Math Problems Vault

Problem 8.138

Let $s$ be an integer, $t$ a positive integer, $(s,t)=1$ . Show that only prime factors of $t$ divide the denominators of the coefficients of the power series with rational coefficients

(1+z)^{\frac st} = \sum _{n=0}^\infty \frac st\left(\frac st-1\right)\cdots \left(\frac st-n+1\right)\frac{z^n}{n!} .

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Problem 8.139

Let $p$ be a prime factor of $t$ and let $p^\alpha ,p^{\alpha_0},p^{\alpha_1},\ldots ,p^{\alpha_n},\ldots$ respectively be the highest powers of $p$ that divide $t,t_0,t_1,\ldots ,t_n,\ldots$ , where $t_n$ denotes the denominator of the $n$ th coefficient in the power series expansion of $(1+z)^{s/t}$ ; $(s,t)=1$ . Calculate $\lim_{n\to \infty }(\alpha_n/n)$ .

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Remark.

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We term the function $f(z)$ an algebraic function, if it satisfies an equation of the form $P_0(z)[f(z)]^l+P_1(z)[f(z)]^{l-1}+\cdots +P_{l-1}(z)f(z)+P_l(z)=0$ where $P_0(z),P_1(z),\ldots ,P_l(z)$ are polynomials, $P_0(z)\neq 0$ .

The proper perspective for evaluating the particular result contained in 138, 139 is given by the following general theorem due to Eisenstein: ``If the power series with rational coefficients $a_1z+a_2z^2+a_3z^3+\cdots +a_nz^n+\cdots$ represents an algebraic function of $z$ , then there exists an integer $T$ , such that $a_1Tz+a_2T^2z^2+a_3T^3z^3+\cdots +a_nT^nz^n\cdots$ has integral coefficients.'' [153.]

Problem 8.140

Find the smallest integer $T$ with the property that all the coefficients of the power series expansion $(1+Tz)^{s/t}$ are integers.

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Problem 8.141

Prove Eisenstein's theorem in the simplest special case, namely for rational functions.

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Problem 8.142

We say that the power series

\frac{s_0}{t_0}+\frac{s_1}{t_1}z+\frac{s_2}{t_2}z^2+\cdots +\frac{s_n}{t_n}z^n+\cdots

with rational coefficients ( $s_n,t_n$ integers, $t_n\geq 1$ , $(s_n,t_n)=1$ ) satisfies the Eisenstein condition, if there exists an integer $T$ such that $T^n$ is divisible by $t_n$ for $n=1,2,3,\ldots$ . If two power series $f(z)$ and $g(z)$ with rational coefficients satisfy the Eisenstein condition, then the series $f(z)+g(z)$ , $f(z)-g(z)$ , $f(z)g(z)$ , also satisfy the condition, and also $f(z)/g(z)$ (assuming $g(0)\gtreqless 0$ ) and $f(g(z))$ (assuming $g(0)=0$ ).

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Problem 8.143

If the two power series with rational coefficients

a_0+a_1z+a_2z^2+\cdots +a_nz^n+\cdots , \qquad b_0+b_1z+b_2z^2+\cdots +b_nz^n+\cdots

satisfy the Eisenstein condition, then the series

a_0b_0+a_1b_1z+a_2b_2z^2+\cdots +a_nb_nz^n+\cdots

also satisfies the condition.

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Problem 8.144

(1)(2)

Show that the Eisenstein condition is equivalent to the following pair of simultaneous conditions:

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(1)

The denominators $t_1,t_2,t_3,\ldots ,t_n,\ldots$ contain only finitely many different prime numbers.

(2)

$(\log t_n)/n$ is bounded, $n=1,2,3,\ldots$ .

Problem 8.145

Expand the two functions

\frac{1}{1-z}+\frac{1}{2-z}+\frac{1}{4-z}+\cdots +\frac{1}{2^n-z}+\cdots ,

\frac{z}{2-z}+\frac{z^2}{2^4-z^2}+\frac{z^3}{2^9-z^3}+\cdots +\frac{z^n}{2^{n^2}-z^n}+\cdots

in powers of $z$ . Do the power series obtained satisfy the two conditions (1), (2) in 144? Are the functions represented algebraic?

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Problem 8.146

By a hypergeometric series we mean a series of the form

F(\alpha ,\beta ,\gamma ;z) = \sum _{n=0}^\infty \frac{\alpha (\alpha +1)\cdots (\alpha +n-1)\cdot \beta (\beta +1)\cdots (\beta +n-1)}{1\cdot 2\cdots n\cdot \gamma (\gamma +1)\cdots (\gamma +n-1)}z^n .

If $\alpha$ and $\beta$ are rational numbers but not both integers, and if $\gamma$ is a positive integer, then it is possible to choose a positive integer $T$ such that all the coefficients in the hypergeometric series $F(\alpha ,\beta ,\gamma ;Tz)$ become integers. (The function

\frac2\pi \int _0^1 \frac{dx}{\sqrt{(1-x^2)(1-zx^2)}} = \sum _{n=0}^\infty \left(\frac{\tfrac 12(\tfrac 12+1)\cdots (\tfrac 12+n-1)}{1\cdot 2\cdots n}\right)^2 z^n

for example is not an algebraic function [Problem 90 of Part I]: the Eisenstein condition is necessary but not sufficient for a function to be algebraic.)

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Problem 8.147

Assume that $\alpha \neq \gamma$ , $\beta \neq \gamma$ and $\alpha ,\beta ,\gamma$ are not all rational, but all the coefficients in the hypergeometric series $F(\alpha ,\beta ,\gamma ;z)$ are rational. Then $\alpha$ and $\beta$ are the roots of an irreducible quadratic equation with rational coefficients.

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Problem 8.148

In the case discussed in 147, $F(\alpha ,\beta ,\gamma ;z)$ does not represent an algebraic function of $z$ . Prove this from Eisenstein's theorem!

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