Rit (Vísindafélag Íslendinga) - 01.06.1951, Page 31

31
[3]
or
1
(p('l+am)J" = j ' v > |y~x) A^/<«cp(i)) (ö)
n=o n=o
X
Further
—>

cpOi+am)*" - ~Y~fc<r i J ?7/<V(h))
n — o /i — o
(7)
(6) is the wellknown Euler transformation of the series
cpCii+am^x'1
n = n
while (7) is a more generalized form of the same trans-
formation. We have derived it from the C- transform of the
basal function. I like to point out that we cannot conclude
from (6) that the original series becomes infinitely large for
jc= 1. However, if cp(ri) is an entire polynomial in rj, (6N
turns out as a finite series and we can then savely con-
clude tiiat the value of the series becomes infinitely great
for x=l. Likewise we see from (7) that the series becomes
infinitely large for x=\/k, if /e ~77/0>cp(ii) is an entire poly-
nomial in tj. Another important transformation is obtained
by means of A3 in that we recall that 1 = 1 —- ,
cp (iid-w^)X'1
n = o
—»
— cp (r)—con)
n — 1
1
Xn
(8)
1 find this transformation very helpful on many occasions,
however, some precaution is advisable when using it.
2. The series
dn cp(i))
rfn"
n — n
d'ly(u) xn
dun n!
(9)