数理解析研究所講究録 1341 巻 2003 年 102-111

102

ON THE GAUSS HYPERGEOMETRICSERIES WITH ROOTS OUTSIDE THE
UNIT DISK
高野勝男 [Takano Katsuo]
茨城大学理学部
岡崎宏光 [Okazaki Hiromitsu] 熊本大学教育学部

1.

INTRODUCTION

It is known in [9] that the normed conjugate product of gamma functions
such as,(1)$\frac{2}{\pi}\Gamma(1-\cdot x)\Gamma(1+ix)$$= \frac{2}{\pi}\frac{1}{\Pi_{n=1}^{\infty}(1+x^{2}/n^{2})}$is an infinitely divisible density. In the process in showing the infinite divisibility of the probability distribution with density (1) afamily of polynomials
with roots outside the unit disk appeared. Prom the infinite divisibility of
the above probability distribution and from numerical analysis of roots of theterminating hypergeometric series we conjectured that the following density
function consisting of normed conjugate product of gamma functions is an
infinitely divisible density.
$c| \frac{\Gamma(m+ix)}{\Gamma(m)}|^{2}=\frac{c}{\Pi_{n\Phi}^{\infty}(1+x^{2}/(m+n)^{2})}$(m $\in N)$(2)(cf. [1. 6.1.25]) In this case the Gauss hypergeometric series appears ingeneral form and it is much more complicated than the case $m=1$ . We arenecessary to study the location of roots of the Gauss hypergeometric series
in showing the infinite divisibility of the probability distribution with density(2). In this paper we will show that many Gauss hypergeometric series haveroots outside the unit disk

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2.

ON THE GAUSS HYPERGEOMETRICSERIESIn what follows, suppose taht $a_{1}=m$ , $a_{2}=m+1,\ldots$ , $a_{n}=m+n-1$ andconsider the following density function instead of (2),(3)$f(x)= \frac{c}{\Pi_{j=1}^{n}(x^{2}+a_{j}^{2})}$

where

$c$

is anormalized constant to be satisfied by the following$\int_{-\infty}^{\infty}f(x)dx=1$.

The probability density function $f(x)$ is an approximation of the above righthand side of (2) in the sense of weak limit. Let us consider acharacteristic
function of the density fucntion (3). It holds that$\phi(t)$$= \int_{-\infty}^{\infty}e^{\dot{\iota}tx}\frac{c}{\square _{j=1}^{n}(x^{2}+a_{j}^{2})}dx$, $-\infty<t<\infty$ .

(4)$= \pi c\sum_{j=1}^{n}\frac{\exp(-a_{j}|t|)}{a_{j}\mathrm{I}\mathrm{I}_{l=1,l\neq j}^{n}(-a_{j}^{2}+a_{l}^{2})}$

If we set $x=\exp(-|t|)$ then we obtain apolynomial such as the followingform,$\phi(t)=\pi c\sum_{j=1}^{n}\frac{x^{a_{\mathrm{j}}}}{a_{j}\Pi_{l=1,l\neq j}^{n}(-a_{j}^{2}+a_{l}^{2})}$,$0\leq x$$\leq 1$,and we have acomplex polynomial,$P_{n-1}(z)=(-1)^{n-1}a_{n} \Pi_{l=1}^{n-1}(-.

We will use the symbol.

a_{n}^{2}+a_{l}^{2})\sum_{j=1}^{n}\frac{z^{a_{f}-m}}{a_{j}\Pi_{l=1,l\neq j}^{n}(-a_{j}^{2}+a_{\mathrm{t}}^{2})}$$g_{n}(z)$(5)

in place of $P_{n}(z)$ . They are concretely as follows.
$g_{0}(z)=1$(6)$g_{1}(z)=1+ \frac{(-1)(2m)}{2m+2}z$(7)$g_{2}(z)=1+ \frac{(-2)(2m)}{2m+3}z+\frac{(-2)(-1)(2m)(2m+1)}{(2m+3)(2m+4)}\frac{z^{2}}{2!}$(8)$g_{3}(z)$$=$$1+ \frac{(-3)(2m)}{2m+4}z+\frac{(-3)(-2)(2m)(2m+1)}{(2m+4)(2m+5)}\frac{z^{2}}{2!}$$+$$\frac{(-3)(-2)(-1)(2m)(2m+1)(2m+2)}{(2m+4)(2m+5)(2m+6)}\frac{z^{3}}{3!}$

(9)

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$g_{n}(z)=1+ \frac{(-n)(2m)}{2m+n+1}z+\frac{(-n)(-n+1)(2m)(2m+1)}{(2m+n+1)(2m+n+2)}\frac{z^{2}}{2!}$$+$$\frac{(-n)(-n+1)(-n+2)(2m)(2m+1)(2m+2)z^{3}}{(2m+n+1)(2m+n+2)(2m+n+3)3!}+\cdots$$+$$\frac{(-n)(-n+1)\cdots(-n+k-1)(2m)(2m+1)(2m+2)\cdots(2m+k-1)}{(2m+n+1)(2m+n+2)(2m+n+3)\cdots(2m+n+k)}$$\frac{z^{k}}{k!}+\cdots$$(-n)(-n+1)\cdots$ $(-2)(-1)(2m)(2m+1)(2m+2)\cdots$ $(2m+n-1)z^{n}$$+$$\overline{(2m+n+1)(2m+n+2)(2m+n+3)\cdots(2m+2n)}\overline{n!}$$=2F1(2m, -n;2m+n+1;z)$(10)

Two trigonometirc sums are coining from the polynomials
the unit circle C:zand$g_{n}(z)$. Consider$=e^{\theta}\dot{.}(0\leq\theta\leq 2\pi)$$g_{n}(e^{\dot{l}\theta}.)$$=2F1(2m,$-n; $2m+n+1;e^{\theta}\dot{.})$ .

(11)

It is often convenient for us to treat the polynomial $z^{m}g_{n}(z)$ in place of $g_{n}(z)$ .
Let us set$u(m, n;\theta)=Ree^{im\theta}g_{n}(e^{:\theta})$ ,(12)$v(m, n;\theta)=Ime^{im\theta}g_{n}(e^{\dot{\iota}\theta})$

.

(13)

We have$u(m,n; \theta)=\cos m\theta+\frac{(-n)(2m)}{2m+n+1}\cos(m+1)\theta$

$+$

$+$

$+$

$(-n)(-n+1)(2m)(2m+1)\cos(m+2)\theta$$\overline{(2m+n+1)(2m+n+2)}\overline{2!}$$(-n)(-n+1)(-n+2)(2m)(2m+1)(2m+2)\cos(m+3)\theta$$\overline{(2m+n+1)(2m+n+2)(2m+n+3)}\overline{3!}+\cdots$$\frac{(-n)(-n+1)\cdots(-n+k-1)(2m)(2m+1)(2m+2)\cdots(2m+k-1)}{(2m+n+1)(2m+n+2)(2m+n+3)\cdots(2m+n+k)}$$\frac{\cos(m+k)\theta}{k!}$$+\cdot$$+$..

$\frac{(-n)(-n+1)\cdots(-2)(-1)(2m)(2m+1)(2m+2)\cdots(2m+n-1)}{(2m+n+1)(2m+n+2)(2m+n+3)\cdots(2m+2n)}$$\cos(m+n)\theta$$\overline{n!}$

(14)

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$v(m,n; \theta)=\sin m\theta+\frac{(-n)(2m)}{2m+n+1}\sin(m+1)\theta$$(-n)(-n+1)(2m)(2m+1)\sin(m+2)\theta$$+$$\overline{(2m+n+1)(2m+n+2)}\overline{2!}$$(-n)(-n+1)(-n+2)(2m)(2m+1)(2m+2)\sin(m+3)\theta$
$+$

$+$

$\overline{(2m+n+1)(2m+n+2)(2m+n+3)}\overline{3!}+\cdots$$\frac{(-n)(-n+1)\cdots(-n+k-1)(2m)(2m+1)(2m+2)\cdots(2m+k-1)}{(2m+n+1)(2m+n+2)(2m+n+3)\cdots(2m+n+k)}$$\sin(m+k)\theta$$\overline{k!}$$+\cdot$$+$..

$\frac{(-n)(-n+1)\cdots(-2)(-1)(2m)(2m+1)(2m+2)\cdots(2m+n-1)}{(2m+n+1)(2m+n+2)(2m+n+3)\cdots(2m+2n)}$$\sin(m+n)\theta$$\overline{n!}$

.

(15)

It can be shown that $u(m,n;\theta)$ and $v(m, n;\theta)$ do not always make aJordan
curve when runs through the interval $[-\pi/2,\pi/2]$ . See the figures after a$\theta$conjecture in the last section.

3.

THE HYPERGEOMETRIC SERIES HASNOT ROOTS ON THE UNIT CIRCLE

It is known in [1] that the Gauss hypergeometric series is asolution ofadifferential equation. That is, $g_{n}(z)$ satisfies the hypergeometric equation.
$z(1-z) \frac{d^{2}}{dz^{2}}g_{n}(z)+(c-(a+b+1)z)\frac{d}{dz}g_{n}(z)-abg_{n}(z)=0$

.

(16)

In the above equation we assume $a=2m$ , $b=-n$ and $c=2m+n+1$ . We
are possibly able to make use of aproperty of two independent solutios ofthe second order differential equations and obtain the following

Theorem 1. If $2\leq m$ and $2\leq n\leq 10$ the Gauss hypergeometric series
has not roots on the unit circle.

$g_{n}(z)$Proof. If $z^{m}g_{n}(z)$ has not roots on the unit circle then $g_{n}(z)$ has not roots

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on the unit circle. In order to show thatcircle we will show that the following relation$z^{m}g_{n}(z)$has not roots on the unit

( , ni ) $=c(m,n)(1-\cos\theta)^{n-1}$(17)
holds, where $c(m, n)$ is positive constant not depending on the variable .
If and only if,then r( 0) $=0$ . But we have $\cos km\theta_{0}=1=x$
and $u(m,n;\theta_{0})=const$ $\phi(0)>0$ and so $v’(m,n;\theta_{0})=0$ and we obtain anidentity$r(\theta)=u(m,n;\theta)v’(m, n;\theta)-u’(m, n;\theta)v$ $m$$\theta$$\theta$$\theta_{0}=0$$2\pi$$\cdot$$m+ \frac{(-n)(2m)(m+1)}{2m+n+1}+\frac{(-n)(-n+1)(2m)(2m+1)}{(2m+n+1)(2m+n+2)}\frac{(m+2)}{2!}$

$+$

$+$

$(-n)(-n+1)(-n+2)(2m)(2m+1)(2m+2)(m+3)$$\overline{(2m+n+1)(2m+n+2)(2m+n+3)}\overline{3!}+\cdots$$\frac{(-n)(-n+1)\cdots(-n+k-1)(2m)(2m+1)(2m+2)\cdots(2m+k-1)}{(2m+n+1)(2m+n+2)(2m+n+3)\cdots(2m+n+k)}$$\frac{(m+k)}{k!}+\cdots$$+$$\frac{(-n)(-n+1)\cdots(-2)(-1)(2m)(2m+1)(2m+2)\cdots(2m+n-1)}{(2m+n+1)(2m+n+2)(2m+n+3)\cdots(2m+2n)}$(18)$\frac{(m+n)}{n!}=0$

for $n=2$ , , 10.
The case of $n=2$ :We have$\ldots$$u(m, 2;\theta)$$= \cos m\theta+\frac{(-2)(2m)}{2m+3}\cos(m+1)\theta$$+$$\frac{(-2)(-1)(2m)(2m+1)}{(2m+3)(2m+4)}\frac{\cos(m+2)\theta}{2!}$

(19)

and
$v(m, 2;\theta)$$= \sin m\theta+\frac{(-2)(2m)}{2m+3}\sin(m+1)\theta$$+$.

$\frac{(-2)(-1)(2m)(2m+1)}{(2m+3)(2m+4)}\frac{\sin(m+2)\theta}{2!}$(20)

We see that$r(\theta)$$=$$Re \{e^{-\dot{\iota}m\theta}g_{2}(e^{-i\theta})\frac{1}{i}\frac{d}{d\theta}\{e.\cdot g_{2}(m\theta e^{\dot{l}\theta})\}\}$$=$$\{1+\frac{(-2)(2m)}{2m+3}\cos\theta+\frac{(-2)(-1)(2m)(2m+1)}{(2m+3)(2m+4)}\frac{\cos 2\theta}{2!}\}$

(21)

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$\{m+\frac{(-2)(2m)}{2m+3}(m+1)\cos\theta$$+$$\frac{(-2)(-1)(2m)(2m+1)}{(2m+3)(2m+4)}(m+2)\frac{\cos 2\theta}{2!}\}$$+$$\{\frac{(-2)(2m)}{2m+3}(m+1)\sin\theta+\frac{(-2)(-1)(2m)(2m+1)}{(2m+3)(2m+4)}(m+2)\frac{\sin 2\theta}{2!}\}$$\{\frac{(-2)(2m)}{2m+3}\sin\theta+\frac{(-2)(-1)(2m)(2m+1)}{(2m+3)(2m+4)}\frac{\sin 2\theta}{2!}\}$

.

(22)

For simplicity, set $y=\cos\theta$ and substitute the following idenstitiee,$2y^{2}-1$ and $\sin 2\theta=\sin\theta\cdot(2y)$ , in the last memeber of (22). Then we seethat$\infty \mathrm{s}2\theta$$r(\theta)$

$=$

$=$

$\{1-\frac{2(2m)}{2m+3}y+\frac{2\cdot 1(2m)(2m+1)(2y^{2}-1)}{(2m+3)(2m+4)2!}\}$$\{m-\frac{2(2m)(m+1)}{2m+3}y+\frac{2\cdot 1(2m)(2m+1)(m+2)(2y^{2}-1)}{(2m+3)(2m+4)2!}\}$$+$$(1-y^{2}) \{-\frac{2(2m)}{2m+3}+\frac{2\cdot 1(2m)(2m+1)(2y)}{(2m+3)(2m+4)2!}\}$$\{-\frac{2(2m)(m+1)}{2m+3}+\frac{2\cdot 1(2m)(2m+1)(m+2)(2y)}{(2m+3)(2m+4)2!}\}$

$=$

(23)

$\frac{2(2m)(2m+1)(2m+2)}{(2m+3)(2m+4)}(1-y)^{2}$and we obtain (17) for the case $n=2$ .
The case of $n=3$ :We have$u(m, 3; \theta)=\cos m\theta+\frac{(-3)(2m)}{2m+4}\cos(m+1)\theta$$+$(-3)(-2)(2m)(2m+l) $\cos(m+2)\theta$$\overline{(2m+4)(2m+5)}\overline{2!}$

$+$

(24)

$\frac{(-3)(-2)(-1)(2m)(2m+1)(2m+2)\cos(m+3)\theta}{(2m+5)(2m+6)(2m+7)3!}$and$v(m, 3; \theta)=\sin m\theta+\frac{(-3)(2m)}{2m+4}\sin(m+1)\theta$

$+$

$+$

(-3)(-2)(2m)(2m+l) $\sin(m+2)\theta$$\overline{(2m+4)(2m+5)}\overline{2!}$(-3)(-2)(-1)(2m)(2m+l)(2m+2) $\sin(m+3)\theta$$\overline{(2m+4)(2m+5)(2m+6)}\overline{3!}$.

(25)

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We see that$r(\theta)$

(26)

$=$

$Re \{e^{-im\theta}g_{3}(e^{-i\theta})\frac{1}{i}\frac{d}{d\theta}\{e^{im\theta}g_{3}(e^{i\theta})\}\}$$=$$\{1+\frac{(-3)(2m)}{2m+4}\cos\theta+\frac{(-3)(-2)(2m)(2m+1)}{(2m+4)(2m+5)}\frac{\cos 2\theta}{2!}$$+$$\frac{(-3)(-2)(-1)(2m)(2m+1)(2m+2)}{(2m+5)(2m+6)(2m+7)}\frac{\cos 3\theta}{3!}\}$$\{m+\frac{(-3)(2m)}{2m+4}(m+1)\cos\theta$(-3)(-2)(2m)(2m+l)$+$$\overline{(2m+4)(2m+5)}(m+2)\frac{\cos 2\theta}{2!}$$+$$\frac{(-3)(-2)(-1)(2m)(2m+1)(2m+2)}{(2m+5)(2m+6)(2m+7)}(m+3)\frac{\cos 3\theta}{3!}\}$$+$$\{\frac{(-3)(2m)}{2m+4}(m+1)\sin\theta$$+$$\frac{(-3)(-2)(2m)(2m+1)}{(2m+4)(2m+5)}(m+2)\frac{\sin 2\theta}{2!}$$+$$\frac{(-3)(-2)(-1)(2m)(2m+1)(2m+2)}{(2m+5)(2m+6)(2m+7)}(m+3)\frac{\sin 3\theta}{3!}\}$$\{\frac{(-3)(2m)}{2m+4}\sin\theta+\frac{(-3)(-2)(2m)(2m+1)}{(2m+4)(2m+5)}\frac{\sin 2\theta}{2!}$$+$$\frac{(-3)(-2)(-1)(2m)(2m+1)(2m+2)}{(2m+5)(2m+6)(2m+7)}\frac{\sin 3\theta}{3!}\}$

.

(27)

Substituting the following idenstities, $\cos 2\theta=2y^{2}-1$ , $\cos 3\theta=4y^{3}-3y$ and, sin39 $=\sin\theta\cdot(4y^{2}-1)$ in the last memeber of (27), thenwe see that$\sin 2\theta=\sin\theta\cdot(2y)$$r(\theta)$$=$$\{1-\frac{3(2m)}{2m+4}y+\frac{3\cdot 2(2m)(2m+1)(2y^{2}-1)}{(2m+4)(2m+5)2!}$-$\frac{3!(2m)(2m+1)(2m+2)(4y^{3}-3y)}{(2m+4)(2m+5)(2m+6)3!}\}$$\{m-\frac{3(2m)(m+1)}{2m+4}y+\frac{3\cdot 2(2m)(2m+1)(m+2)(2y^{2}-1)}{(2m+4)(2m+5)2!}$-$\frac{3!(2m)(2m+1)(2m+2)(m+3)(m+3)(4y^{3}-3y)}{(2m+4)(2m+5)(2m+6)3!}\}$$+$$(1-y^{2}) \{-\frac{3(2m)}{2m+4}+\frac{3\cdot 2(2m)(2m+1)(2y)}{(2m+4)(2m+5)2!}$

109$\frac{3!(2m)(2m+1)(2m+2)(4y^{2}-1)}{(2m+4)(2m+5)(2m+6)3!}\}$$\{-\frac{3(2m)(m+1)}{2m+4}.+\frac{3\cdot 2(2m)(2m+1)(m+2)(2y)}{(2m+4)(2m+5)2!}$$\frac{3!(2m)(2m+1)(2m+2)(m+3)(m+3)\{4y^{2}-1\}}{(2m+4)(2m+5)(2m+6)3!}\}$(28)$\frac{2^{2}(2m)(2m+1)(2m+2)(2m+3)}{(2m+4)(2m+5)(2m+6)}(1-y)^{3}$$=$and we obtain (17) for the case $n=3$ . Repeating this method for the cases$n=4,5,6$ , , 10 we obtain the assertion of theorem, q.e.d.
$\ldots$

THE HYPERGEOMETRIC SERIES HASROOTS OUTSIDE THE UNIT DISK

4.

appears outside theIf $m=1$ it is known in [8] that the roots ofis $z_{1}=m+1/m$ and if $n=2$ theclosed unit disk. If $n=1$ the root ofaretwo roots of$g_{n}(z)$$g_{1}(z)$$g_{2}(z)$$z_{1}= \frac{m+1}{2m+1}(\frac{m+2}{m+1}+i\sqrt{\frac{3(m+2)}{m}})$,$z_{2}= \frac{m+1}{2m+1}(\frac{m+2}{m+1}-i\sqrt{\frac{3(m+2)}{m}})$for all m $\in N$. These roots are outside the unit disk. We obtain the following
computational result.
Conjecture: If $2\leq m\leq 20$ and $3\leq n\leq 17$ the Gauss hypergeometricseries $g_{n}(z)$ has roots outside the closed unit disk.

By the following graphs which were drawn with the computer, we are able
is most near 0,to conclude that this conjecture is true. If the value ofthen is apoint on the unit circle since it can be seen from the curve thatthe origin is outside the range of the hypergeometric series with the domain
is not equal to 0.
of the unit disk and$g_{n}(z)$$z$$g_{n}(z)$

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References[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,New York, Dover, 1970.

[2] L. Bondesson, On the infinite divisibility of the half-Cauchy and otherdecreasing densities and probability functions on the nonnegative line,
Scand. Acturial J., (1985), 225-247.
[3] M. J. Goovaerts, L. D’Hooge, and N. De Pril, On the infinite divisibility of the product of two -distributed stochastical variables, Appliedmathematics and computation, 3(1977), 127-135.
$\Gamma$[4] D. H. Kelker, Infinite divisibility and variance mixtures of the normal
distribution, Ann. Math. Statist, 42 (1971), 802-808.
[5] G. Sansone&J. Gerretsen, Lectures on the theory of functions of acomplex variable, 1. Holomorphic functions, P. Noordhoff-Groningen, 1960

[6] K. Sato, Class L of multivariate distributions and its subclasses,
J Multivaria Anal. 10 (1980), 207-232.
[7] F. W. Steutel, Preservation of infinite divisib 垣 ity under mixing and related topics, Math. Centre Tracts, Math. Centre, Amsterdam, 33. 1970

[8] K. Takano, On afamily of polynomials with zeros outside the unit disk,
htemational J. comput. Numer. Anai. AppJ., 1. No.l (2012),369-382.
[9] K. Takano, On infinite divisibility of normed product of Cauchy densities,
J. Comput. Applied Math., 150(2003),253-263.

[10] O. Thorin, On the infinite divisibility of the Pareto distribution, Scand.
ActuriaL J., (1977), 3140.

Katsuo TAKANO
Department of Mathematics
Faculty of Science, Ibaraki University
MitO-city, Ibaraki 310, JAPAN-mail ktaka@mx.ibaraki.ac.jp$\mathrm{e}$Hiromitsu OKAZAKI
Department of Mathematics
Faculty of Education, Kumomoto University
KumomotO-city, Kumomoto 860, JAPAN-mail:okazaki@educ.kum0m0t0-u.ac.jp$\mathrm{e}$