Math Problems Vault

Problem 7.17

Let the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots$ represent a rational function whose denominator is of precise degree $q$ and whose numerator is of precise degree $p-1$ . Assume that the numerator and the denominator have no common divisor. Setting $d = \max (0, p-q)$ we have

A_{d}^{(q+1)} = A_{d+1}^{(q+1)} = A_{d+2}^{(q+1)} = \cdots = 0.

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Remark.

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In Problems 17--34 we consider the Hankel (or recurrent) determinants

\begin{vmatrix} a_{n} & a_{n+1} & a_{n+2} & \ldots & a_{n+r-1} \\ a_{n+1} & a_{n+2} & a_{n+3} & \ldots & a_{n+r} \\ a_{n+2} & a_{n+3} & a_{n+4} & \ldots & a_{n+r+1} \\ \multicolumn {5}{c}{\dotfill } \\ a_{n+r-1} & a_{n+r} & a_{n+r+1} & \ldots & a_{n+2r-2} \end{vmatrix} = A_{n}^{(r)},

formed from the coefficients of the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ ; $a_{n}$ is the leading entry, $r$ is the number of rows.

Problem 7.18

Let $d, q$ be non-negative integers. If we have

A_{d}^{(q)} \neq 0, \quad A_{d+1}^{(q)} \neq 0, \quad A_{d+2}^{(q)} \neq 0, \quad A_{d+3}^{(q)} \neq 0, \quad \ldots

A_{d}^{(q+1)} = 0, \quad A_{d+1}^{(q+1)} = 0, \quad A_{d+2}^{(q+1)} = 0, \quad \ldots

then the power series may be represented as the quotient of two polynomials, where the denominator is of degree $q$ and the numerator is of degree $\leq q+d-1$ .

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Remark.

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Investigate the dependence of the linear forms

L_{n}(x) = a_{n} x_{0} + a_{n+1} x_{1} + a_{n+2} x_{2} + \cdots + a_{n+q} x_{q}.

Problem 7.19

A_{n}^{(r)} A_{n+2}^{(r)} - A_{n}^{(r+1)} A_{n+2}^{(r-1)} = (A_{n+1}^{(r)})^{2}.

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Problem 7.20

If we have

A_{m}^{(q+1)} = A_{m+1}^{(q+1)} = A_{m+2}^{(q+1)} = A_{m+3}^{(q+1)} = \cdots = A_{m+t-1}^{(q+1)} = 0,

then the $t$ determinants $A_{m+1}^{(q)}, A_{m+2}^{(q)}, A_{m+3}^{(q)}, \ldots , A_{m+t}^{(q)}$ are either all $=0$ , or all $\neq 0$ .

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Problem 7.21

The triangular array

\begin{array}{ccccccccc} A_{0}^{(1)} & A_{1}^{(1)} & A_{2}^{(1)} & A_{3}^{(1)} & A_{4}^{(1)} & * & * & * & * \\ ]0.5ex] & A_{0}^{(2)} & A_{1}^{(2)} & A_{2}^{(2)} & * & * & * & * & * \\[0.5ex]& & A_{0}^{(3)} & * & * & * & A_{n+2}^{(r-1)} & * & * \\ ]0.5ex] & & & * & * & A_{n}^{(r)} & A_{n+1}^{(r)} & A_{n+2}^{(r)} & * \\[0.5ex]& & & & * & * & A_{n}^{(r+1)} & * & * \end{array}

is arranged so that a right angle opening upwards and bisected by the vertical contains only minors of the determinant standing at the vertex of the angle.

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Problem 7.22

(1)(2)(3)

In the array of Problem 21, $A_{n}^{(r)}$ and $A_{n+1}^{(r)}$ constitute a horizontal pair, $A_{n+1}^{(r)}$ and $A_{n}^{(r+1)}$ constitute a vertical pair, $A_{n}^{(r)}$ and $A_{n+1}^{(r+1)}$ constitute a diagonal pair, $A_{n+1}^{(r)}$ and $A_{n-1}^{(r+1)}$ also form a diagonal pair, cross-wise to $A_{n}^{(r)}$ and $A_{n+1}^{(r+1)}$ .

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(1)

If a diagonal pair vanishes, then the diagonal pair cross-wise to it also vanishes.

(2)

If a horizontal pair vanishes, then either the horizontal pair lying above it or the one lying below it also vanishes.

(3)

If a vertical pair vanishes, then either the neighbouring vertical pair on the left or the one on the right also vanishes.

Theorems (1), (2), (3) also hold on the slanting boundary line of the array of Problem 21.

Problem 7.23

If of the infinitely many determinants

A_{0}^{(k+1)}, \quad A_{1}^{(k+1)}, \quad A_{2}^{(k+1)}, \quad \ldots , \quad A_{n}^{(k+1)}, \quad \ldots

only finitely many are different from $0$ , then the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ represents a rational function whose denominator is of degree not greater than $k$ . [Problems 20, 18.]

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Problem 7.24

If of the infinitely many determinants

A_{0}^{(1)}, \quad A_{0}^{(2)}, \quad A_{0}^{(3)}, \quad \ldots , \quad A_{0}^{(n)}, \quad \ldots

only finitely many are different from $0$ , then the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ represents a rational function.

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Problem 7.25

If of the infinitely many determinants

A_{0}^{(1)}, \quad A_{1}^{(2)}, \quad A_{2}^{(3)}, \quad A_{3}^{(4)}, \quad \ldots , \quad A_{n}^{(n+1)}, \quad \ldots

A_{1}^{(1)}, \quad A_{2}^{(2)}, \quad A_{3}^{(3)}, \quad \ldots , \quad A_{n}^{(n)}, \quad \ldots

only finitely many are different from $0$ , then the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ represents a rational function.

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Problem 7.26

Show that the sufficient criteria given in Problems 23, 24, 25 for the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ to represent a rational function are also necessary.

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Problem 7.27

The rank of the matrix $\mathfrak {H}$ is finite if and only if the power series $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ represents a rational function.

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Remark.

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An infinite matrix

\begin{pmatrix} a_{00} & a_{01} & a_{02} & \ldots \\ a_{10} & a_{11} & a_{12} & \ldots \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}

is said to be of finite rank $r$ if all determinants of order $r+1$ contained in it vanish but not all those of order $r$ .

In the infinite Hankel matrix $\mathfrak {H}$

\mathfrak {H} = \begin{pmatrix} a_{0} & a_{1} & a_{2} & \ldots \\ a_{1} & a_{2} & a_{3} & \ldots \\ a_{2} & a_{3} & a_{4} & \ldots \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}

( $a_{\lambda \mu } = a_{\lambda +\mu }$ ) we term the rank just defined also the gross rank of $\mathfrak {H}$ . By striking out the first $k$ columns (or rows) we obtain from $\mathfrak {H}$ the matrix $\mathfrak {H}_{k}$

\mathfrak {H}_{k} = \begin{pmatrix} a_{k} & a_{k+1} & a_{k+2} & \ldots \\ a_{k+1} & a_{k+2} & a_{k+3} & \ldots \\ a_{k+2} & a_{k+3} & a_{k+4} & \ldots \\ \multicolumn {4}{c}{\dotfill } \end{pmatrix}

whose rank we shall denote by $r_{k}$ . We have $\mathfrak {H}_{0} = \mathfrak {H}$ , $r_{0} = r$ , $r_{0} \geq r_{1} \geq r_{2} \geq \cdots$ . The minimum of the sequence of numbers $r_{0}, r_{1}, r_{2}, \ldots$ , which is necessarily attained after finitely many steps, we term the net rank of $\mathfrak {H}$ .

Problem 7.28

The gross rank of $\mathfrak {H}$ is equal to the order of the last non-vanishing determinant in the infinite sequence

A_{0}^{(1)}, \quad A_{0}^{(2)}, \quad A_{0}^{(3)}, \quad A_{0}^{(4)}, \quad \ldots .

If this is $A_{0}^{(p)}$ then the net rank is equal to the order of the first non-vanishing determinant in the finite sequence

A_{1}^{(p)}, \quad A_{2}^{(p-1)}, \quad A_{3}^{(p-2)}, \quad \ldots , \quad A_{p}^{(1)}.

(The net rank $=0$ if these $p$ determinants all vanish.)

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Problem 7.29

The net rank of $\mathfrak {H}$ (assumed finite) is equal to the degree of the denominator of the rational function represented by $a_{0} + a_{1} z + a_{2} z^{2} + \cdots + a_{n} z^{n} + \cdots$ . The rational function is a proper fraction if and only if the gross rank is equal to the net rank. If the gross rank is greater than the net rank, then the gross rank exceeds the degree of the numerator by one. (The numerator and the denominator of the rational function are assumed to have no common divisors; the degree here is the precise degree.)

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Problem 7.30

The power series

a_{0} + \frac{a_{1} z}{1!} + \frac{a_{2} z^{2}}{2!} + \cdots + \frac{a_{n} z^{n}}{n!} + \cdots

satisfies a linear homogeneous differential equation with constant coefficients if and only if the determinants

A_{0}^{(1)}, \quad A_{0}^{(2)}, \quad A_{0}^{(3)}, \quad \ldots , \quad A_{0}^{(n)}, \quad \ldots ,

apart from a finite number, all vanish.

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Problem 7.31

Let $Q_{n}(z)$ be a polynomial of degree $n$ , $Q_{n}(0) = 1$ , $n = 0, 1, 2, \ldots$ . For brevity set

a_{0} + a_{1} z + a_{2} z^{2} + \cdots = f(z), \qquad Q_{k}(z) f(z) = \square _{k} a_{0} + \square _{k} a_{1} z + \square _{k} a_{2} z^{2} + \cdots ,

Q_{k}(z) Q_{l}(z) f(z) = \square _{k} \square _{l} a_{0} + \square _{k} \square _{l} a_{1} z + \square _{k} \square _{l} a_{2} z^{2} + \cdots

( $\square_{k} \square_{l} a_{n}$ is a homogeneous linear expression in $a_{n}, a_{n-1}, a_{n-2}, \ldots , a_{n-k-l}$ ).

We have

A_{0}^{(n+1)} = \begin{vmatrix} a_{0} & \square _{1} a_{1} & \ldots & \square _{n} a_{n} \\ \square _{1} a_{1} & \square _{1} \square _{1} a_{2} & \ldots & \square _{1} \square _{n} a_{n+1} \\ \multicolumn {4}{c}{\dotfill } \\ \square _{n} a_{n} & \square _{n} \square _{1} a_{n+1} & \ldots & \square _{n} \square _{n} a_{2n} \end{vmatrix}.

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Problem 7.32

Let $a_{0}/z + a_{1}/z^{2} + a_{2}/z^{3} + \cdots$ be the power series expansion of a rational function whose denominator (with no common divisors with the numerator) is $z^{q} - c_{1} z^{q-1} - c_{2} z^{q-2} - \cdots - c_{q}$ . Between the matrices

\mathfrak {A}_{m} = \begin{pmatrix} a_{m} & a_{m+1} & \ldots & a_{m+q-1} \\ a_{m+1} & a_{m+2} & \ldots & a_{m+q} \\ \multicolumn {4}{c}{\dotfill } \\ a_{m+q-1} & a_{m+q} & \ldots & a_{m+2q-2} \end{pmatrix}, \qquad \mathfrak {C} = \begin{pmatrix} 0 & 0 & 0 & 0 & \ldots & 0 & c_{q} \\ 1 & 0 & 0 & 0 & \ldots & 0 & c_{q-1} \\ 0 & 1 & 0 & 0 & \ldots & 0 & c_{q-2} \\ 0 & 0 & 1 & 0 & \ldots & 0 & c_{q-3} \\ \multicolumn {7}{c}{\dotfill } \\ 0 & 0 & 0 & 0 & \ldots & 1 & c_{1} \end{pmatrix}

there exists the relation

\mathfrak {A}_{m} = \mathfrak {A}_{0} \mathfrak {C}^{m}, \qquad m = 0, 1, 2, \ldots .

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Problem 7.33

In a system of finitely many power series there are $r$ linearly independent series if the rank of the corresponding matrix is $r$ , and any power series of the system is linearly dependent on those $r$ power series.

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Remark.

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Let the rank of the infinite matrix $\mathfrak {M}$ with finitely many ( $=m$ ) rows

\mathfrak {M} = \begin{pmatrix} a_{10} & a_{11} & a_{12} & \ldots & a_{1n} & \ldots \\ a_{20} & a_{21} & a_{22} & \ldots & a_{2n} & \ldots \\ \multicolumn {6}{c}{\dotfill } \\ a_{m0} & a_{m1} & a_{m2} & \ldots & a_{mn} & \ldots \end{pmatrix}

be denoted by $r_{0}$ and let the rank of the matrix that is formed from $\mathfrak {M}$ by deleting the first $n$ columns (in whose left upper corner we have the element $a_{1n}$ ) be $r_{n}$ . We have $r_{0} \geq r_{1} \geq r_{2} \geq \cdots$ , and we term $\lim_{n \rightarrow \infty } r_{n}$ the net rank of $\mathfrak {M}$ .

The matrix $\mathfrak {M}$ corresponds to the system of $m$ power series

f_{1}(z) = a_{10} + a_{11} z + a_{12} z^{2} + \cdots + a_{1n} z^{n} + \cdots

f_{2}(z) = a_{20} + a_{21} z + a_{22} z^{2} + \cdots + a_{2n} z^{n} + \cdots

f_{m}(z) = a_{m0} + a_{m1} z + a_{m2} z^{2} + \cdots + a_{mn} z^{n} + \cdots .

We say that these are linearly dependent if there exist constants $c_{1}, c_{2}, \ldots , c_{m}$ satisfying $\left|c_{1}\right| + \left|c_{2}\right| + \cdots + \left|c_{m}\right| > 0$ , such that we have

c_{1} f_{1}(z) + c_{2} f_{2}(z) + \cdots + c_{m} f_{m}(z) = 0

identically in $z$ . They are said to be quasilinearly dependent if there exist constants $c_{1}, c_{2}, \ldots , c_{m}$ satisfying $\left|c_{1}\right| + \left|c_{2}\right| + \cdots + \left|c_{m}\right| > 0$ , such that we have

c_{1} f_{1}(z) + c_{2} f_{2}(z) + \cdots + c_{m} f_{m}(z) = P(z)

identically in $z$ , where $P(z)$ is a polynomial.

The power series $f(z), zf(z), z^{2} f(z), \ldots , z^{m} f(z)$ are quasilinearly dependent for sufficiently large $m$'' and $f(z)$ represents a rational function'': these two statements are equivalent. We say that $f(z)$ represents an algebraic function if the $(m+1)^{2}$ power series $z^{\mu } [f(z)]^{\nu }$ ( $\mu , \nu = 0, 1, 2, \ldots , m$ ) are linearly dependent for sufficiently large $m$ . We say that $f(z)$ satisfies an algebraic differential equation of the $r$ th order, if the $(m+1)^{r+2}$ power series

z^{\mu } [f(z)]^{\nu } [f'(z)]^{\nu _{1}} [f''(z)]^{\nu _{2}} \ldots [f^{(r)}(z)]^{\nu _{r}} \qquad (\mu , \nu , \nu _{1}, \nu _{2}, \ldots , \nu _{r} = 0, 1, 2, \ldots , m)

are linearly dependent for sufficiently large $m$ .

Problem 7.34

In a system of finitely many power series there are $r$ quasilinearly independent series if the net rank of the corresponding matrix is $r$ , and any power series of the system is quasilinearly dependent on those $r$ power series.

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