On the zeros of certain confluent hypergeometric functions

Abstract:

The theory of continued fractions is used to derive the following results which hold for $- \tfrac {1}{2} < \alpha < \infty$: (1) If \[ _1{F_1}(\alpha ;2\alpha + 1;z) = 0\quad [{}_1{F_1}(\alpha + 1;2\alpha + 1;z) = 0]\] then $\operatorname {Re} (z) > 0[ < 0]$. (It is deduced from this result that if ${I_{{\alpha ^{ - 1/2}}}}(z) + {I_{{\alpha ^{ + 1/2}}}}(z) = 0$ then $\operatorname {Re} (z) > 0$, and it is shown that if $\alpha$ is an integer, an unbounded number of roots of this equation exists.) (2) The roots of the equ tions \[ _1{F_1}(\alpha ;2\alpha ;z){ = _1}{F_1}(\alpha ;2\alpha + 1;z)\] and \[ _1{F_1}(\alpha ;2\alpha ;z){ = _1}{F_1}(\alpha + 1;2\alpha + 1;z)\] are identical, pure imaginary, symmetrically distributed about the origin and unbounded in number. (3) Let ${C_n}(z)(n = 0,1, \cdots )$ be the successive convergents of the continued fraction associated with $g(z){ = _1}{F_1}(\alpha ;2\alpha ;z){/_1}{F_1}(\alpha ;2\alpha + 1;z)$. Te oots $z = iy_v^{(n)}(v = 1,2, \cdots )$ of the eqation$g(z) = {C_n}(z)$ have the same properties as those described in (2). Furthermore, they interlace: subject to a suitable ordering, $y_v^{(n)} < y_v^{(n + 1)} < y_{v + 1}^{(n)}(v = 1,2, \cdots )$. (A special case of this result concerns the function ${e^z}$ and the convergents of its continued fraction expansion, and is an extension of the forula ${e^{2iv\pi }} = 1(v = 1,2, \cdots )$.)