Funkcialaj Ekvacioj, 6 (1964), 37?46
Reciprocals of Inverse Factorial Series
By W. A. HARRIS,
Jr.* and H. L. TURRITTIN
(University of Minnesota)
1. Introduction
relating to the manipulation of convergent power series,
asymptotic series and convergent
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
defined by a power
The
$¥mathrm{d}¥mathrm{e}¥mathrm{t}¥mathrm{a}¥dot{¥mathrm{i}}¥mathrm{l}¥mathrm{s}$
$¥dot{¥mathrm{i}}¥mathrm{n}¥mathrm{v}¥mathrm{e}¥mathrm{r}¥mathrm{s}¥mathrm{e}$
$G¥mathrm{b}.¥mathrm{e}$
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
represented either by a convergent or asymptotic series of the form
(2)
$F(z)=¥sum_{v=1}^{¥infty}¥beta_{v}z^{-v}$
$F$
be
,
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
(3)
$F$
is defined by the factorial series
$F(z)=¥sum_{s=0}^{¥infty}¥overline{z(z+1}^{¥frac{!}{)}}s.a_{s}.¥cdot(z¥overline{+s)}$
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 MilneThompson [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
are represented by factorial series
and abscissa of convergence
and coefficients
of type (3), with coefficients
and abscissa convergence
respectively; then the product $F(z)F_{2}(z)$ has
factorial series representation of type (3) convergent in the half-plane $Re(z)>$
with coefficien
, where
$F_{2}$
$¥lambda$
$a_{s}$
$b_{s}$
$a$
$¥lambda_{2}$
$¥max¥{0, ¥lambda,¥lambda_{2}¥}$
$ts$
$c_{s}$
$c_{n+1}=¥sum_{s=0}^{¥infty}(n-1)!b_{n-s}c_{n-s,s}/(n+1)1$
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
and
were all positive, the coefficients
of the product would also be positive.
Theorem 1 implies that $F(z)$ and all its powers
,
can be
represented as factorial series
$a_{s}$
$b_{s}$
$c_{s}$
$F^{2}(t),F^{¥theta}(t)$
(4)
$ F^{¥mathrm{V}}(z)=¥sum_{s=0}^{¥infty}¥frac{s^{1}b_{vs}}{z(z+1)(z+s)}¥cdots$
;
$¥nu=1,2$
,
$¥cdots$
$¥cdots$
;
convergent in a common half-plane $Re(z)>maz¥{0, ¥lambda¥}$ . Furthermore, if the
coefficients
are positive, then all the
are positive.
Let us formally replace
in (1) by series (4) and rearrange the order
of the terms to obtain a new formal series
$a_{s}=b_{1S}$
$b_{vs}$
$t^{v}$
(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
is the abscissa of convergence. Let a $=$
¥ ¥
¥
¥
and select an
and set $w=$ a
. By hypothesis series (3)
converges when $=w$ and so the individual terms in the series must
$¥lambda$
$ max {0,
$¥epsilon>0$
lambda }$
$+¥epsilon$
$z$
*
This is really a finite sum since
$b_{vs}=0$
for
$¥nu>s+¥mathrm{t}$
.
Reciprocals
approach zero as
.
equal
to
be
value
absolute
$ s¥rightarrow¥infty$
39
of Inverse Factorial Series
This means there is a largest term; let its
$M/w$ . 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
takes the form
termwise now
$F(z)$
$ 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 $Re(z)>w+1$ .
important features; the .coefficient dominance
There are two
$d_{Jl}¥geq|a_{n}|$
and the analiticity of
$w+1$
) in the neighborhood of infinity; .
$¥mathrm{i}$
$L_{¥backslash }^{(}z$
.
$¥mathrm{e}.$
, for
,
$|z|¥backslash $
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 $Re(z)>w+1$ .
so that $¥sigma>w+1+M/r$ ; then
Let $ Re(z)=¥sigma$ and restrict
$¥sigma$
$L^{¥gamma}(¥sigma)=(¥frac{M}{¥sigma-w-1})^{v}=¥sum_{s=0}^{¥infty}¥frac{s!d_{¥mathrm{v}s}}{¥sigma(¥sigma+1)(¥sigma+s)}¥cdots<¥mathrm{r}^{¥gamma}$
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 halfplane, 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
If
Corollary.
$H(z)-H(¥infty)$
, then
is analytic in the neighborhood of
can be represented by an inverse factorial series convergent in some
$ z=¥infty$
$H(z)=¥sum_{¥mathrm{v}=0}^{¥infty}gz^{-¥mathrm{V}}$
right half-plane.
Proof. Take
3.
$G(t)=H(1/t)$
and
.
$F(z,¥$=¥frac{1}{z}$
The inverse of transformation
$(z, z+m)$
Again consider the function
) and its factorial series representation
with
an abscissa of convergence and let $m$ be a given positive integer.
(4)
It is known, see [1] or [2], that a certain transformation $(z, z+m)$ will
yield a new representation of function $F(z)$ , namely
$F^{/}¥backslash z$
$¥lambda$
(7)
$ F(z)=¥sum_{s=0}^{¥infty}¥frac{s^{1}f_{ms}}{(z+m)(z+m+1)(z+m+s)}¥cdots$
’
where the constants
(8)
$f_{ms}=¥sum_{v=0}^{s}$
$¥left(¥begin{array}{l}m+¥nu-1¥¥¥nu¥end{array}¥right)$
$a_{s-v}$
.
It is also known that the abscissa of convergence
of the new series (7)
does not exceed if
. However, if
, the situation is more obscure.
Milne-Thompson ([1], p. 293), states that in general” (meaning sometimes),
. No upper bound appears to have been given for
when
.
Indeed in this case, as we shall show,
is never larger than zero and
can even be minus infinity.
, assume temporarily that $m=1$ . Then in
To obtain this bound on
$f_{1S}=a_{0}+a_{1}+
¥
cdots+a_{s}$
this special case
. Let
$¥lambda_{m}$
$¥lambda$
$¥lambda<0$
$¥lambda¥geq 0$
“
$¥lambda_{m}¥geq 0$
$¥lambda<0$
$¥lambda_{m}$
$¥lambda_{m}$
$¥lambda_{m}$
$¥lambda_{m}$
a
$=¥lim_{n¥rightarrow¥infty}¥sup¥log|¥sum_{s=0}^{n}a_{s_{1}}|/¥log n$
.
It is known
and a $=0$ if
a if
. Set $w=$ a , where
is
any chosen positive number. Corresponding to
there exists a positive
$M$
that
we
have
the
weak
number
such
dominance
$¥lambda=$
$¥lambda<0$
$¥lambda¥geq 0$
$+¥epsilon$
$¥epsilon$
$¥epsilon$
$|f_{1n}|=|¥sum_{s=0}^{n}a_{s}|<M$
$¥left(¥begin{array}{l}w+n¥¥n¥end{array}¥right)$
for all
$n$
.
Hence
$|¥sum_{s=0}^{n}f_{1S}|<M¥sum_{s=0}^{n}$
$¥left(¥begin{array}{l}w+s¥¥s¥end{array}¥right)=M¥left(¥begin{array}{l}w+n+1¥¥n¥end{array}¥right)¥sim¥frac{Mn^{w+1}}{¥Gamma(w+2)}$
asymptotically as
, one
. Using the $¥lim¥sup$ formula applied to
$m=1$
convergence
finds that the abscissa of
for series (7) when
can not
,
, and
exceed $w$ . But in the case under consideration
.
Since 8 is arbitrary,
does not exceed zero. This argument can be
repeated when $m=2$ , $m=3$ , etc., and so we have proved
$ n¥rightarrow¥infty$
$f_{1s}$
$¥lambda_{1}$
$¥lambda<0$
$¥lambda_{1}$
$¥alpha=0$
$w=¥mathcal{E}$
Reciprocals
of Inverse Factorial Series
41
Theorem 3. If $m$ is a positive integer and the abscissa of convergence
for series (3) is negative, the abscissa of convergence for series (7) does not
.
exceed zero; . ,
For example, if
$¥lambda$
$¥lambda_{m}¥leq 0$
$¥mathrm{i}$
$¥mathrm{e}.$
$F(z)=¥underline{1}-¥underline{1}$
$z$
and
then
and
$m=1$ ,
$¥lambda=-¥infty$
$¥lambda=¥lambda_{1}=-¥infty$
$¥lambda_{1}=0$
.
$z(z+1)$
Norlund, [2, p. 196], gives an example where
.
For present purposes we need to consider the inverse of transformation
$(z, z+m)$ ;
. , suppose series (7) is given and the abscissa of convergence
is any finite number or
. 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$
and select
and set
. Series (7) converges when
$z=w_{1}$ .
be the largest of the quantities
Let
$¥mathrm{i}$
$¥mathrm{e}.$
$-¥infty$
$¥lambda_{m}$
$¥epsilon>0$
$¥{0,¥lambda_{1}+1¥}$
$ w_{1}=¥alpha_{1}+¥epsilon$
$M_{1}/w_{1}$
$s!|f_{1s}|/w_{1}(w_{1}+1)¥cdots(w_{1}+s)$
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
. Therefore series (3) converges if $Re(z)>¥max¥{0, ¥lambda_{1}+1¥}$ .
Repeating this analysis $m$ times one obtains
$Re(z)>w_{1}=¥mathrm{a}_{1}+¥mathcal{E}$
, for every
$¥epsilon>0$
Theorem 4. If series (7), when $m$ is a positive integer, is used to define
the analytic function $F(z)$ and series (7) has an abscissa of convergence
, then
$F(z)$ can also be represented by a unique series (3) which is convergent in at
least the half-plane
$¥lambda_{m}$
$Re(z)>¥max¥{m-1, ¥lambda_{m}+m¥}$
.
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$
.
W. A. HARRIS, Jr. & H. L. TURRITTIN
42
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 convergent 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
$S(z)=(1/Q(z))-1/m!a_{m}$ ,
$S(z)$
also can be represented by a convergent factorial series.
then
Finally noting by Theorem 1 that a product of two convergent factorial
series is a convergent factorial 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
$¥mathrm{s}¥dot{¥mathrm{e}}$
$J(z)=1/z^{m+2}F(z)$
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)
where
$F(z, u)=G_{0}(z)+¥sum_{v=1}^{¥infty}$
$a_{v}$
are constants;
$¥alpha_{1}¥neq 0$
(a
$v+G_{¥gamma}(z)$
)
$u^{¥mathrm{v}}$
,
; 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$
’
$¥nu=1,2$
,
$¥cdots$
,
, } and select an
all converge in the half-plane $ Re(z)>¥lambda$ . Let
. Set $w=$ a . Then, as has been indicated, there exists a sequence
, ,
such that
of positive constants
$¥alpha=¥max¥langle 0$
$¥epsilon>0$
$+¥epsilon$
$M_{0}$
$M_{1}$
$¥cdots$
$|g_{vs}|<M_{v}$
$¥left(¥begin{array}{l}w+s¥¥s¥end{array}¥right)$
$¥lambda$
Reciprocals
and the respective series
(11)
$G_{¥gamma}(z)$
43
of Inverse Factorial Series
are dominated termwise by the series
$L_{¥nu}(z)=¥frac{M_{v}}{z-w-1}=¥sum_{s=0}^{¥infty}¥frac{s^{1}d_{v}}{z(z+1)}¥ldots s¥overline{(z+s)}$
and these series are absolutely convergent when $Re(z)>w+1$ .
each
.
We also presume that the series
Moreover
$d_{¥mathcal{V}S}¥geq|g_{¥mathcal{V}¥mathrm{S}}|$
$ M_{0}+M_{1}u+¥cdots+M_{v}u^{v}+¥cdots$
is small enough, say
Let $Re(z)=¥sigma>w+1$ ; then
converges if
$|u|¥leq r$
$u$
.
$|G_{v}(z)|¥leq¥sum_{s=0}^{¥infty}¥frac{s^{1}|g_{vs}|}{|z||z+1||z+s|}¥cdots¥leq¥sum_{s=0}^{¥infty}¥frac{s!d_{¥mathrm{v}s}}{¥sigma(¥sigma+1)(¥sigma+s)}¥cdots=¥frac{M_{¥nu}}{¥sigma-w-1}$
and
$|F(z,u)|¥leq|G_{0}(z)|+¥sum_{¥mathrm{v}=1}^{¥infty}$ $( | ¥mathrm{a}_{¥gamma}|+|G_{¥nu}(z)|)$
$|u|^{¥gamma}$
$¥leq¥frac{1}{¥sigma-w-1}¥sum_{¥mathrm{v}=0}^{¥infty}M_{v}|u|^{¥gamma}+¥sum_{v=1}^{¥infty}|¥mathrm{a}_{¥gamma}||u|^{v}$
.
and $Re(z)>w+1$ , series (10) converges and defines a
Hence, if
function $F(z, u)$ .
We seek a solution $u=S(z)$ of the equation $F(z, u)=0$ of the form
$|u|¥leq ¥mathrm{r}$
(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$
$=¥sum_{s=v-1}^{¥infty}¥frac{s!h_{m¥mathrm{v}s}}{(z+m)(z+m+1)(z+¥mathrm{n}¥iota+s)}¥cdots$
’
and
axe computed as though the series for were known
where
to be convergent in a right half-plane.
Set
and $h_{mv0}=h_{mv1}=¥cdots=h_{m,¥mathrm{v}.v2}¥_=0$
(13)
and $m=1,2$ ,
. It follows at once from the formulas in
for $¥nu=2,3$ ,
,
,
that if
Theorem 1, from (13), and an easy induction on
are all non-negative, then all the coefficients
; $n=1,2$ ,
for $¥nu=1,2$ ,
,
,
(14)
$S$
$h_{mvs}$
$h_{vs}$
$h_{¥gamma 0}=h_{¥gamma 1}=¥cdots=h_{¥mathcal{V},¥mathrm{V}-2}=0$
$¥cdots$
$¥cdots$
$¥nu$
$h_{¥gamma,¥gamma_{¥_}1},h_{vv}$
$¥cdots$
$h_{¥gamma,n+v-2}¥geq 0$
$¥cdots$
$h_{10},h_{11}$
$¥cdots$
$ h_{1,,¥iota 1}¥_$
$¥cdots$
and, by virtue of (8)
(15)
$h_{m,v,v-1}$
,
$h_{m¥mathrm{v}v}$
,
$¥cdots$
,
$h_{m,v,n+¥mathcal{V}-2}¥geq 0$
; $¥nu=1,2$ ,
and $n=1,2$ ,
In order to compute formally the
equation $F(z,u)=0$ ,
for
$m=1,2$ ,
$¥cdots$
$¥cdots$
$¥cdots$
$h_{1S}$
.
substitute series (12) into the
W. A. HARRIS, Jr. & H. L. IURRIT IN
44
(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
one finds
$1/z(z+1)¥cdots(z+s)$
$¥alpha_{1}h_{10}+g_{00}=0$
$¥mathrm{a}_{1}h_{11}+g_{01}+g_{10}h_{10}+¥mathrm{a}_{2}h_{21}=0$
,
equal to zero,
,
where
$h_{21}=h_{10}^{2}$
;
and in general
(17)
s†
$¥alpha_{1}h_{1S}+s!g_{0s}+¥sum_{¥mathrm{v}=2}^{s+1}s!¥alpha_{v}h_{vs}$
$+¥gamma¥sum_{=1}^{s}¥sum_{¥eta=0}^{s-¥nu}¥eta!g_{¥gamma¥eta}(s-1-¥eta)!h_{¥eta+1,¥mathrm{V},S-1-¥eta}=0$
.
, and that
for $¥nu=2,3$ ,
As a special case suppose that a $1>0$ ;
non-negative
and that
is
for all
and ; then it is clear that
$s=2,3$
,
,
is
also the
non-negative
,
terms. Indeed
is the sum of
sum of only non-negative terms, as can be verified from (17), (14) and (15)
by induction on
These facts suggest that we should consider the dominant equation
$¥cdots$
$¥alpha¥simeq^{¥prime}0$
$g_{vs}¥leq 0$
$h_{10}$
$s$
$¥nu$
$¥cdots$
$h_{1s}$
$h_{11}$
$s_{¥sim}$
(18)
and let
(19)
$K(z,u)=( | ¥mathrm{a}_{1}|-L_{1}(z))u-L_{0}(z)-¥sum_{v=2}^{¥infty}(| ¥mathrm{a}_{v}|+L_{v}(z))$ $u^{v}=0$
$z=1/t$
and
$K(1/t,u)=N(t,u)$
to obtain a new equation
$N(t,u)=¥frac{M_{0}t}{(w+1)t-1}+(|a_{1}|+¥frac{M_{1}t}{(w+1)t-1})¥mathrm{u}$
?
$¥sum_{v=2}^{¥infty}|a_{v}|u^{v}+¥sum_{v=2}^{¥infty}¥frac{M_{¥mathrm{v}}tu^{v}}{(w+1)t-1}=0$
.
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 neighborhood 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
45
of Inverse Factorial Series
, ,
can then be computed in turn from a system of
analogous
equations
to (17). The dominance has been so chosen that
¥
for all . 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 rearrangewere
ment of terms is therefore permissible. Since originally the
chosen to formally make $F(z, S(z))¥equiv 0$, it is now clear that rigorously
the coefficients
$e_{s} geq|h_{1S}|$
$e_{1}$
$e_{¥mathrm{z}}$
$¥cdots$
$s$
$h_{s}$
$F(z, S(z))¥equiv 0$
.
This yields our final
, $(¥nu=0,1, ¥cdots)$ , of inverse factorial series
Theorem 6. Let a sequence
and a $=¥max$
be given and all convergent in a half-plane $ Re(z)>¥lambda$ . Let
be $M_{v}/(z-¥alpha-¥mathcal{E}-1)$ . Let
. Let the termwise dominant function of
$G_{¥gamma}(z)$
$¥epsilon>0$
$G_{¥gamma}(z)$
$¥{0,¥lambda¥}$
,
verges
$¥mathrm{a}_{1},¥alpha_{2}$
$¥cdots$
for
be a given sequence of constants, a
$|u|¥leq r$ , $r>0$ and suppose that
, such that
also converges when
converges for $|u|¥leq r$ and
$1¥neq 0$
$¥sum_{v=1}^{¥infty}|¥alpha_{v}|u^{V}$
con-
$|u|¥leq ¥mathrm{r}$
$¥sum_{v=0}^{¥infty}M_{¥nu}¥mathrm{u}^{¥nu}$
.
¥
¥
$Re(z)$
Then the series
$>$ a
which can
. The equation $F(z,u)=0$ has an analytic solution
be represented by a convergent inverse factorial series in some right half-plane.
$F(z,u)=G_{0}(z)+ sum_{v=1}^{ infty}$
$(¥mathrm{a}_{¥gamma}+G_{¥mathcal{Y}}(z))¥mathrm{u}^{¥nu}$
$¥mathrm{u}=S(z)$
$+¥mathcal{E}+1$
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$
where each
uniformly for
the form
$g_{s}(u)$
is analytic for
$|u|¥leq r$
,
$ Re(z)>¥lambda$
.
$|g_{s}(u)|<M¥left(¥begin{array}{l}w+s¥¥s¥end{array}¥right)$
’
$¥mathrm{a}_{1}¥neq 0$
,
and the indicated series converged
In this case there would exist bounds of
$|u|¥leq r$
and
$|g_{v¥mathrm{s}}|<M¥left(¥begin{array}{l}w+s¥¥s¥end{array}¥right)/¥mathrm{r}^{v}$
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 calculus 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)