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On the first zero of the confluent hypergeometric function. (Italian. English summary) Zbl 0573.33003
The question of determining the zeros of the confluent hypergeometric function \({}_ 1F_ 1(-n-b;a+1;x)\) in the real domain where a and b are non-negative, has hitherto presented some difficulty. Here this important problem is tackled in an ingenious manner by making use of a result of F. Tricomi [Comment. Math. Helv. 25, 196-204 (1951; Zbl 0054.033)], where the rth zero of a certain function \({}_ 2F_ 1\) is connected with the rth zero of an associated function \({}_ 1F_ 1\). The authors deduce a most convenient asymptotic formula for the lower-order zeros of the confluent hypergeometric function under consideration. In conclusion, the formula is tested by putting a and b both equal to zero, when the known zeros of the Laguerre polynomial are compared with the values of the zeros calculated using the method of this paper.
Reviewer: H.Exton
33C05 Classical hypergeometric functions, \({}_2F_1\)
65D20 Computation of special functions and constants, construction of tables
41A30 Approximation by other special function classes