Math Problems Vault

Problem 5.183

Along with the polynomial

f(z) = a_0+a_1z+a_2z^2+\cdots +a_mz^m

of degree $m$ with real coefficients $a_0, a_1, \ldots , a_m$ , $a_m \geq 0$ , we consider the following three associated polynomials:

P(z,\omega ) = a_0+a_1z+a_2z(z-\omega )+\cdots +a_mz(z-\omega )\ldots (z-\overline{m-1}\, \omega ),

\begin{aligned} Q(z,\omega ) = \; & a_0 \quad (1+z-\overline{m-1}\, \omega )(1+z-\overline{m-2}\, \omega )\ldots (1+z-\omega )(1+z) \\ + \; & a_1 \quad (1+z-\overline{m-1}\, \omega )(1+z-\overline{m-2}\, \omega )\ldots (1+z-\omega )z \\ + \; & a_2 \quad (1+z-\overline{m-1}\, \omega )(1+z-\overline{m-2}\, \omega )\ldots (z-\omega )z \\ & \dotfill \\ + \; & a_{m-1}(1+z-\overline{m-1}\, \omega )(z-\overline{m-2}\, \omega ) \quad \ldots (z-\omega )z \\ + \; & a_m \quad (z-\overline{m-1}\, \omega ) \quad (z-\overline{m-2}\, \omega ) \quad \ldots (z-\omega )z, \end{aligned}

\begin{aligned} R(z,\omega ) = \; & a_0 \quad (1-z+\overline{m-1}\, \omega )(1-z+\overline{m-2}\, \omega )\ldots (1-z+\omega )(1-z) \\ + \; & a_1 \quad (1-z+\overline{m-1}\, \omega )(1-z+\overline{m-2}\, \omega )\ldots (1-z+\omega )z \\ + \; & a_2 \quad (1-z+\overline{m-1}\, \omega )(1-z+\overline{m-2}\, \omega )\ldots (z-\omega )z \\ & \dotfill \\ + \; & a_{m-1}(1-z+\overline{m-1}\, \omega )(z-\overline{m-2}\, \omega ) \quad \ldots (z-\omega )z \\ + \; & a_m \quad (z-\overline{m-1}\, \omega ) \quad (z-\overline{m-2}\, \omega ) \quad \ldots (z-\omega )z. \end{aligned}

The three associated polynomials depend on a parameter $\omega$ . If we simultaneously replace

z \text{ by } -z, \qquad \omega \text{ by } -\omega , \qquad a_{\nu } \text{ by } (-1)^{\nu }a_{\nu }, \quad \nu =0,1,2,\ldots ,m, \qquad \text{then}

Q(z,\omega ) \text{ is transformed into } R(z,\omega ).

If we set

f(z)e^{kz} = \sum _{n=0}^{\infty } \frac{k^n}{n!}A_n^{(k)}z^n, \qquad k>0,

f(z)(1-z)^{-k-1} = \sum _{n=0}^{\infty } \frac{\Gamma (k+n+1-m)}{\Gamma (k+1)}\frac{k^m}{n!}B_n^{(k)}z^n, \qquad k \text{ integral}, \quad k>m-1,

f(z)(1+z)^{k-1} = \sum _{n=0}^{k+m-1} \frac{(k-1)!k^m}{n!(k-n+m-1)!}C_n^{(k)}z^n, \qquad k \text{ integral}, \quad k \geq 1,

then the coefficients $A_n^{(k)}$ , $B_n^{(k)}$ and $C_n^{(k)}$ may be expressed in terms of the associated polynomials of $f(z)$ , $P(z,\omega )$ , $Q(z,\omega )$ , and $R(z,\omega )$ .

?

Problem 5.184

Prove the formula, valid for $\omega >0$ , $-1+(m-1)\omega <\mathrm{Re}\left(z\right)<0$ ,

\frac{Q(z,\omega )}{\omega ^m} = \frac{\Gamma \left(\dfrac {1}{\omega }+1\right)}{\Gamma \left(\dfrac {1+z}{\omega }-m+1\right)\Gamma \left(-\dfrac {z}{\omega }\right)} \int _0^1 (1-t)^{\frac{1+z}{\omega }}t^{-\frac{z+\omega }{\omega }}f\left(\frac{t}{t-1}\right)\, dt

and find analogous formulae for $P(z,\omega )$ and $R(z,\omega )$ .

?

Problem 5.185

Denote by

\mathfrak {F}_a^b, \quad \mathfrak {P}_a^b, \quad \mathfrak {Q}_a^b, \quad \mathfrak {R}_a^b

the number of zeros of the polynomials

f(z), \quad P(z,\omega ), \quad Q(z,\omega ), \quad R(z,\omega ),

respectively, in the open interval $a<z<b$ . We have the following inequalities:

\mathfrak {F}_0^{\infty } \leq \mathfrak {P}_0^{\infty }, \qquad \mathfrak {F}_{-\infty }^0 \geq \mathfrak {P}_{-\infty }^0,

\mathfrak {F}_0^1 \leq \mathfrak {Q}_0^{\infty }, \qquad \mathfrak {F}_{-\infty }^0 \geq \mathfrak {Q}_{-1+(m-1)\omega }^0,

\mathfrak {F}_0^{\infty } \leq \mathfrak {R}_0^{1+(m-1)\omega }, \qquad \mathfrak {F}_{-1}^0 \geq \mathfrak {R}_{-\infty }^0, \qquad \mathfrak {F}_{-\infty }^{-1} \geq \mathfrak {R}_{1+(m-1)\omega }^{\infty }.

Here we have assumed throughout that $\omega >0$ ; in the inequalities in the second line we require moreover that $(m-1)\omega <1$ , and in the last inequality of the first column we require that $\omega^{-1}$ be an integer. [38, 80.]

?

Problem 5.186

Assuming that $f(z)$ does not vanish at the end-points of the corresponding interval (that is at $z=0, z=1, z=-1$ respectively) we can add to the above inequalities the statement that the difference between the two sides is an even number (possibly $0$ ).---Why must no additional assumptions be made concerning the point $z=+\infty$ ?

?

Problem 5.187

The three power series considered in 183 have the following properties:

(1) They have not fewer changes of sign than the polynomial $f(z)$ has zeros in the interior of the intervals

(0,+\infty ), \qquad (0,1), \qquad (0,+\infty ),

respectively.

(2) The number of changes of sign decreases or remains unaltered as $k$ increases.

(3) The number of changes of sign attains the number of zeros of $f(z)$ in the corresponding interval, if $k$ is sufficiently large.

?

Problem 5.188

Along with the real polynomial $f(z)$ of degree $m$ , consider the fourth associated polynomial

J(z,\omega ) = f(m\omega )-\binom {m}{1}f(\overline{m-1}\, \omega )z+\binom {m}{2}f(\overline{m-2}\, \omega )z^2-\cdots +(-1)^mf(0)z^m,

$\omega >0$ . Analogously to the notation of 185, let $\mathfrak {J}_a^b$ denote the number of zeros of $J(z,\omega )$ in the open interval $a<z<b$ . Then we have the inequalities

\mathfrak {F}_{m\omega }^{\infty } \leq \mathfrak {J}_0^1, \qquad \mathfrak {F}_{-\infty }^0 \leq \mathfrak {J}_1^{\infty }, \qquad \mathfrak {F}_0^{m\omega } \geq \mathfrak {J}_{-\infty }^0.

If we assume that $f(0) \geq 0$ , $f(m\omega ) \geq 0$ , then in addition we can assert that the difference between the two sides is an even number.

?

Problem 5.189

The number of zeros of the real polynomial $f(z)$ of degree $m$ in the interval $m\omega <z<\infty$ is not greater than the number of zeros of the polynomial

\Delta ^mf(0)+\binom {m}{1}\Delta ^{m-1}f(0)z+\binom {m}{2}\Delta ^{m-2}f(0)z^2+\cdots +f(0)z^m

in the interval $0<z<1$ ; here we use the notation

\Delta ^{\nu }f(0) = f(\nu \omega )-\binom {\nu }{1}f(\overline{\nu -1}\, \omega )+\binom {\nu }{2}f(\overline{\nu -2}\, \omega )-\cdots +(-1)^{\nu }f(0), \qquad \nu =0,1,2,\ldots ,m.

?

Problem 5.189.1

The polynomial

f(x) = a_0+a_1x+a_2x^2+a_3x^3+\cdots +a_mx^m

with real coefficients $a_0, a_1, \ldots , a_m$ has not more zeros in the interval $0<x<1$ than the polynomial

S(x) = a_0+a_1x+a_2x^4+a_3x^9+\cdots +a_mx^{m^2}

has in the interval $0<x<1$ , even if in the case of the latter only distinct zeros with odd multiplicity are taken into account.

?

Problem 5.189.2

The entire function

D(x) = a_11^{-x}+a_22^{-x}+a_33^{-x}+\cdots +a_mm^{-x}

has not more zeros in the interval $0<x<\infty$ than the polynomial $f(x)-a_0$ has in the interval $0<x<1$ , even if in the case of the latter only distinct zeros with odd multiplicity are taken into account.

?

Problem 5.189.3

Let $F(x_1, x_2, \ldots , x_m)$ denote a homogeneous polynomial with real coefficients in the variables $x_1, x_2, \ldots , x_m$ and assume that

F(x_1, x_2, \ldots , x_m)>0

where

x_1 \geq 0, \quad x_2 \geq 0, \quad \ldots , \quad x_m \geq 0, \quad x_1+x_2+\cdots +x_m>0.

Then the coefficients of the polynomial

(x_1+x_2+\cdots +x_m)^kF(x_1, x_2, \ldots , x_m)

are non-negative for all sufficiently large positive integers $k$ . (The particular case $m=2$ coincides with a particular case of 187 (3) concerning the third expression considered in 183 when $\mathfrak {F}_0^{\infty }=0$ .)

?