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On the zeros of certain confluent hypergeometric functions


Author: P. Wynn
Journal: Proc. Amer. Math. Soc. 40 (1973), 173-182
MSC: Primary 33A30
DOI: https://doi.org/10.1090/S0002-9939-1973-0318529-7
MathSciNet review: 0318529
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Abstract: The theory of continued fractions is used to derive the following results which hold for $ - \tfrac{1}{2} < \alpha < \infty $: (1) If

$\displaystyle _1{F_1}(\alpha ;2\alpha + 1;z) = 0\quad [{}_1{F_1}(\alpha + 1;2\alpha + 1;z) = 0]$

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

$\displaystyle _1{F_1}(\alpha ;2\alpha ;z){ = _1}{F_1}(\alpha ;2\alpha + 1;z)$

and

$\displaystyle _1{F_1}(\alpha ;2\alpha ;z){ = _1}{F_1}(\alpha + 1;2\alpha + 1;z)$

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 )$.)

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1973-0318529-7
Article copyright: © Copyright 1973 American Mathematical Society