# §12.11(i) Distribution of Real Zeros

If $a\geq-\tfrac{1}{2},$ then $\mathop{U\/}\nolimits\!\left(a,x\right)$ has no real zeros. If $-\tfrac{3}{2}, then $\mathop{U\/}\nolimits\!\left(a,x\right)$ has no positive real zeros. If $-2n-\tfrac{3}{2}, $n=1,2,\dots$, then $\mathop{U\/}\nolimits\!\left(a,x\right)$ has $n$ positive real zeros. Lastly, when $a=-n-\tfrac{1}{2}$, $n=1,2,\dots$ (Hermite polynomial case) $\mathop{U\/}\nolimits\!\left(a,x\right)$ has $n$ zeros and they lie in the interval $[-2\sqrt{-a},2\sqrt{-a}\,]$. For further information on these cases see Dean (1966).

If $a>-\tfrac{1}{2},$ then $\mathop{V\/}\nolimits\!\left(a,x\right)$ has no positive real zeros, and if $a=\tfrac{3}{2}-2n$, $n\in\Integer$, then $\mathop{V\/}\nolimits\!\left(a,x\right)$ has a zero at $x=0$.

# §12.11(ii) Asymptotic Expansions of Large Zeros

When $a>-\frac{1}{2}$, $\mathop{U\/}\nolimits\!\left(a,z\right)$ has a string of complex zeros that approaches the ray $\mathop{\mathrm{ph}\/}\nolimits z=\frac{3}{4}\pi$ as $z\to\infty$, and a conjugate string. When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\bar{z}_{a,s}$, where $s$ is a large positive integer and

 12.11.1 $z_{a,s}=e^{\frac{3}{4}\pi i}\sqrt{2\tau_{s}}\left(1-\frac{ia\lambda_{s}}{2\tau% _{s}}+\frac{2a^{2}\lambda_{s}^{2}-8a^{2}\lambda_{s}+4a^{2}+3}{16\tau_{s}^{2}}+% \mathop{O\/}\nolimits\!\left(\lambda_{s}^{3}\tau_{s}^{-3}\right)\right),$

with

 12.11.2 $\tau_{s}=\left(2s+\tfrac{1}{2}-a\right)\pi+i\mathop{\ln\/}\nolimits\!\left(\pi% ^{-\frac{1}{2}}2^{-a-\frac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2% }+a\right)\right),$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function, $\mathop{\ln\/}\nolimits z$: principal branch of logarithm function, $s$: nonnegative integer, $a$: real or complex parameter and $\tau_{s}$ Permalink: http://dlmf.nist.gov/12.11.E2 Encodings: TeX, pMML, png

and

 12.11.3 $\lambda_{s}=\mathop{\ln\/}\nolimits\tau_{s}-\tfrac{1}{2}\pi i.$ Symbols: $\mathop{\ln\/}\nolimits z$: principal branch of logarithm function, $s$: nonnegative integer, $\tau_{s}$ and $\lambda_{s}$ Permalink: http://dlmf.nist.gov/12.11.E3 Encodings: TeX, pMML, png

When $a=\tfrac{1}{2}$ these zeros are the same as the zeros of the complementary error function $\mathop{\mathrm{erfc}\/}\nolimits(z/\sqrt{2})$; compare (12.7.5). Numerical calculations in this case show that $z_{\frac{1}{2},s}$ corresponds to the $s$th zero on the string; compare §7.13(ii).

# §12.11(iii) Asymptotic Expansions for Large Parameter

For large negative values of $a$ the real zeros of $\mathop{U\/}\nolimits\!\left(a,x\right)$, ${\mathop{U\/}\nolimits^{\prime}}\!\left(a,x\right)$, $\mathop{V\/}\nolimits\!\left(a,x\right)$, and ${\mathop{V\/}\nolimits^{\prime}}\!\left(a,x\right)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the $s$th real zeros of $\mathop{U\/}\nolimits\!\left(a,x\right)$ and ${\mathop{U\/}\nolimits^{\prime}}\!\left(a,x\right)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u^{\prime}_{a,s}$, respectively. Then

 12.11.4 $u_{a,s}\sim 2^{\frac{1}{2}}\mu\left(p_{0}(\alpha)+\frac{p_{1}(\alpha)}{\mu^{4}% }+\frac{p_{2}(\alpha)}{\mu^{8}}+\cdots\right),$ Symbols: $\sim$: asymptotic equality, $s$: nonnegative integer, $a$: real or complex parameter, $u_{a,s}$: zeros, $\alpha$ and $p_{n}(\zeta)$: coefficients Referenced by: §12.11(iii) Permalink: http://dlmf.nist.gov/12.11.E4 Encodings: TeX, pMML, png

as $\mu$ ($=\sqrt{-2a}$) $\to\infty$, $s$ fixed. Here $\alpha=\mu^{-\frac{4}{3}}a_{s}$, $a_{s}$ denoting the $s$th negative zero of the function $\mathop{\mathrm{Ai}\/}\nolimits$ (see §9.9(i)). The first two coefficients are given by

 12.11.5 $p_{0}(\zeta)=t(\zeta),$ Symbols: $\zeta$: change of variable and $p_{n}(\zeta)$: coefficients Permalink: http://dlmf.nist.gov/12.11.E5 Encodings: TeX, pMML, png

where $t(\zeta)$ is the function inverse to $\zeta(t)$, defined by (12.10.39) (see also (12.10.41)), and

 12.11.6 $p_{1}(\zeta)=\frac{t^{3}-6t}{24(t^{2}-1)^{2}}+\frac{5}{48((t^{2}-1)\zeta^{3})^% {\frac{1}{2}}}.$ Symbols: $\zeta$: change of variable and $p_{n}(\zeta)$: coefficients Permalink: http://dlmf.nist.gov/12.11.E6 Encodings: TeX, pMML, png

Similarly, for the zeros of ${\mathop{U\/}\nolimits^{\prime}}\!\left(a,x\right)$ we have

 12.11.7 $u^{\prime}_{a,s}\sim 2^{\frac{1}{2}}\mu\left(q_{0}(\beta)+\frac{q_{1}(\beta)}{% \mu^{4}}+\frac{q_{2}(\beta)}{\mu^{8}}+\cdots\right),$ Symbols: $\sim$: asymptotic equality, $s$: nonnegative integer, $a$: real or complex parameter, $u^{\prime}_{a,s}$: zeros and $\beta$ Permalink: http://dlmf.nist.gov/12.11.E7 Encodings: TeX, pMML, png

where $\beta=\mu^{-\frac{4}{3}}a^{\prime}_{s}$, $a^{\prime}_{s}$ denoting the $s$th negative zero of the function ${\mathop{\mathrm{Ai}\/}\nolimits^{\prime}}$ and

 12.11.8 $q_{0}(\zeta)=t(\zeta).$ Symbols: $\zeta$: change of variable Permalink: http://dlmf.nist.gov/12.11.E8 Encodings: TeX, pMML, png

For the first zero of $\mathop{U\/}\nolimits\!\left(a,x\right)$ we also have

 12.11.9 $u_{a,1}\sim 2^{\frac{1}{2}}\mu\left(1-1.85575\;708\mu^{-4/3}-0.34438\;34\mu^{-% 8/3}-0.16871\;5\mu^{-4}-0.11414\mu^{-16/3}-0.0808\mu^{-20/3}-\cdots\right),$ Symbols: $\sim$: asymptotic equality, $a$: real or complex parameter and $u_{a,s}$: zeros Referenced by: §12.11(iii) Permalink: http://dlmf.nist.gov/12.11.E9 Encodings: TeX, pMML, png

where the numerical coefficients have been rounded off.

For further information, including associated functions, see Olver (1959).