MATHEMATICS

I. V. VITEN'KO, A. N. KOSTOVSKII

DIVISION AND FACTORIZATION OF LAURENT SERIES

(Presented by Academician M. A. Lavrent'ev, 20 XI 1964)

Let a Laurent series be given

\[ f(z)=\sum_{m=-\infty}^{\infty} a_m z^m, \tag{1} \]

convergent in the annulus

\[ r<|z|<R. \tag{2} \]

Consider a system of annuli

\[ \{r_{p_i}<|z|<r_{p_{i+1}}\}. \tag{3} \]

Each of the annuli

\[ r_{p_i}<|z|<r_{p_{i+1}} \tag{4} \]

of this system satisfies the conditions: a) \(r\le r_{p_i}<|z|<r_{p_{i+1}}\le R\); b) the function \(f(z)\) has no zeros in (4); c) the function \(f(z)\) has at least one zero on each of the circles \(|z|=r_{p_i}\) and \(|z|=r_{p_{i+1}}\) (where \(r_{p_i}\ne r,\ r_{p_{i+1}}\ne R\)).

To each annulus (4) we assign an integer (number) \(p_i\), equal to

\[ p_i=I(\rho)=\int_{|z|=\rho}\frac{f'(z)}{f(z)}\,dz,\qquad r_{p_i}<\rho<r_{p_{i+1}}. \tag{5} \]

In (¹) this number \(p_i\) is called the principal index of the series (1). Obviously, the difference \(p_i-p_{i-1}=I(r_{p_i}+0)-I(r_{p_i}-0)\) is equal to the number of zeros of \(f(z)\) on the circle \(|z|=r_{p_i}\).

In each of the annuli (4) the quotient \(\varphi(z):f(z)\) is uniquely expanded into a Laurent series

\[ \frac{\varphi(z)}{f(z)}=\sum_{m=-\infty}^{\infty} d_{m,i}z^m, \tag{6} \]

where \(\varphi(z)\) is an arbitrary Laurent series convergent in (2):

\[ \varphi(z)=\sum_{m=-\infty}^{\infty} c_m z^m. \tag{7} \]

Even for polynomials, the numerical determination of the coefficients of the series (6) in the annulus (4) is usually connected with great difficulties.

In this note a method is proposed for dividing Laurent series, by means of which one can numerically determine the coefficients \(d_{m,i}\) in any of the annuli (4) of the system; a new algorithm is also proposed for the factorization of Laurent series (in particular, of power series and polynomials). For the series (1), the system of annuli (3) may consist of a countable or finite number of annuli (4); in particular, it consists of a single annulus (2) if \(f(z)\) has no zeros in the annulus of convergence.

Lemma. The function \(f(z)\), given by the Laurent series (1), in each annulus (4) can be represented in the form

\[ f(z)=z^{p_i}q_{p_i}\exp \sum_{\substack{m=-\infty\\ m\ne0}}^{\infty}\frac{b_{m,i}}{m}z^m, \]

where in (6) we have put \(\varphi(z)\equiv f'(z)\) and

\[ \frac{f'(z)}{f(z)}=\sum_{m=-\infty}^{\infty} b_{m,i}z^{m-1}; \tag{8} \]

\(p_i\) is the number of the annulus (4); \(p_i\) is determined by formulas (5); \(q_{p_i}\) is a certain constant depending on \(p_i\).

Integrating the left- and right-hand sides of (8) from \(z_0\) to \(z\) \(\bigl(r_{p_i}<|z|<r_{p_i+1}\bigr)\), we obtain:

\[ \ln \frac{f(z)}{z^{p_i}}-\sum_{\substack{m=-\infty\\ m\ne 0}}^{\infty}\frac{b_{m,i}}{m}z^m = \ln \frac{f(z_0)}{z_0^{p_i}}-\sum_{\substack{m=-\infty\\ m\ne 0}}^{\infty}\frac{b_{m,i}}{m}z_0^m . \]

It follows immediately from this that the assertion of the lemma holds, with

\[ q_{p_i}=\frac{f(z_0)}{z_0^{p_i}}\exp\left(-\sum_{\substack{m=-\infty\\ m\ne 0}}^{\infty}\frac{b_{m,i}}{m}z_0^m\right). \]

Theorem 1. The function \(f(z)\), given by the series (1), can be represented in the annulus (2) in the form

\[ f(z)=z^{p_i}q_{p_i}f_{p_i}^{-}(z)f_{p_i}^{+}(z), \tag{9} \]

where \(f_{p_i}^{-}(z)\) is a function holomorphic in the circle \(|z|>r\), \(f_{p_i}^{-}(\infty)=1\); \(f_{p_i}^{+}(z)\) is holomorphic in \(|z|<R\), \(f_{p_i}^{+}(0)=1\); \(p_i\) is the principal index of \(f(z)\). For a given \(p_i\), the representation (9) has the uniqueness property.

Let us now consider the following functions:

\[ f_k(z)=\prod_{j=0}^{k-1} f\bigl(\omega_j^{(k)}z\bigr)=\sum_{m=-\infty}^{\infty} a_m^{(k)}z^{km}, \tag{10} \]

\[ F_k(z)=\varphi(z)\prod_{j=1}^{k-1} f\bigl(\omega_j^{(k)}z\bigr)=\sum_{m=-\infty}^{\infty} c_m^{(k)}z^m, \tag{11} \]

where \(\omega_j^{(k)}=\exp\frac{2\pi i}{k}j;\ j=0,1,2,\ldots,k-1;\ k=1,2,\ldots\)

The function (11) can be represented in the form

\[ F_k(z)=(-1)^{p_i(k-1)}q_{p_i}^{\,k}z^{kp_i}\frac{\varphi(z)}{f(z)} \exp\sum_{\substack{m=-\infty\\ m\ne 0}}^{\infty}\frac{b_{km,i}}{m}z^{km}. \tag{12} \]

In (?) it is shown that, in the annulus \(r_{p_i}+\varepsilon<|z|<r_{p_i+1}-\varepsilon\), the sequence of functions

\[ U(z)=\exp\sum_{\substack{m=-\infty\\ m\ne 0}}^{\infty}\frac{b_{km,i}}{m}z^m \]

tends uniformly to one as \(k\to\infty\), however small \(\varepsilon>0\) may be. Therefore (12) may be written as

\[ \frac{\varphi(z)}{f(z)}=(-1)^{p_i(k-1)}\lim_{k\to\infty}\frac{F_k(z)}{q_{p_i}^{\,k}z^{kp_i}}. \tag{13} \]

Putting \(\varphi(z)\equiv f'(z)\) in (13), we obtain the coefficients of the expansion (8), and hence can find the factors

\[ f_{p_i}^{-}(z)=\exp\sum_{m=-\infty}^{-1}\frac{b_{m,i}}{m}z^m,\qquad f_{p_i}^{+}(z)=\exp\sum_{m=1}^{\infty}\frac{b_{mi}}{m}z^{-m}. \tag{14} \]

Introduce the functions

\[ M_k(z)=\prod_{j=1}^{k} f(\tau_j^{(k)}z),\qquad N_k(z)=\prod_{j=1}^{k} f[(\tau_j^{(k)})^{-1}z], \tag{15} \]

where \(\tau_j^{(k)}\) are the roots of the polynomial \(\psi(z)=z^k-z^{k-1}+1\). For the roots of the polynomial \(\psi(z)\) it is not difficult to establish the inequalities

\[ (1/2)^{1/k-1}<|\tau_j^{(k)}|<k^{1/k},\qquad (1/k)^{1/k}<|(\tau_j^{(k)})^{-1}|<2^{1/k-1},\qquad j=1,2,\ldots,k. \]

Using the same methods as in the derivation of (13), we obtain

\[ f_{p_i}^{-}(z)=\lim_{k\to\infty}\frac{N_k(z)}{[q_{p_i}(-z)^{p_i}]^k},\qquad f_{p_i}^{+}(z)=\lim_{k\to\infty}\frac{M_k(z)}{[q_{p_i}(-z)^{p_i}]^k}, \tag{16} \]

where

\[ S_m^{(k)}=\sum_{j=1}^{k}(\tau_j^{(k)})^{-1} \begin{cases} 0, & \text{if } -k+1\le m\le -1,\\ 1, & \text{if } 1\le m\le k+1. \end{cases} \]

The same formula also holds for \(N_k(z)\) (instead of \(S_m^{(k)}\) there is the factor \(S_{-m}^{(k)}\)).

The processes (13) and (16) converge uniformly inside the annulus (4). For a variable \(z\) satisfying the condition \(r_{p_i}<\rho_i<|z|<\rho_{i+1}<r_{p_{i+1}}\), convergence in (13) is characterized by the relation

\[ \left| \frac{F_k(z)}{q_{p_i}^{k}(-z)^{kp_i}(-1)^{p_i}} -\frac{\varphi(z)}{f(z)} \right| = o\left( \left|\frac{r_{p_i}}{\rho_i}\right|^k + \left|\frac{\rho_{i+1}}{r_{p_{i+1}}}\right|^k \right). \]

The same convergence also holds in (16).

For computing the coefficients of the series (10), recurrence formulas were proposed in \((^{3,4})\) for \(k\ge2\). Analogous formulas for \(k\ge2\) can also be proposed for determining the coefficients of the series (11). These formulas will be simplest when \(k\) takes the discrete values \(1,2,4,8,16,\ldots\),

\[ f_{2k}(z)=f_k(z)f_k[(-1)^{1/k}z],\qquad F_{2k}(z)=F_k(z)f_k[(-1)^{1/k}z], \]

where

\[ a_m^{(2k)}=(-1)^m\left[(a_m^{(k)})^2+2\sum_{j=1}^{\infty}(-1)^j a_{m-j}^{(k)}a_{m+j}^{(k)}\right], \]

\[ c_m^{(2k)}=\sum_{j=-\infty}^{\infty}(-1)^j a_j^{(k)}c_{m-kj}^{(k)}, \tag{17} \]

\[ a_m^{(1)}=a_m,\qquad c_m^{(1)}=c_m;\qquad m=0,\pm1,\pm2,\ldots;\qquad k=1,2,\ldots . \]

From (6), (13), and (17) we can write

\[ |d_{m,i}-c_{m+kp_i}^{(k)}/a_{p_i}^{(k)}|=o(\rho^k),\qquad r_{p_i}/r_{p_{i+1}}<\rho<1. \]

If \(r_{p_i}\ne r\) and \(r_{p_{i+1}}\ne R\), then \(o(\rho^k)\) in the last equality may be replaced by \(o(|r_{p_i}/r_{p_{i+1}}|^k)\).

The coefficients of the series (11) can also be computed by the formulas

\[ F_k^{(m)}(z)=\frac{\varphi(z)}{z^m}\prod_{j=1}^{k-1} f(\omega_j^{(k)}z),\qquad Q_k^{(m)}(z)=\sum_{j=0}^{k-1} F_k^{(m)}(\omega_j^{(k)}z), \]

\[ Q_{2k}^{(m)}(z)=Q_k^{(m)}(z)f_k[(-1)^{1/k}z]+Q_k^{(m)}[(-1)^{1/k}z]f_k(z), \]

where

\[ c_{m+kp}^{(2k)} = (-1)^p \left[ a_p^{(k)}c_{m+kp}^{(k)} + \sum_{j=1}^{\infty}(-1)^j \left( a_{p-j}^{(k)}c_{m+(p+j)k}^{(k)} + a_{p+j}^{(k)}c_{m+(p-j)k}^{(k)} \right) \right]. \tag{18} \]

By formulas (18) it is convenient to compute the coefficients for fixed \(m\) and variable \(p\).

Formulas (17) and (18) can also be used for computing the coefficients of the factors \(f_{p_i}^{-}, f_{p_i}^{+}\) in (9). For this it is necessary to put \(c_m^{(1)}=(m+1)a_{m+1}\) in (7) and (11) to compute, by the method described above, the coefficients of the logarithmic derivative (8) in the annulus (4), using formulas (13) and (17). Then, by the formulas

\[ t_{m,i}^{+}=\frac{1}{m}\sum_{j=1}^{m} t_{m-j,i}^{+} b_{j,i},\qquad t_{0,i}^{+}=1, \]

\[ t_{m,i}^{-}=-\frac{1}{m}\sum_{j=1}^{m} t_{m-j,i}^{-} b_{j,i},\qquad t_{0,i}^{-}=1, \tag{19} \]

\[ m=0,1,2,\ldots, \]

one can find the expansion (14)

\[ f_{p_i}^{+}(z)=\exp \sum_{m=1}^{\infty}\frac{b_{m,i}}{m}z^m =\sum_{m=1}^{\infty} t_{m,i}^{+}z^m, \]

\[ f_{p_i}^{-}(z)=\exp \sum_{m=-\infty}^{-1}\frac{b_{m,i}}{m}z^m =\sum_{m=-\infty}^{-1} t_{m,i}^{-}z^m. \]

In formula (18) it is important that \(p\) be a principal index. Necessary and sufficient conditions for a principal index were given in \((^{1,5})\).

Suppose that in an arbitrarily chosen annulus

\[ r<r_1<|z|<R_1<R \tag{20} \]

the function \(f(z)\) has zeros \(z_1,z_2,\ldots,z_n\).

Theorem 2. The function \(f(z)\) is represented in the annulus (2) in the form

\[ f(z)=z^{p_i}q_{p_i}\prod_{i=1}^{n}\left(1-\frac{z}{z_i}\right)f_{p_i+n}^{+}f_{p_i}^{-} = z^{p_i+n}q_{p_i+n}\prod_{j=1}^{n}\left(1-\frac{z_i}{z}\right)f_{p_i}^{-}f_{p_i+n}^{+}, \tag{21} \]

where \(p_i=I(r_1+0)\), \(p_{i+n}=I(R_1-0)=p_k\). The polynomial in formula (21) with roots from the annulus (20) can be determined by the formulas

\[ \prod_{j=1}^{n}\left(1-\frac{z}{z_i}\right) = f_{p_i}^{+}(z):f_{p_i+n}^{+}(z), \qquad \prod_{j=1}^{n}\left(1-\frac{z_i}{z}\right) = f_{p_i+n}^{-}(z):f_{p_i}^{-}(z), \]

or one may use Newton’s formulas, computing the power sums of the zeros of the polynomial by the formulas

\[ S_{-m}=\sum_{j=1}^{n} z_j^{-m}=b_{m,k}-b_{m,i},\qquad m=\pm1,\pm2,\ldots . \]

In \((^{2})\) it is shown that the principal indices \(p_i\) can be computed in the process of transforming the given series by formulas (10), rather than using formulas (5) for this purpose.

Lviv State University
named after Iv. Franko

Received
20 VIII 1964

REFERENCES

\({}^{1}\) I. V. Viten’ko, A. N. Kostovskii, DAN, 155, No. 4 (1964).
\({}^{2}\) I. V. Viten’ko, Dop. AN UkrSSR, 9, No. 1 (1963).
\({}^{3}\) I. V. Viten’ko, O. M. Kostovs’kyi, Theoretical and Applied Mathematics, Publishing House of Lviv University, issue 2, 31 (1963).
\({}^{4}\) A. N. Kostovskii, Journal of Computational Mathematics and Mathematical Physics, 1, No. 2, 345 (1961).
\({}^{5}\) A. N. Kostovskii, DAN, 147, No. 2 (1962).