=40~

Expansion of a product in which the a's form a

geogetric sequence; the O-funciion,

Let II = II(1 + a,x) where the o's form a geometric sequence
ar, ar®, ar®, *+++, in which the ratio r may be a complex number®®
but such that |r| < 1. This product can be written II(1 4 orx) ,

and it is absolutely~uniformly convergent, For this product,

 

ar
Sy=ar 4+ ar? 4+ ar® 4 cecees o ;
ler
28
e a’y
sp=a’r? 4 a4 %O 4 cereer = AT
1=-9

O BIIR L TN R, e e e T 8

: n
Sp = a4 a"r® 4 afSh g eeees o T e

-r

Using equations (12) the coefficients of the series are

O 2
O el ¢ =_-i:._-n_:___. secos
R T R Y R

3 .
a"r r? . N

P r g
and IT = 1 + 3Cua™ .

If x in the above product is replaced by eXx' then x" in
the series is replaced by e”A’ and the coefficients Cp remain
unchanged, This product can be written TI{1 + ar® eX’] sand, it dis
absolutely-uniformly convergent if |r| < 1, This permits us to
apply the formulas just given, to another class of preducts,

As an example, expand the generalized C-function of Heine.®?

; Dot e
ng i, 2niz b
(agw; - ang)m_ _— —
Glgyis TTH I 0e e 2 ¢ Wy e Wy Y

nt 2Ny
8(“1‘*’1 ~ agllp )=

 

2ni
sy r=e 1 apd A==2
Wy

In this product a=-

~

et ettt .

80 Goursat, sec, 8,

 

———ray
31 Thie generalization was develored by Professor C. H. Ashton of the

University of Kansas. See list of references.