Matbematics. - On the G-function. I. By C. S. MEIJER. (Communicated
by Prof. J. G. VAN DER CORPUT.)
(Communicated at the meeting of January 26, 1946.)
REFERENCES.
W. N . BAILEY.
1. Generalized hypergeometric series (Cambridge Tracts in Mathematics and
Mathematical Physics, 32 (1935)) .
E . W . BARN ES.
2. The asymptotic expansion of integral functions defined by TA YL:OR' s series.
Phil. Trans. Royal Soc. London, (A) 206,249-297 (1906) .
3. Th~ asymptotic expansion of integral functions defined by generalized hypergeometrie series. Proc. London Math. Soc., (2) 5, 59-116 (1907).
4. On functions defined by simple types of hypergeometric series. Trans.
Cambridge Phil. Soc., 20, 253-279 (1908) .
5. A new development of the theory of the hypergeometrjc functions. Proc. London
Math. Soc., (2) 6, 141-177 (1908) .
J. G . VAN DER CORPUT.
6. On KUMMER's solutions of the hypergeometrie dif~rential equation. Proc. Kon.
Akad. v . Wetenseh., Amsterdam, 39, 105&-1059 ' (1936) .
P. DEBYE.
7. Semdkonvergente Entwickelungen für die Zylinderfunktionen und ihre Ausdehnung ins Komplexe. Sitzungsberichte der math.-phys. Klasse der K. B. Akad.
der W 'issensch. zu München, 40, no. 5 (1910).
A. R. FORSYTIl.
8. A treatise on differential equations (fifth edition, 1921) .
C. Fox.
9. The asymptotic expansion of generalized hypergeometrie functions. Proc.
Londo!:! Math. Soc., (2) 27,389-400 (1928) .
C . F . GAUSS.
10. Determinatio seriei nostrae per aequationem differentialem secundi ordinls.
Werke, 3, 207-229 (1866) .
:Ë. GoURSAT,.
11. Sur l'équation di;férentielle linéaire qui admet pour 'intégrale la sérle hypergéométrique. Ann. de l'école normale, (2) 10, (1881), Supplément, 3-142.
F. KLEIN .
12. Vorlesungen über die hyper geometrische Funktion (1933) .
E. E. KUMMER.
13. Ober die hypergeometrische Reihe .. .. .. ...... .. .. .... .......... ... Journ. für die reine
und angewandte Math., 15, 39-83 and 127-172 (1836).
T . M. MAcRoBERT.
H. Induction proofs of the relations between certain asymptotic expansions and
corresponding generalized hypergeometrie series. Proc. Royal Soc. Edinburgh,
58, 1-13 (1938).
15. Aproof by induction of the analytic continuations of certain generalized hypergeometrie functions. Philosophical Magazine, (7) 25, 848-851 (1938).
16. Solution in multiple series of a type of generalized hypergeometrie equation.
Proc. Royal Soc. Edinburgh, 59, 49-54 (1939).
17. Expressions for generalized hypergeometric functions in multiple series. Ibidem.
59, 141-144 (1939).
C. S. MEIJER.
18. Ober WHITIAKERsche bezw. BESS,ELsche Funktionen und deren Produkte.
Nieuw Archief VQOr. Wiskunde, (2) 18, (4tes Heft) 1~39 (1936).
228
19. Einige IntegraldarsteUungen aus der Theorie der BESSELschen und WHITTAKERschen Fu.,ktionen. Proc. Kon. Akad. v. Wetensch., Amsterdam, 39, 394-403
and 519-527 (1936).
20. Uber Produkte von WHITTAKERschen Funktionen. Ibidem, 40, 133-141 and
259-263 (1937).
21. Uber BESSELsche, STRUVEsche und LOMMELsche Funktjonen. Ibidem, 43,
198-210 and 366--378 (1940).
22. Neue Integraldarstellungen für WHITTAKERsche Funktionen. Ibidem, 44, 81-92,
186--194,298-307,442-451 and 590-598 (1941).
23. btegraldarstellungen für WHITT,AKERsche Funktionen und ihre Produkte.
Ibidem, 44, 435-441 and 599-605 (1941) .
24. Multiplikationstheoreme für die Funktion G~': (z). Ibidem, 44, 1062-1070
(1941 ).
25. Integraldarstellungen für STRUVEsche und BESSELsche Funktionen. Compositio
Mathematica, 6, 348-367 (1939) .
L. PoCHHAMME.R.
26. Uber die Differentialgleichung der allgemeineren hypergeometrischen Reihe mit
zwei endlichen singulären Punk ten. Joum. für die reine u.,d angewandte Math.,
102,76--159 (1888) .
F. C. SMITH.
27. Relations among the fundamental solutions of the generalized hypergeometric
equation wh en p
q
I. I. Non-Iogarithmic cases. Bulletin American Math.
Soc., 44, 429-433 (1938).
G. G. STOKES.
28. Note on the determination of arbitrary consta:J.ts which appear as multipliers
ofl semi-convergent series. Proc. Cambridge Phi!. Soc., 6, 362-366 (1889) .
[Math. and Phys. Papers, V, 221-225 (1905).)
J. THOMAB.
29. Uber die höheren hypergeometrischen Reihen, insbesondere über die Reihe :
.... .. ...... ...... ... Math. Ann., 2, 427-444 0.870).
G. N. WATSON.
30. A c1ass of integral functions defined by TAYLOR's series. Trans. Cam bridge
Phi!. Soc., 22, 15-37 (1923).
31. A treatise on the theory of BESSEL fU:J.ctions (1922).
E . T . WHITI',AKER and G. N . WATSON.
32. A course of modern analysis (fourth edition, 1927).
E. WINKLER.
33. Uber die hypergeometrische Differentialgleichung n ter Ordnung mit zwei endIichen Singulären Punkten. Dissertation München (1931) .
34. Die Unbestimmtheitstelle der ausgearteten hypergeometrischen Differentialgleichung. Sitzungsberichte der math.-naturwissensch. Abteilung di!r Bayerischen
Akad. der Wissensch. zu München, 1933, 237-256.
E. M . WRiIGHT.
35. The asymptotic expansion of the generalized hypergeometric function . Joum.
London Math. Soc., 10, 286--293 (1935).
36. The asymptotic expansion of integral functions defined by TA VLOR series.
Phi!. Trans. Royal. Soc. London, (A) 238,423-451 (1940) .
D. WRINCH.
37. An asymptotic formula for the hypergeometric function 064 (z). Philosophical
Magazine, (6) 41, 161-173 (1921) .
38. A generalized hypergeometric function with n parameters. Ibidem, (6) 41,
174-186 (1921).
39. Some approximations to hypergeometric functions. Ibidem, (6) 45, 818-827
(1923) .
= +
229
§ 1.
Definition of the function
G;: ~ (z ) .
Suppose that m, n, pand q are integers with
q =- 1. 0 -== n -== p -== q and 0 -== m -== q ;
suppose further that the number z satisfies the inequalities
z -=j=. 0 and I z I < 1 if q
z -=j=. 0 if q
p;
>
= p.
moreover that the numbers al' " ', an and bI' " ', bm fulfil the condition
Then the function
m,n
G p,
q
a
a )
( IbI····· b
q
P
Z
I· •.••
G;; ~
(1)
(j=l ..... n; h=l •.... m).
f
(z) is defined as follows 1)
II r(bj-s) j=1
IÎ
1
_ _
- 2'
j=1
r(l-aj
q
+ s)
Z
p
sd
(2)
S.
j=Ifz+t(1-b r + s) j!!+1 r(a}-s)
;tz
c
The contour C runs from co and encloses all the poles
i
+ i i Cr
to co
i
is a positive number)
(3)
(j= 1. ...• m).
b}.b j +1.b}+2 ... .
but none of ·the poles
u= 1•...• n)
a}-I.aj-2.a}-3 •...
.
(4)
of the integrand 2).
It follows from the asymptotic expansion of the Gamma function that the
integral (2) is convergent and independent of i .
IE m = 0, the integrand in (2) is analytic inside C. Hence we Eind
O.n (
Gp ,q
Z
lalb ..... ba
p)-
t ••.•
q
t
. (5)
- O.
More generally: Suppose m :;:;: 0 and
(j= 1. ...•
m;
h
= 1. ...• m;j-=j=. hl;
(6)
th en the in~egrand has simple poles at the points (3) and the value of the
integral (2) is equal to minus the sum of the residu es of the integrand at
these poles. By evaluating these residues we obtain 3)
;"g
=
1) If p
O. the function G is denoted by G
(z I bI • ...• b q ) .
2) Such a contour C can always be drawn, since, because of (1), none of the poles (3)
coincides with O:1e of the poles (4) .
3) We employ the usual notation for the generalized hypergeometric function:
P
+
+
IJ laj (aj
1) . •. (a}
h-l)j
pFq (a l ••••• ap;{3I ..... {3q;z)=.L: _J<-.-·=_q'---I_ _ _ _ _ _ _ _ __
<Xl
h =0
Zh
h! IJ 1,8} (f3}
}=I
+ 1) ... ({3} + h-l)1
230
(7)
XpFq-dl+bh-al ..... I+bh-ap; l+bh-bl ....... I+bh-bq;(-I)p-m-n z);
the asterisk denotes that the number 1 + bh - bh is to be omitted in the
sequence 1 + h - blo ...• 1 + bh - b q• This expansion holds. provided
that aj and b j satisfy not only (1) but also the additional condition (6) 4).
I may remark here that in my former papers 5) the function G;: ~ (z)
has been defined by means of (7).
Prom (7) it is apparent that
with a branch~point at z = O.
G;; ~(z)
is a many~valued function of z
The function G;"~ (z) is a symmetrie function of al • . ..• an, of
an+lo .. .• apo of b 1 , "', bm and of b m+ 1 , •••• bq; this follows immediately
from the definition (2) .
When there is no risk of ambiguity. we shall in our notation omit the
parameters al, "', ap and blo "', bq. The function G defined by (2) will
then be denoted by G;;~ (z). Wh en the parameters are not explicitly
mentioned, it will always be supposed that they are al' •.• , ap and b 1 • ••. , b q•
Por instanee by the symbols G~:~ (z) and G~: ~ (z) we shall denote the
functions
GZ'~ (z Ial' ...• ap ) .
,
bI •...• b q
respect.
G;:~ (z Ia
.
bI' ...• abP)
q
l
• ....
4) The generalized hypergeometric function pFq(al, ... ,ap;flt, .. . ,Pq;z) is meaningless if Pj = 0, -1. -2, ... (1 -:;;;, j ~ q). The function
q
pFq (a l , ... ,ap;f3I, ...• f3q;z)
11 r(f3J)
j=s+1
is only meaningless when Pj = 0, -1, -2, ... (1 ~ j ~ 5). If one or more of the numbers
Pj (5 1 ~ j ~ q) is equal to zero or a negative integer, the last function is in the usual
·way defined as follows:
+
q
pFq (al' ... , ap; f31' ...• f3q; z)
IJ r(f3J)
J=S+I
p
+ I) ... (aJ + h-l)1
L.J
s q
· '
h=O hl IJ, If3J (f3J + I) ... (f3J + h-I)I IJ r(f3j + h)
J=I
J=s+1
Zh IJ laJ (aJ
Ol)
~
=
j=1
+
•
wh en b j
0, -1. -2, ... (5
1 ~ i ~ q), then some initial terms of the series on the
right-hand side of this relation vanish.
6) MEIJER, [18]- [2S].
231
Now let
q ==-- 1. 0 -== 1- 1 -== n
-== p -== q
and 0 -== k -== q ;
we consider
In th is function we change the order of the parameters al • .... ap in a
special way; we replace the sequence
al' ..•• ap by an-I+2 • ...• apo al' .•.• an-I+I'
Then we obtain
Ian-I+2' .•. , apo al' .•• , an-I+I)
k.l-I (
G p, q
b1•••• bq
Z
t
• (8)
.
This function we wilt briefly, denote by G~;~-I,n (z).
We now suppose
I-==I-==n-==p-==q, I-==t-==n-I+ 1. O-==k-==q
and consider
In th is function we change the order of the parameters al, ...• ap in the
following way: we replace the sequence
al • ...• ap by at. an-I+2 • ...• apo al • .•.• 8t-J, at+l • •..• an-I+I.
Then we get
k,l (
Gp
,
q
Z
Iat. an-I+2 •. ..• apo 81"
•••
at-I. at+l • . ..• a n- I + I )
(9)
•
bI.' ..• b q
This function shall be denoted by G~;~n (zllat).
Since G~;~(z) is a symmetrie function of a1+1o " ', ap, the functions (8)
and (9) are independent of n if 1 = 1. Hence we find
kOn
Gp:ti (z)
=
kO
Gp;q
al.···.ap )
I b J•••• , bq =
I
(
Z
kO
Gp;q (z)
.
(10)
and
The function
Iat. al' ...• at-J, at+J, ••. ,ap )
k,1 (
Gp •q
Z
bI ••••• b q
(1
-== t -== p)
shall henceforth be denoted by G~;~ (zllat). Sa we obtain
G~;~n (z
11 at)
= G~:~ (z 11 at).
•
• (11)
232
§ 2.
Asymptotic expansions of
BARNES •
type for G;: ~ (z) (q
> p).
In some cases an asymptotic expansion of G;:~ (z) (q > p) for large
values of Izl can be deduced by means of the methods of BA'RNES 6). I will
communicate here the results. First I give two definitions.
Definition 1. Suppose that p, q and t are integers with 1 :s; t ::;:;; p < q:
suppose further that the numbers at and b l •. ". b q satisfy the condition
at-b j =j=.1.2.3 •...
(j=1. ... q).
Then. for brevity. we formally 7) write
(12)
00
Ep,q(zllat)=z-I+at } ;
p
hl II r(1
h=O
.
+ aj-at + h)
(13)
j=1
j*t
ff the divergent series on the right~hand side of (13) is written by means
of the hypergeometric notation. the formal expansion (13) takes ihe form
q
+
z-l+at II [ (1
b j-at)
j=1
F
bi-at . ...•
bq-at:)
p
q p-I
.
.
•.
IJ r(1 +aj-at)
l+al-at •....• I+ap- a t;-Z-1
j=1
j*t
the asterisk denoting that the number 1 + at - a t is to be omitted in the
sequence 1 + al - at, .. .. 1 + ap - at.
(1 +
1+
From this definition it follows
Ep,q(ze2yni Ilat)=e2priat Ep,q(zllat)
Definition 2.
(y=O.± 1.±2....).
(15)
Suppose that m. n, q and t are integers with
1 -= t -= n and 0 -= m -= q ;
suppose further that the numbers al .... , an satisfy the condition
aj-at =j=. O.
(j = 1. ...• n; j =j=. t).
± 1. ± 2. . . .
T hen the coefficient D m, ~ (t) is defined as follows:
q
IJ sin (b j-at) n
6m,~(t)=nm+n-q-1 j=mn+1
IJ sin (aj-at) n
.
(16)
.
(17)
J~/t
This coefficient mayalso be written in the form 8)
n
III r(at-aj) r(l
1=1
+ aj-at)!
6m ,n (t) = {_I)m+n-q-1 ,*t
q
q
II Ir{at-b j)r{1 +bj-at)l
j=m+1
6)
BARNES, (3). Some of BARNES' formulae are a1so proved by WINKLER, (34) and
MACRoBERT (14).
7)
8)
The seriês on the right-hand side of (13) is divergent because of q
(17) follows from (16) by means of
. n zn = -
SlO
r{-z) r(1
+ z).
> p.
(ti)
233
The results obtained by
BARNES
can now be stated as follows 9):
Theorem A. The function G~:~ (z llat) with 1 -;;. t -;;. p < q admits for
large values of I z I with I arg z I < (! q - ! p + 1) n the asymptotic
expansion 10)
(18)
More generally
Theorem B.
11 ) :
Suppose that m, n, pand q are integers with
suppose further that the numbers
conditions
Bt-b j
Bj-at
=f 1. 2. 3 •...
=f 0, ± 1. ± 2, ...
al •...•
an and bI' .... b m satisfy the
't= 1. ... , n:j= 1. ... , m).
(j
(19)
= 1. .... n: t = 1. .... n: j =f t).
(20)
Then for large values of Izl with largzl«m+n-!p-!q)n
the following asymptotic expansion holds 12)
G;:~
(z)
V)
En
e(m +n-q-I)niat ~m, ~
(t) Ep. q (z e(q-m-n + I)"j
I1 at)
(21)
t=1
9)
BARNES. [2].295-296 and [3]. 65. The formulae of BARN ES are written in another
notation. The symbols C;; ~. Ep. q and 6 m. ~ do not occur i:1 his papers.
10) In (18) it is of course supposed that the numbers at and ÓJ. ..... b q satisfy the
con dit ion (12).
.
Formula (18) is not true uniformly for I arg z 1< (t q - t p
1 )n; it is true uniformly
only for I arg z I -;;. (t q - t p
1) n----a for, every ,,> O. A similar remark holds with '
regard to the asymptotic expansions (21) and (22) .
11) BARNES. [3]. 70. The formulae (21) and (18) are proved by evaluating the
residues of the integrand in (2) at the poles outside the contour C.
The result ot' BARoNES with regard to formula (21) is not entirely complete; he seems to
consider, only the case with m ;;;;: q - p.
12)
The expansion of the term
+
+
e(m
+ n-q-I)"i at
6 m. ~ (t) Ep. q (z e(q-m-n + I)"i 11 at)
can according to (17) and (13) be written in the form
The coeff:icients in this divergent series are significant.
bI. , .. . b m satisfy the conditions (19) and (20).
jf
the numbers al . , ,,. a n and
234
BARN ES has also investigated the behaviour of G~: ~(z) for I z I tending
to infinity, His result runs as follows 13):
The function G~:~ (z) possesses Eor large values of I z
expansion of the form
I an
asymptotic
where
{} = _1_
~t (p-q + I) + I bh-.É a) , ,
q~p (
~
h=l
This expansion holds for Iarg z I < (q e
=t
iE q
= p + I,
e
p
,
(23)
.,
(24)
h=l
+ ê) n;
herein is
= 1 if q ~ p + 2"
,
The coefficients MI' M2' .. , do not depend on z, but are complicated
functions of the parameters al' .. " ap and bI' .. " b q which I will not
record here 14),
In what follows we shall, for brevity, formally write
Hp,q (z) = [exp ((p_q)
zq~p)] z{}
q-p-l
(2 n)-2l V q-p
!
+ ~l + ~2 +". ,
zq-p
(25)
zq-p
where {} is defined by (23).
The results contained in (22) and (23) can then be formulated as
follows:
Theorem C. The function GZ:~ (z) possesses for large V'alues of
with I arg z I < (q - p + e) n the asymptotic expansion
Gg:~ (z)
C/)
Hp,q (z).
Iz I
. (26)
It appears from (25) and (23) that the forma I function Hp,q(z)
satisfies the relation
(_l)P-q+l exp ~ 2ni (
§ 3,
htl bh- h~ áh) ~ Hp,q (ze(2 P-2 q)ni) = Hp,q (z).
(27)
The analytic continuation of the function G;"; (z) (simple cases).
The function G;";(z) has been defined in § 1 only for Iz 1< 1. The
analytic continuation oE this function when I z I ~ 1 will be denoted too ·
by the symbol G;";(z).
13)
BARN ES, [2]. 296-297; [3].80 and 108-110,
The proof of (22) is much more difficult than that of (18) and (21),
14) The reader. is referred to BARNES' paper [3].
235
Now it is well~known 15) that the generaliz~d hypergeometric function
= 1. and that.
if across~cut is made along the real axis of the z~plane from 1 to co . then
the function is analytic and o,ne~valued throughout the cut plane.
pF p- 1 (al • •. .• ap ; {31 •. .. • {3P-1;' Z) has a branch~point at z
Considering the function G~'; (z ). which can by (7) he written as a
linear combination of functions of the type zÀ pF p_d(-l)p-m-nz). we
shall make a cross~cut in th~ z~plane along the real axis from (-1) m+n-p
to co . (-1) m+n_p; we may expect that G~';(z) is analytic (not one~
valued in general) in the cut plane.
1 will now shew that a somewhat more general result holds when
m + n ~ p + 2.
For this purpose I consider the integral
1
2 ni
f
fi r(bj-s) !i r(l-aj + s)
J=1
p
n
j=m+l
J=1
p
r(l-b j
+ s) j=n+l
II r(aj-s)
z
sd
s •.
. (28)
L
where 1 -== m -== p. 1 -== n -== p . m + n =- p + '1 and Bj-b h =f L 2. 3.. ..
1. . . .. n; h = 1. . . .. m).
The contour L runs from - co i + a to co i + a (a is an arbitrary rea I
number) and is curved. if necessary. so that the points
(j
=
b j. b j
+ 1. b j + 2.. . .
(j
= 1. . . .. m)
He on the right and the points
aj-1. aj-2. aj-3, . ..
(j = 1. ... , n)
He on the left of the contour.
The integral (28) is a function of z which is analytic for each value of
z whether I z I is greater than. equal tO, or less than unity. provided that
z ~ 0 and I arg z I < (m, + n - p) Jl 1 6 ) .
IE I z I < 1. we may 17) . without altering the value of the int~gral. bend
round the contour L so as to coincide with the contour C defined in § 1.
We therefore get in view of (2) .
Theorem D. The function G~';(z) with m + n ~ p + 1 is an
analgtic function of z in the sector I arg z I < (m + n - p) Jl; this
function mag be represented bg the integral (28).
Now. when m
+ n ~ p + 2.
15 )
POCHHAMMER.
MACROBER"r. [17] .
(26) ;
the point (-1)
WINKLER.
(33) ;
m +R-P
lies in the interior of
MACRoBERT.•
.(16) ,
Comp.
aIso
16) The reader may compare BAIL'EY. [1). 5-6 or WHITTAKER and WATSON. [32].
286-288. where a proof for a special case is elaborated. The proof of the general case
runs along the same Hnes.
.
17 )
Comp. BAILEY. Ioc. cito or WHITrAKER and WATSON. Ioc. cit~ .
236
the sedor
of z for
I arg z I < (m + n - p ).n. So, if we restrict ourselves to values
which I arg z I < (m + n - p).n, the function G~'; (z) with
=
+ n ;;;;' p + 2 has no singularity at z (-1) m+n-p.
Hence, if m + n;;;;' p + 2 and at the same time larg z I < (m + n-p).n,
the cross-cut from (-1) m + n - p to 00 • (-1) m + n - p is superfluous.
.",m
In all other cases, in continuing the function G~'; (z) outside the circle
I, we will make a cross-cut along the real axis from (-1) m + n - p to
Iz I =
00 •
(-1) m+n_p.
Replacing s by -s in (28) we get
1 J~
2.ni
.n
J=I
p
. IJ
r(l-aj-s).ll r(b j +s)
J=I
p
r(aj s) IJ r(I-bj-s)
+
J=n+1
(1)5 ds.
_
z
j=m+1
L
On comparing this integral with (28) we find in virtue of theorem D
that it is equal to
I
n,m.(~ I-bi' ... , I-b p ) .
G~p
z
I-al, ... , I-ap
So we obtain
Theorem E. The function G';,'; (z) with m + n ;;;;, p + 1 possesses in
the sector I arg z I < (m + n - p ).n the following analytic continuation
1
outside the circle I z I
=
m,n ( Ial' ... ,a p ) _
n,m (~ I I-bi' ... , I-b p )
G p, p z. b I' ... , b p - G p,P
·
z I-al,I
...• -ap
. (29)
liD order to obtain a convenient notation for the continuation of the
function G~:~ (z ll at) I .give still another definition.
Definition 3. (Comp. Definition 1.) If pand · t are integers with
1 :s; t :s; pand if the numbers at and bI' ... , b p satisfy the condition
(j= 1• .. .• p).
then the function Ep, p (z 11 at) is defined as foUows :
Ep,p (z
11
at)
lp ( 1 I
I-bi .. . .. I-b
)
= Gp,p
.
z I-at. I-al, . .. ,I-at-I. l-at+1, . . .• l-ap
p
(30)
This function may because of (7) also be written (comp. (14))
(31)
237
the asterisk denotes, as usual, that the number 1 + at - a t is to be omitted
in the sequence 1+ al-at , " ', 1 + ap-at 18).
Now it follows from (7) . (17) and (3I), if the numbers al • . ..• an and
bI ..... b m satisfy the conditions (19) and (20).
n. m
G p•P
=
(J... \I-bI'
.. . . I-b
I
I
z
p)
-al ... . • -ap
n
~ e(m+n-p-l)nlat [:::.m . n
t=1
p
(t) E p •p (ze(P-m-n+l) n lll at).
Theorem E may therefore be stated in another way. namely
Theorem E*. (Comp. Theorem B. ) ff m + n ;;;: p + 1 and the numbers
an and bI ' ... . b m satisfy the conditions (19) and (20). then the
al • . .. •
function G';.·;(z) possesses in the sector largz l «m+n-p) n an
analytic continuation outside the unit-circle which can be expressed in the
form
6;';
(z)
=
n
L:
e(m+n-p-I)nlat [:::.m.~
(t) E p• p (ze(p-m-n+1) n l
ll at) .
(32)
t=1
The particular case with m = pand n = 1 of formula (29) (which is.
of course. also the corresponding particular case of (32)) is worth while to
mention it separately; because of theorem E and (30) we obtain:
Theorem F.
(Comp. Theorem A.)
The function G~:~ (z 11 at)
possesses in the sector 1 arg z 1 < n the following analytic continuation
outside the unit-circle
G~:1 (z
11 at)
= E p•p (z 11 at).
(33)
This formula and the particular case with m = 1 and n -:- p of (32) are
well-known. For p = 2 the formulae are proved by GAUSS. KUMMER.
GOURSAT. BARNES and VAN DER CORPUT 19). Tlhe general case has been
treated by THOMAE. WINKLER. MACRoBERT and F . C. SMITH 20 ) .
18 )
Comp. footnote 4) .
GAUSS. [10]. 220; KUMMER. [13]. 60; GoURSAT. [11J . 29; BARN ES. [5J.
146-152; VAN DER CORPUT. [.6J . Comp. also: FOR9YTH . [8J . Chapter VI ; KLEIN . [12J .
§ 10; WHITTAKER and WATSON. (32). § 14. 51. VAN DER CORPUT and WHITTAKER and
WATSON give the formula in question with some misprints.
19 )
20) THOMAE. [29J; WINKLER, [33J; MAcRoBERT. [15J; SMITH . [27J . Comp. also
MEIJER. [19J. formula (40) .