Math Problems Vault

Problem 5.28

The polynomials $P(x)$ and $P(\alpha x)$ have the same number of changes of sign if $\alpha$ is positive.

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Remark.

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By the changes of sign and the indices of change of the polynomial $a_0+a_1x+a_2x^2+\cdots +a_nx^n$ or the power series $a_0+a_1x+a_2x^2+a_3x^3+\cdots$ we mean the changes of sign and indices of change of the finite or the infinite sequence of coefficients $a_0, a_1, a_2, \ldots , a_n$ or $a_0, a_1, a_2, a_3, \ldots$ respectively.

Problem 5.29

Denote the number of changes of sign of the polynomials

\begin{aligned} P(x) & = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n, \\ P(-x) & = a_0 - a_1 x + a_2 x^2 - \cdots + (-1)^n a_n x^n \end{aligned}

by $C^+$ and $C^-$ respectively. We then have

C^+ + C^- \leq n.

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Problem 5.30

Let $\alpha > 0$ . Passing from the polynomial

a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n

to the polynomial

\begin{aligned} (\alpha - x)(a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n) & \\ = \alpha a_0 + (\alpha a_1 - a_0) x + (\alpha a_2 - a_1) x^2 + \cdots & - a_n x^{n+1} \end{aligned}

the number of changes of sign increases; in fact it increases by an odd number.

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Remark.

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For the case $\alpha = 1$ cf. 15.

Problem 5.31

Let $\alpha > 0$ . Passing from the power series

a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

to the power series

(\alpha - x)(a_0 + a_1 x + a_2 x^2 + \cdots ) = \alpha a_0 + (\alpha a_1 - a_0) x + (\alpha a_2 - a_1) x^2 + \cdots

the number of changes of sign does not decrease. In fact it increases if the original series $a_0 + a_1 x + a_2 x^2 + \cdots$ converges for $x = \alpha$ .

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Problem 5.32

Let $\alpha > 0$ . Passing from the power series

a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

to the power series

(\alpha + x)(a_0 + a_1 x + a_2 x^2 + \cdots ) = \alpha a_0 + (\alpha a_1 + a_0) x + (\alpha a_2 + a_1) x^2 + \cdots

the number of changes of sign does not increase.

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Problem 5.33

Passing from the power series

a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

to the power series

\frac{a_0 + a_1 x + a_2 x^2 + \cdots }{1-x} = a_0 + (a_0+a_1) x + (a_0+a_1+a_2) x^2 + \cdots

the number of changes of sign does not increase.

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Problem 5.34

Let $\alpha > 0$ . Passing from the power series

a_0 + \frac{a_1}{1!} x + \frac{a_2}{2!} x^2 + \frac{a_3}{3!} x^3 + \cdots

to the power series

e^{\alpha x} \left(a_0 + \frac{a_1}{1!} x + \frac{a_2}{2!} x^2 + \cdots \right) = \sum _{n=0}^{\infty } \frac{a_0 \alpha ^n + \binom {n}{1} a_1 \alpha ^{n-1} + \cdots + a_n}{n!} x^n

the number of changes of sign does not increase.

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Remark.

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Laguerre, op. cit. 1, p. 22. The proof has gaps.

Problem 5.35

Let $p_1, p_2, \ldots , p_n$ be positive numbers. Set

\begin{aligned} a_0 + \frac{a_1 x}{p_1-x} + \frac{a_2 x^2}{(p_1-x)(p_2-x)} + \cdots + \frac{a_n x^n}{(p_1-x)(p_2-x)\cdots (p_n-x)} & \\ = A_0 + A_1 x + A_2 x^2 + \cdots & \end{aligned}

(the power series expansion is convergent for sufficiently small values of $x$ ). The number of changes of sign of the finite sequence $a_0, a_1, a_2, \ldots , a_n$ is not less than the number of changes of sign of the infinite sequence $A_0, A_1, A_2, \ldots$ .

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Remark.

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Mathematical induction, 31.

Problem 5.36

Let $Z$ be the number of positive zeros of the polynomial $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$ and $C$ the number of changes of sign of the sequence of its coefficients. We then have

C - Z \geq 0.

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Remark.

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Problem 5.37

Continuation of 36. $C-Z$ is an even number.

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Problem 5.38

Let the radius of convergence of the power series $a_0 + a_1 x + a_2 x^2 + \cdots$ be $\rho$ , let the number of its zeros in the interval $0 < x < \rho$ be $Z$ and let the number of changes of sign in the sequence of its coefficients be $C$ . Then

Z \leq C.

Hence, in particular: If $C$ is finite then $Z$ is also finite.

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Remark.

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Besides 31, the theory of functions of a complex variable must be used.

Problem 5.39

The power series

2 - \frac{x}{1\cdot 2} - \frac{x^2}{2\cdot 3} - \frac{x^3}{3\cdot 4} - \cdots

has no zeros in its circle of convergence. (The sequence of its coefficients has one change of sign---37 cannot be extended to power series without some essential additional condition.)

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Problem 5.40

If $\rho = \infty$ or if $\rho$ is finite and $a_0+a_1 \rho +a_2 \rho^2+\cdots$ is divergent, then, provided $C$ is finite, $C-Z$ is a non-negative even number.

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Problem 5.41

Let the minimum of the real numbers $\xi_1, \xi_2, \ldots , \xi_n$ be denoted by $\xi_{\alpha }$ and their maximum by $\xi_{\omega }$ . The number of zeros of the polynomial

a_0+a_1(x-\xi _1)+a_2(x-\xi _1)(x-\xi _2)+\cdots +a_n(x-\xi _1)(x-\xi _2)\ldots (x-\xi _n)

in the interval $\xi_{\omega } < x < \infty$ is equal to, or less by an even number than, the number of changes of sign of the sequence

a_0, a_1, a_2, \ldots , a_n,

and the number of zeros in the interval $-\infty < x < \xi_{\alpha }$ is equal to, or less by an even number than, the number of changes of sign of the sequence

a_0, -a_1, a_2, -a_3, \ldots , (-1)^n a_n.

(For $\xi_1 = \xi_2 = \cdots = \xi_n$ equivalent to 36, 37.)

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Remark.

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