§13.9 Zeros

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

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

When and $b>0$ let ${\varphi }_r$, $r=1,2,3,\mathrm{\dots }$, be the positive zeros of $M(a,b,x)$ 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

as $a\to -\mathrm{\infty }$ with $r$ fixed.

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

For fixed $a,b\in ℂ$ the large $z$-zeros of $M(a,b,z)$ satisfy

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 $M(a,b,z)$ has no zeros in the regions $P_{b/a}$, if ; $P_1$, if $1\le a\le b$; $P_{\alpha }$, where $\alpha =(2a-b+ab)/(a(a+1))$, if and . The same results apply for the $n$th partial sums of the Maclaurin series (13.2.2) of $M(a,b,z)$.

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 $ℂ$ the large $a$-zeros of $M(a,b,z)$ are given by

where $n$ is a large positive integer.

For fixed $a$ and $z$ in $ℂ$ the function $M(a,b,z)$ has only a finite number of $b$-zeros.

For fixed $a$ and $b$ in $ℂ$, $U(a,b,z)$ has a finite number of $z$-zeros in the sector . Let $T(a,b)$ be the total number of zeros in the sector , $P(a,b)$ be the corresponding number of positive zeros, and $a$, $b$, and $a-b+1$ be nonintegers. For the case $b\le 1$

and

For the case $b\ge 1$ 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 $a,b\in ℝ$ and $2a-b>-1$, then $U(a,b,z)$ has no zeros in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi$.

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

For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $U(a,b,z)$ are given by

where $n$ is a large positive integer.

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