<<

Funkcialaj Ekvacioj, 6 (1964), 37?46

Reciprocals of Inverse

By W. A. HARRIS, Jr.* and H. L. TURRITTIN

(University of Minnesota)

1. Introduction

The $¥mathrm{d}¥mathrm{e}¥mathrm{t}¥mathrm{a}¥dot{¥mathrm{i}}¥mathrm{l}¥mathrm{s}$ relating to the manipulation of convergent , asymptotic series and convergent $¥dot{¥mathrm{i}}¥mathrm{n}¥mathrm{v}¥mathrm{e}¥mathrm{r}¥mathrm{s}¥mathrm{e}$ factorial series run quite parallel up to a certain stage and then gaps appear in the theory relating to factorial series. For instance let the function $G¥mathrm{b}.¥mathrm{e}$ defined by a power series

(1) $G(t)=¥sum_{v=0}^{¥infty}g_{v}t^{v}$ absolutely convergent in a disk $|t|¥leq r$ , $r>1$ . Let the function $F$ be represented either by a convergent or asymptotic series of the form

(2) $F(z)=¥sum_{v=1}^{¥infty}¥beta_{v}z^{-v}$ , then it is known that $H(z)=G(F(z))-g_{0}$ can also be represented by a convergent or asymptotic series of type (2). However, if $F$ is defined by the factorial series

(3) $F(z)=¥sum_{s=0}^{¥infty}¥overline{z(z+1}^{¥frac{!}{)}}s.a_{s}.¥cdot(z¥overline{+s)}$

convergent in a half-plane $ Re(z)>¥lambda$, it appears not to have been shown that $H(z)$ has a factorial series representation of type (3) convergent in some right half-plane. In this paper we shall fill some of these gaps and show in particular that $H(z)$ does possess a convergent factorial series representation. To obtain our results the concept of termwise dominance of a factorial series is introduced. This concept, so useful in the treatment of power series, does not appear to have been applied (to the best of our knowledge) to factorial series. This termwise dominance is also used to prove an implicit function theorem for factorial series. Background information may be found in the treatises by Milne- Thompson [4] and Norlund [6].

$*$ Supported in part by the National Science Foundation under Grant $¥mathrm{G}$ -18918. 38 W. A. HARRIS, Jr. & H. L. TURRITTIN

The results of this paper are of interest in their own right and should be particularly useful in the theory of difference equations, see Harris [3] and [4], where factorial series play a role analogous to power series in the theory of differential equations.

2. Termwise dominance

Nielsen [5] has proved the following

Theorem 1. If two functions $F$ and $F_{2}$ are represented by factorial series

$¥lambda$ of type (3), with coefficients $a_{s}$ and abscissa of convergence and coefficients

$b_{s}$ ¥ and abscissa convergence $ lambda_{2}$ respectively; then the product $F(z)F_{2}(z)$ has $a$ factorial series representation of type (3) convergent in the half-plane $Re(z)>$

$¥max¥{0, ¥lambda,¥lambda_{2}¥}$ $ts$ with coefficien $c_{s}$ , where

$c_{n+1}=¥sum_{s=0}^{¥infty}(n-1)!b_{n-s}c_{n-s,s}/(n+1)1$ and

$C_{r,s}=¥sum_{p=0}^{s}$ $¥left(¥begin{array}{l}¥mathrm{r}-p¥¥p¥end{array}¥right)$ $a_{s-p}$ and $¥left(¥begin{array}{l}¥mathrm{r}¥¥p¥end{array}¥right)=¥Gamma(¥mathrm{r}+1)/¥Gamma(p+1)¥Gamma(r-p+1)$ .

Note that, if the coefficients $a_{s}$ and $b_{s}$ were all positive, the coefficients

$c_{s}$ of the product would also be positive.

Theorem 1 implies that $F(z)$ and all its powers $F^{2}(t),F^{¥theta}(t)$ , $¥cdots$ can be represented as factorial series

(4) $ F^{¥mathrm{V}}(z)=¥sum_{s=0}^{¥infty}¥frac{s^{1}b_{vs}}{z(z+1)(z+s)}¥cdots$ ; $¥nu=1,2$ , $¥cdots$ ; convergent in a common half-plane $Re(z)>maz¥{0, ¥lambda¥}$ . Furthermore, if the coefficients $a_{s}=b_{1S}$ are positive, then all the $b_{vs}$ are positive. Let us formally replace $t^{v}$ in (1) by series (4) and rearrange the order of the terms to obtain a new formal series

(5) $ G(F(z))=g_{0}+¥sum_{s=0}^{¥infty}¥frac{s^{1}ds}{z(z+1)(z+s)}¥cdots$ where

(6) $d_{s}=¥sum_{v=1}^{¥infty}g_{¥gamma}b_{vs}^{*}$.

We would like to prove that series (5) converges by the technique of dominant series.

Consider series (3) where $¥lambda$ is the abscissa of convergence. Let a $=$ $¥max¥{0, ¥lambda¥}$ and select an $¥epsilon>0$ and set $w=$ a $+¥epsilon$ . By hypothesis series (3) converges when $z$ $=w$ and so the individual terms in the series must

* ¥ ¥ This is really a finite sum since $b_{vs}=0$ for $ nu>s+ mathrm{t}$ . Reciprocals of Inverse Factorial Series 39 approach zero as $ s¥rightarrow¥infty$ . This means there is a largest term; let its absolute value be equal to $M/w$ . Then for all $s$

$|a_{s}|¥leq M¥left(¥begin{array}{l}w+s¥¥s¥end{array}¥right)$ .

The series $L(z)$ which is to dominate the series for $F(z)$ termwise now takes the form

$ L(z)=^{M}¥overline{z-}w-1--=¥frac{M}{z}+¥frac{M(w+1)}{z(z+1)}+¥frac{M(w+1)(w+2)}{z(z+1)(z+2)}+¥cdots$

$=¥sum_{s=0}^{¥infty}¥frac{s!d_{s}}{z(z+1)(z+s)}¥cdots$ ’ where

$d_{s}=M¥left(¥begin{array}{l}w+s¥¥s¥end{array}¥right)$ , and this series converges absolutely if $Re(z)>w+1$ . There are two important features; the .coefficient dominance

$d_{Jl}¥geq|a_{n}|$

$¥mathrm{i}$ $¥mathrm{e}.$ $|z|¥backslash $ and the analiticity of $L_{¥backslash }^{(}z$ ) in the neighborhood of infinity; . , for , $w+1$ . It follows at once that all the series

$ L^{¥gamma}(z)=¥frac{M^{v}}{(z-w-1)^{¥gamma}}=¥sum_{s=0}^{¥infty}¥frac{s^{1}d_{¥mathrm{V}S}}{z(z+1^{¥backslash /}¥backslash z+s)},¥cdots$

have positive coefficients $d_{vs}>|b_{vs}|$ and converge absolutely if $Re(z)>w+1$ .

Let $ Re(z)=¥sigma$ and restrict $¥sigma$ so that $¥sigma>w+1+M/r$; then

$L^{¥gamma}(¥sigma)=(¥frac{M}{¥sigma-w-1})^{v}=¥sum_{s=0}^{¥infty}¥frac{s!d_{¥mathrm{v}s}}{¥sigma(¥sigma+1)(¥sigma+s)}¥cdots<¥mathrm{r}^{¥gamma}$ .

Since by hypothesis $¥sum_{v=0}^{¥infty}g_{v}r^{v}$ converges, the double series of positive terms

$¥sum_{¥mathrm{v}=0}^{¥infty}¥sum_{s=0}^{¥infty}¥frac{s!|g_{v}|d_{v}}{¥sigma(¥sigma+1)(}¥cdots¥sigma s¥overline{+s)}$

converges and dominates term by term the double series

$¥gamma¥sum_{=0}^{¥infty}¥sum_{s=0}^{¥infty}¥frac{s!g_{v}b_{¥gamma}}{z(z+1)(}¥cdots s¥overline{z+s)}$ .

Any desired rearrangement of terms in this latter series is therefore admissible and convergence is preserved. Hence series (5) converges if $Re(z)>w+1+M/r$ . This proves Theorem 2. If $G(t)$ is analytic in a neighborhood of the origin and $F(z)$ can be represented by an inverse factorial series convergent in a right half- plane, then $¥{G(F(z))-G(0)¥}$ also has a factorial series representation in some right half-plane. 40 W. A. HARRIS, Jr. & H. L. TURRITTIN

We have as a

Corollary. If $H(z)=¥sum_{¥mathrm{v}=0}^{¥infty}gz^{-¥mathrm{V}}$ is analytic in the neighborhood of $ z=¥infty$ , then $H(z)-H(¥infty)$ can be represented by an inverse factorial series convergent in some right half-plane.

Proof. Take $G(t)=H(1/t)$ and $F(z,¥$=¥frac{1}{z}$ .

3. The inverse of transformation $(z, z+m)$

Again consider the function $F^{/}¥backslash z$) and its factorial series representation

(4) with an abscissa of convergence $¥lambda$ and let $m$ be a given positive . It is known, see [1] or [2], that a certain transformation $(z, z+m)$ will yield a new representation of function $F(z)$ , namely

$ F(z)=¥sum_{s=0}^{¥infty}¥frac{s^{1}f_{ms}}{(z+m)(z+m+1)(z+m+s)}¥cdots$ (7) ’ where the constants

¥ $¥left(¥begin{array}{l}m+¥nu-1¥¥¥nu¥end{array}¥right)$ (8) $f_{ms}= sum_{v=0}^{s}$ $a_{s-v}$ .

It is also known that the abscissa of convergence $¥lambda_{m}$ of the new series (7) does not exceed $¥lambda$ if $¥lambda¥geq 0$ . However, if $¥lambda<0$ , the situation is more obscure. Milne-Thompson ([1], p. 293), states that “ in general” (meaning sometimes),

$¥lambda_{m}¥geq 0$ . No upper bound appears to have been given for $¥lambda_{m}$ when $¥lambda<0$ .

$¥lambda_{m}$ Indeed in this case, as we shall show, is never larger than zero and $¥lambda_{m}$ can even be minus infinity.

To obtain this bound on $¥lambda_{m}$ , assume temporarily that $m=1$ . Then in this special case $f_{1S}=a_{0}+a_{1}+¥cdots+a_{s}$ . Let

a $=¥lim_{n¥rightarrow¥infty}¥sup¥log|¥sum_{s=0}^{n}a_{s_{1}}|/¥log n$ .

¥ $ lambda=$ ¥ ¥ ¥ ¥ It is known a if $ lambda geq 0$ and a $=0$ if $¥lambda<0$ . Set $w=$ a $+ epsilon$ , where $ epsilon$ is any chosen positive number. Corresponding to $¥epsilon$ there exists a positive number $M$ such that we have the weak dominance

$|f_{1n}|=|¥sum_{s=0}^{n}a_{s}|

$|¥sum_{s=0}^{n}f_{1S}|

finds that the abscissa of convergence $¥lambda_{1}$ for series (7) when $m=1$ can not

¥ exceed $w$ . But in the case under consideration $¥lambda<0$ , $ alpha=0$ , and $w=¥mathcal{E}$ .

Since 8 is arbitrary, $¥lambda_{1}$ does not exceed zero. This argument can be repeated when $m=2$ , $m=3$ , etc., and so we have proved Reciprocals of Inverse Factorial Series 41

Theorem 3. If $m$ is a positive integer and the abscissa of convergence $¥lambda$ for series (3) is negative, the abscissa of convergence for series (7) does not

$¥mathrm{i}$ ¥ ¥ exceed zero; . $¥mathrm{e}.$ , $ lambda_{m} leq 0$ . For example, if

$F(z)=¥underline{1}-¥underline{1}$

$z$ $z(z+1)$

and $m=1$ , then $¥lambda=¥lambda_{1}=-¥infty$ . Norlund, [2, p. 196], gives an example where

¥ ¥ $ lambda=- infty$ and $¥lambda_{1}=0$ . For present purposes we need to consider the inverse of transformation

$¥mathrm{i}$ $(z, z+m)$ ; . $¥mathrm{e}.$ , suppose series (7) is given and the abscissa of convergence

$¥lambda_{m}$ is any finite number or $-¥infty$ . Then formally compute the unique associated series (3). Does this associated series (3) have a half-plane of convergence ? To answer this question begin again by taking $m=1$ . Let a$ 1=¥max$ $¥{0,¥lambda_{1}+1¥}$ and select $¥epsilon>0$ and set $ w_{1}=¥alpha_{1}+¥epsilon$ . Series (7) converges when $z=w_{1}$ . Let $M_{1}/w_{1}$ be the largest of the quantities

$s!|f_{1s}|/w_{1}(w_{1}+1)¥cdots(w_{1}+s)$ ,

so that for all $s$

$|f_{1s}|¥leq M_{1}$ $¥left(¥begin{array}{l}w_{1}+s¥¥s¥end{array}¥right)$ . Thus

$|¥sum_{s=0}^{n}a_{s}|=|f_{1n}|¥leq M_{1}$ $¥left(¥begin{array}{l}w_{1}+n¥¥n¥end{array}¥right)¥sim M_{1}n^{w_{1}}/¥Gamma(w_{1}+1)¥vee$

It follows that the formal series (3) converges if $Re(z)>w_{1}=¥mathrm{a}_{1}+¥mathcal{E}$ , for every $¥epsilon>0$ . Therefore series (3) converges if $Re(z)>¥max¥{0, ¥lambda_{1}+1¥}$ . Repeating this analysis $m$ times one obtains Theorem 4. If series (7), when $m$ is a positive integer, is used to define the $F(z)$ and series (7) has an abscissa of convergence $¥lambda_{m}$ , then $F(z)$ can also be represented by a unique series (3) which is convergent in at least the half-plane $Re(z)>¥max¥{m-1, ¥lambda_{m}+m¥}$ .

4. Reciprocals of factorial series

Let a convergent factorial seris of type (3) be given; but suppose that some of the lead coefficients vanish so that for the case on hand

(9) $F(z)=¥sum_{s=m}^{¥infty}¥frac{s^{1}a_{s}}{z(z+1)¥cdot¥cdot,(z+s)}$ ; $a_{n},¥neq 0$ ; $ Re(z)>¥lambda$ . 42 W. A. HARRIS, Jr. & H. L. TURRITTIN

Write $1/F(z)=z^{m+2}P(z)/Q(z)$ , where $P(z)=¥frac{1}{z}(1+¥frac{1}{z})(1+¥frac{2}{z})¥cdots(1+¥frac{m}{z})$ and

$ Q(z)=m!a_{m}+¥sum_{s=m+1}^{¥infty}¥frac{s!a_{s}}{(z+m+1)(z+m+2)(z+s)}¥cdots$ .

The product $P(z)$ is analytic at infinity and consequently has a con- vergent factorial series representation. The convergent series in $Q(z)$ can be replaced by a convergent factorial series of type (3) by virtue of Theorem 4. Once this has been done an application of Theorem 2 with $G(t)=1/(m!a_{m}+t)$ shows that, if we define the function $S(z)$ by the equation $S(z)=(1/Q(z))-1/m!a_{m}$ , then $S(z)$ also can be represented by a convergent factorial series. Finally noting by Theorem 1 that a product of two convergent factorial series is a convergent factorial $¥mathrm{s}¥dot{¥mathrm{e}}$ ries and recalling the fact that a constant times a convergent factorial series is again a convergent series and that the sum of two such series is also a convergent factorial series, we have demonstrated Theorem 5. If $F(s)$ is a convergent inverse factorial series of type (9) then $J(z)=1/z^{m+2}F(z)$ is representable as a convergent inverse factorial series in some right half-plane. 5. An implicit function theorem

Consider a function of the form

(10) $F(z, u)=G_{0}(z)+¥sum_{v=1}^{¥infty}$ (a $v+G_{¥gamma}(z)$) $u^{¥mathrm{v}}$ ,

$¥alpha_{1}¥neq 0$ where $a_{v}$ are constants; ; and the series

$¥sum_{v=1}^{¥infty}¥alpha_{v}u^{v}$ converges for $|u|¥leq r$ , $r>0$ . Also suppose the series

$ G_{¥mathrm{v}}(z)=¥sum_{s=0}^{¥infty}¥frac{s!g_{¥mathrm{V}S}}{z(z+1)(z+s)}¥cdots$ $¥nu=1,2$ $¥cdots$ ’ , , all converge in the half-plane $ Re(z)>¥lambda$ . Let $¥alpha=¥max¥langle 0$ , $¥lambda$ } and select an $¥epsilon>0$ . Set $w=$ a $+¥epsilon$ . Then, as has been indicated, there exists a of positive constants $M_{0}$ , $M_{1}$ , $¥cdots$ such that

$|g_{vs}|

and the respective series $G_{¥gamma}(z)$ are dominated termwise by the series

(11) $L_{¥nu}(z)=¥frac{M_{v}}{z-w-1}=¥sum_{s=0}^{¥infty}¥frac{s^{1}d_{v}}{z(z+1)}¥ldots s¥overline{(z+s)}$ and these series are absolutely convergent when $Re(z)>w+1$ . Moreover each $d_{¥mathcal{V}S}¥geq|g_{¥mathcal{V}¥mathrm{S}}|$ . We also presume that the series

$ M_{0}+M_{1}u+¥cdots+M_{v}u^{v}+¥cdots$

¥ converges if $u$ is small enough, say $|u| leq r$ . Let $Re(z)=¥sigma>w+1$ ; then

$|G_{v}(z)|¥leq¥sum_{s=0}^{¥infty}¥frac{s^{1}|g_{vs}|}{|z||z+1||z+s|}¥cdots¥leq¥sum_{s=0}^{¥infty}¥frac{s!d_{¥mathrm{v}s}}{¥sigma(¥sigma+1)(¥sigma+s)}¥cdots=¥frac{M_{¥nu}}{¥sigma-w-1}$ and

¥ $|F(z,u)|¥leq|G_{0}(z)|+¥sum_{¥mathrm{v}=1}^{¥infty}$ $( | ¥mathrm{a}_{¥gamma}|+|G_{¥nu}(z)|)$ $|u|^{ gamma}$

$¥leq¥frac{1}{¥sigma-w-1}¥sum_{¥mathrm{v}=0}^{¥infty}M_{v}|u|^{¥gamma}+¥sum_{v=1}^{¥infty}|¥mathrm{a}_{¥gamma}||u|^{v}$ .

Hence, if $|u|¥leq ¥mathrm{r}$ and $Re(z)>w+1$ , series (10) converges and defines a function $F(z, u)$ . We seek a solution $u=S(z)$ of the equation $F(z, u)=0$ of the form

(12) $u=S(z)=¥sum_{s=0}^{¥infty}¥frac{s!h_{1}}{z(z+1)}¥ldots s(¥overline{z+s)}$ , which is convergent in some right half-plane. At first let us proceed formally setting

$ S^{v}(z)=¥sum_{s=v-1}^{¥infty}¥frac{s!h_{vs}}{z(z+1)(z+s)}¥cdots$

$=¥sum_{s=v-1}^{¥infty}¥frac{s!h_{m¥mathrm{v}s}}{(z+m)(z+m+1)(z+¥mathrm{n}¥iota+s)}¥cdots$ ’

$S$ where $h_{vs}$ and $h_{mvs}$ axe computed as though the series for were known to be convergent in a right half-plane. Set (13) $h_{¥gamma 0}=h_{¥gamma 1}=¥cdots=h_{¥mathcal{V},¥mathrm{V}-2}=0$ and $h_{mv0}=h_{mv1}=¥cdots=h_{m,¥mathrm{v}.v2}¥_=0$

for $¥nu=2,3$ , $¥cdots$ and $m=1,2$ , $¥cdots$ . It follows at once from the formulas in

$¥cdots$ $ h_{1,,¥iota 1}¥_$ Theorem 1, from (13), and an easy induction on $¥nu$ that if $h_{10},h_{11}$ , , are all non-negative, then all the coefficients

¥ ¥ ¥ ¥ ¥ $¥cdots$ $¥cdots$ (14) $h_{ gamma, gamma_{ _}1},h_{vv}$ , $ cdots$ , $h_{¥gamma,n+v-2}¥geq 0$ for $ nu=1,2$ , ; $n=1,2$ , and, by virtue of (8)

¥ ¥ ¥ ¥ (15) $h_{m,v,v-1}$ , $h_{m mathrm{v}v}$ , $ cdots$ , $h_{m,v,n+ mathcal{V}-2} geq 0$

for $m=1,2$ , $¥cdots$ ; $¥nu=1,2$ , $¥cdots$ and $n=1,2$ , $¥cdots$ .

In order to compute formally the $h_{1S}$ substitute series (12) into the equation $F(z,u)=0$ , 44 W. A. HARRIS, Jr. & H. L. IURRIT IN

(16) $¥sum_{s=0}^{¥infty}¥frac{s!g_{0s}}{z(z+1)(z+s)}¥cdots+¥sum_{¥mathrm{v}=0}^{¥infty}¥alpha_{v}¥sum_{s=v_{-}1}^{¥infty}¥frac{s!h_{vs}}{z(z+1)(z+s)}¥cdots$

$+¥sum_{n=1}^{¥infty}¥sum_{s=1}^{¥infty}¥frac{s!g_{ns}}{z(z+1)(z+s)}¥cdots¥sum_{v=n-1}^{¥infty}¥frac{¥nu!h_{s+1,n,¥mathrm{v}}}{(z+s+1)(z+s+2)(z+s+1+¥nu)}¥cdots$

$=0$ . Setting the sum of the coefficients of $1/z(z+1)¥cdots(z+s)$ equal to zero, one finds $¥alpha_{1}h_{10}+g_{00}=0$ ,

$¥mathrm{a}_{1}h_{11}+g_{01}+g_{10}h_{10}+¥mathrm{a}_{2}h_{21}=0$ , where $h_{21}=h_{10}^{2}$ ; and in general

(17) s † $¥alpha_{1}h_{1S}+s!g_{0s}+¥sum_{¥mathrm{v}=2}^{s+1}s!¥alpha_{v}h_{vs}$

$+¥gamma¥sum_{=1}^{s}¥sum_{¥eta=0}^{s-¥nu}¥eta!g_{¥gamma¥eta}(s-1-¥eta)!h_{¥eta+1,¥mathrm{V},S-1-¥eta}=0$ .

As a special case suppose that a$1>0$ ; $¥alpha¥simeq^{¥prime}0$ for $¥nu=2,3$ , $¥cdots$ , and that

¥ ¥ $h_{10}$ $g_{vs} leq 0$ for all $ nu$ and $s$ ; then it is clear that is non-negative and that

¥ $h_{11}$ is the sum of non-negative terms. Indeed $h_{1s}$ , $s=2,3$ , $ cdots$ , is also the sum of only non-negative terms, as can be verified from (17), (14) and (15)

by induction on $s_{¥sim}$ These facts suggest that we should consider the dominant equation

(18) $K(z,u)=( | ¥mathrm{a}_{1}|-L_{1}(z))u-L_{0}(z)-¥sum_{v=2}^{¥infty}(| ¥mathrm{a}_{v}|+L_{v}(z))$$u^{v}=0$

and let $z=1/t$ and $K(1/t,u)=N(t,u)$ to obtain a new equation

(19) $N(t,u)=¥frac{M_{0}t}{(w+1)t-1}+(|a_{1}|+¥frac{M_{1}t}{(w+1)t-1})¥mathrm{u}$

¥ ¥ ¥ ¥ ¥ ¥ ? $ sum_{v=2}^{ infty}|a_{v}|u^{v}+ sum_{v=2}^{ infty} frac{M_{ mathrm{v}}tu^{v}}{(w+1)t-1}=0$ . Note the indicated series here are absolutely convergent if $|t|<1/(w+1)$ and $|u|¥leq r$ . The standard implicit function theorem, see Bieberbach [1], guarantees the existence of a solution $u=Q(t)$ of equation (19), analytic in a neighbor- hood of $t=0$ , such that $Q(0)=0$ . Hence $u=Q(1/z)=T(z)$ is a solution of our dominant equation (18) analytic in the neighborhood of infinity and such that $T(¥infty)=0$ . Furthermore by a previous corollary $T(z)$ has a convergent inverse factorial series representation

(20) $ T(z)=¥sum_{s=0}^{¥infty}¥frac{s^{1}e_{s}}{z(z+1)(z+s)}¥cdots$

in some right half-plane. This series for $T(z)$ could be determined by substituting the factorial series (11) and (20) into (18). When this is done Reciprocals of Inverse Factorial Series 45

¥ ¥ $ cdots$ the coefficients $e_{1}$ , $e_{ mathrm{z}}$ , can then be computed in turn from a system of equations analogous to (17). The dominance has been so chosen that

¥ $e_{s} geq|h_{1S}|$ for all $s$ . Hence the formal factorial series given in (12) does converge absolutely and represent an analytic function $S(z)$ in at least the same half-plane in which $T(z)$ converges absolutely. If $Re¥{z$) is sufficiently large, not only is the series representing $S(z)$ absolutely convergent; but also the single, double, and triple series in the left member of equation (16) are all absolutely convergent and rearrange- ment of terms is therefore permissible. Since originally the $h_{s}$ were chosen to formally make $F(z, S(z))¥equiv 0$, it is now clear that rigorously $F(z, S(z))¥equiv 0$ . This yields our final

Theorem 6. Let a sequence $G_{¥gamma}(z)$ , $(¥nu=0,1, ¥cdots)$ , of inverse factorial series be given and all convergent in a half-plane $ Re(z)>¥lambda$ . Let $¥epsilon>0$ and a $=¥max$

¥ ¥ ¥ $¥{0,¥lambda¥}$ . Let the termwise dominant function of $G_{ gamma}(z)$ be $M_{v}/(z- alpha- mathcal{E}-1)$ . Let

$¥cdots$ $1¥neq 0$ $¥sum_{v=1}^{¥infty}|¥alpha_{v}|u^{V}$ $¥mathrm{a}_{1},¥alpha_{2}$ , be a given sequence of constants, a , such that con-

¥ ¥ verges for $|u|¥leq r$ , $r>0$ and suppose that $¥sum_{v=0}^{¥infty}M_{¥nu}¥mathrm{u}^{¥nu}$ also converges when $|u| leq mathrm{r}$ . Then the series $F(z,u)=G_{0}(z)+¥sum_{v=1}^{¥infty}$ $(¥mathrm{a}_{¥gamma}+G_{¥mathcal{Y}}(z))¥mathrm{u}^{¥nu}$ converges for $|u|¥leq r$ and $Re(z)$ $>$ a $+¥mathcal{E}+1$ . The equation $F(z,u)=0$ has an analytic solution $¥mathrm{u}=S(z)$ which can be represented by a convergent inverse factorial series in some right half-plane. Remark. As pointed out by Y. Sibuya, $F(z,u)=0$ would also have a solution representable as a convergent factorial series if $F$ had the form

$ F(z,u)=¥sum_{¥mathcal{Y}=1}^{¥infty}¥alpha_{¥gamma}u^{¥mathrm{v}}+¥sum_{s=0}^{¥infty}¥frac{s¥dagger g_{s}(u)}{z(z+1)(z¥dashv- s)}¥cdots$ $¥mathrm{a}_{1}¥neq 0$ ’ ,

where each $g_{s}(u)$ is analytic for $|u|¥leq r$ and the indicated series converged uniformly for $|u|¥leq r$ , $ Re(z)>¥lambda$ . In this case there would exist bounds of the form

$|g_{s}(u)|

and Theorem 6 is applicable.

References

[1] L. Bieberbach, Lehrbuch der Funktionentheorie, B. G. Teubner, Leipzig und Berlin, 1 (1930), pp. 197. [2] W. A. Harris, Jr., Linear systems of difference equations, Contributions to Differential Equations, 1 (1963), p. 489-518. [3] W. A. Harris, Jr., Equivalent classes of difference equations, Contributions to Differential Equations, (to appear). W. A. HARRIS, Jr. & H. L. TURRITTIN

[4] L. M. Milne-Thompson, The of finite differences, Chap. 10, MacMillan and Co., London, 1951. [5] N. Nielsen, Sur la multiplication de deux series de factorielles, Rendiconti della R. Ace. dei Lincei (5), 13 (1904), p. 517-524. [6] N. E. Norlund, Legons sur les series d’interpolation, Chap. 6, Gauthiers-Villars et Cie, Paris, 1926.

(Ricevita la 22-an de augusto, 1963)