§13.9 Zeros

§13.9(i) Zeros of $M\left(a,b,z\right)$

If $a$ and $b-a\neq 0,-1,-2,\dots$, then $M\left(a,b,z\right)$ has infinitely many $z$-zeros in $\mathbb{C}$. When $a,b\in\mathbb{R}$ the number of real zeros is finite. Let $p(a,b)$ be the number of positive zeros. Then

 13.9.1 $\displaystyle p(a,b)$ $\displaystyle=\left\lceil-a\right\rceil,$ $a<0$, $b\geq 0$, ⓘ Symbols: $\left\lceil\NVar{x}\right\rceil$: ceiling of $x$ and $p(a,b)$: number of positive zeros Permalink: http://dlmf.nist.gov/13.9.E1 Encodings: TeX, pMML, png See also: Annotations for §13.9(i), §13.9 and Ch.13 13.9.2 $\displaystyle p(a,b)$ $\displaystyle=0,$ $a\geq 0$, $b\geq 0$, ⓘ Symbols: $p(a,b)$: number of positive zeros Permalink: http://dlmf.nist.gov/13.9.E2 Encodings: TeX, pMML, png See also: Annotations for §13.9(i), §13.9 and Ch.13 13.9.3 $\displaystyle p(a,b)$ $\displaystyle=1,$ $a\geq 0$, $-1, ⓘ Symbols: $p(a,b)$: number of positive zeros Permalink: http://dlmf.nist.gov/13.9.E3 Encodings: TeX, pMML, png See also: Annotations for §13.9(i), §13.9 and Ch.13
 13.9.4 $p(a,b)=\left\lfloor-\tfrac{1}{2}b\right\rfloor-\left\lfloor-\tfrac{1}{2}(b+1)% \right\rfloor,$ $a\geq 0$, $b\leq-1$. ⓘ Symbols: $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$ and $p(a,b)$: number of positive zeros Permalink: http://dlmf.nist.gov/13.9.E4 Encodings: TeX, pMML, png See also: Annotations for §13.9(i), §13.9 and Ch.13
 13.9.5 $p(a,b)=\left\lceil-a\right\rceil-\left\lceil-b\right\rceil,$ $\left\lceil-a\right\rceil\geq\left\lceil-b\right\rceil$, $a<0$, $b<0$, ⓘ Symbols: $\left\lceil\NVar{x}\right\rceil$: ceiling of $x$ and $p(a,b)$: number of positive zeros Permalink: http://dlmf.nist.gov/13.9.E5 Encodings: TeX, pMML, png See also: Annotations for §13.9(i), §13.9 and Ch.13
 13.9.6 $p(a,b)=\left\lfloor\tfrac{1}{2}\left(\left\lceil-b\right\rceil-\left\lceil-a% \right\rceil+1\right)\right\rfloor-\left\lfloor\tfrac{1}{2}\left(\left\lceil-b% \right\rceil-\left\lceil-a\right\rceil\right)\right\rfloor,$ $\left\lceil-b\right\rceil>\left\lceil-a\right\rceil>0$.

The number of negative real zeros $n(a,b)$ is given by

 13.9.7 $n(a,b)=p(b-a,b).$ ⓘ Defines: $n(a,b)$: number of negative real zeros (locally) Symbols: $p(a,b)$: number of positive zeros Permalink: http://dlmf.nist.gov/13.9.E7 Encodings: TeX, pMML, png See also: Annotations for §13.9(i), §13.9 and Ch.13

When $a<0$ and $b>0$ let $\phi_{r}$, $r=1,2,3,\dots$, be the positive zeros of $M\left(a,b,x\right)$ arranged in increasing order of magnitude, and let $j_{b-1,r}$ be the $r$th positive zero of the Bessel function $J_{b-1}\left(x\right)$10.21(i)). Then

 13.9.8 $\phi_{r}=\frac{j_{b-1,r}^{2}}{2b-4a}\left(1+\frac{2b(b-2)+j_{b-1,r}^{2}}{3(2b-% 4a)^{2}}\right)+O\left(\frac{1}{a^{5}}\right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $r=1,2,3,\dots$, $\phi_{r}$: positive zeros and $j_{b,r}$: positive zero of Bessel Referenced by: §13.22, §13.9(i) Permalink: http://dlmf.nist.gov/13.9.E8 Encodings: TeX, pMML, png See also: Annotations for §13.9(i), §13.9 and Ch.13

as $a\to-\infty$ with $r$ fixed.

Inequalities for $\phi_{r}$ are given in Gatteschi (1990), and identities involving infinite series of all of the complex zeros of $M\left(a,b,x\right)$ are given in Ahmed and Muldoon (1980).

For fixed $a,b\in\mathbb{C}$ the large $z$-zeros of $M\left(a,b,z\right)$ satisfy

 13.9.9 $z=\pm(2n+a)\pi\mathrm{i}+\ln\left(-\frac{\Gamma\left(a\right)}{\Gamma\left(b-a% \right)}\left(\pm 2n\pi\mathrm{i}\right)^{b-2a}\right)+O\left(n^{-1}\ln n% \right),$

where $n$ is a large positive integer, and the logarithm takes its principal value (§4.2(i)).

Let $P_{\alpha}$ denote the closure of the domain that is bounded by the parabola $y^{2}=4\alpha(x+\alpha)$ and contains the origin. Then $M\left(a,b,z\right)$ has no zeros in the regions $P_{\ifrac{b}{a}}$, if $0; $P_{1}$, if $1\leq a\leq b$; $P_{\alpha}$, where $\alpha=\ifrac{(2a-b+ab)}{(a(a+1))}$, if $0 and $a\leq b<\ifrac{2a}{(1-a)}$. The same results apply for the $n$th partial sums of the Maclaurin series (13.2.2) of $M\left(a,b,z\right)$.

More information on the location of real zeros can be found in Zarzo et al. (1995) and Segura (2008).

For fixed $b$ and $z$ in $\mathbb{C}$ the large $a$-zeros of $M\left(a,b,z\right)$ are given by

 13.9.10 $a=-\frac{\pi^{2}}{4z}\left(n^{2}+(b-\tfrac{3}{2})n\right)-\frac{1}{16z}\left((% b-\tfrac{3}{2})^{2}\pi^{2}+\tfrac{4}{3}z^{2}-8b(z-1)-4b^{2}-3\right)+O\left(n^% {-1}\right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $z$: complex variable Referenced by: §13.9(i) Permalink: http://dlmf.nist.gov/13.9.E10 Encodings: TeX, pMML, png See also: Annotations for §13.9(i), §13.9 and Ch.13

where $n$ is a large positive integer.

For fixed $a$ and $z$ in $\mathbb{C}$ the function $M\left(a,b,z\right)$ has only a finite number of $b$-zeros.

§13.9(ii) Zeros of $U\left(a,b,z\right)$

For fixed $a$ and $b$ in $\mathbb{C}$, $U\left(a,b,z\right)$ has a finite number of $z$-zeros in the sector $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta(<\tfrac{3}{2}\pi)$. Let $T(a,b)$ be the total number of zeros in the sector $|\operatorname{ph}z|<\pi$, $P(a,b)$ be the corresponding number of positive zeros, and $a$, $b$, and $a-b+1$ be nonintegers. For the case $b\leq 1$

 13.9.11 $T(a,b)=\left\lfloor-a\right\rfloor+1,$ $a<0$, $\Gamma\left(a\right)\Gamma\left(a-b+1\right)>0$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$ and $T(a,b)$: number of zeros Permalink: http://dlmf.nist.gov/13.9.E11 Encodings: TeX, pMML, png See also: Annotations for §13.9(ii), §13.9 and Ch.13
 13.9.12 $T(a,b)=\left\lfloor-a\right\rfloor,$ $a<0$, $\Gamma\left(a\right)\Gamma\left(a-b+1\right)<0$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$ and $T(a,b)$: number of zeros Permalink: http://dlmf.nist.gov/13.9.E12 Encodings: TeX, pMML, png See also: Annotations for §13.9(ii), §13.9 and Ch.13
 13.9.13 $T(a,b)=0,$ $a>0$, ⓘ Symbols: $T(a,b)$: number of zeros Permalink: http://dlmf.nist.gov/13.9.E13 Encodings: TeX, pMML, png See also: Annotations for §13.9(ii), §13.9 and Ch.13

and

 13.9.14 $P(a,b)=\left\lceil b-a-1\right\rceil,$ $a+1, ⓘ Symbols: $\left\lceil\NVar{x}\right\rceil$: ceiling of $x$ and $P(a,b)$: number of positive zeros Permalink: http://dlmf.nist.gov/13.9.E14 Encodings: TeX, pMML, png See also: Annotations for §13.9(ii), §13.9 and Ch.13
 13.9.15 $P(a,b)=0,$ $a+1\geq b$. ⓘ Symbols: $P(a,b)$: number of positive zeros Permalink: http://dlmf.nist.gov/13.9.E15 Encodings: TeX, pMML, png See also: Annotations for §13.9(ii), §13.9 and Ch.13

For the case $b\geq 1$ we can use $T(a,b)=T(a-b+1,2-b)$ and $P(a,b)=P(a-b+1,2-b)$.

In Wimp (1965) it is shown that if $a,b\in\mathbb{R}$ and $2a-b>-1$, then $U\left(a,b,z\right)$ has no zeros in the sector $|\operatorname{ph}{z}|\leq\frac{1}{2}\pi$.

Inequalities for the zeros of $U\left(a,b,x\right)$ are given in Gatteschi (1990). See also Segura (2008).

For fixed $b$ and $z$ in $\mathbb{C}$ the large $a$-zeros of $U\left(a,b,z\right)$ are given by

 13.9.16 $a=-n-\frac{2}{\pi}\sqrt{zn}-\frac{2z}{\pi^{2}}+\tfrac{1}{2}b+\tfrac{1}{4}+% \frac{z^{2}\left(\frac{1}{3}-4\pi^{-2}\right)+z-(b-1)^{2}+\frac{1}{4}}{4\pi% \sqrt{zn}}+O\left(\frac{1}{n}\right),$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $z$: complex variable Referenced by: §13.9(ii), DLMF Update; Version 1.0.13, Erratum (V1.0.12) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16) Permalink: http://dlmf.nist.gov/13.9.E16 Encodings: TeX, pMML, png Errata (effective with 1.0.13): In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected. Reported 2016-09-12 by Adri Olde Daalhuis Errata (effective with 1.0.12): Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right hand side, $\sim$ was replaced by $=$. Reported 2016-07-11 by Rudi Weikard See also: Annotations for §13.9(ii), §13.9 and Ch.13

where $n$ is a large positive integer.

For fixed $a$ and $z$ in $\mathbb{C}$, $U\left(a,b,z\right)$ has two infinite strings of $b$-zeros that are asymptotic to the imaginary axis as $|b|\to\infty$.