed.
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