Part TII,

Applications of the formulas to the expansion of
special types of products,

Expansions of products having binomial or trinomial factors,

Let TI[1 + £,2" be expanded into series. This product is of
the general type II[1+ ayx+ byx® 4 cyx®+ ***** ] where only one ele-
mwent in each sequence f{ag}, {b;}, {eq}, **** is. not zero. Written in
full  TI00 + 60a™ = (14 bix 4.0 4 =+-1[1 40 4 £gx® 40 4 *+Joovees ,

A product of this kind is easily expanded by computing the C's in
terms of the #'s. We have #i1 =1y, #o; = ta, Poo1 = te, P21 = tyt,,

#1021 = t4tg, °**°* . All the p's are zero which have a subscript
different from zero or one, Using equaticns (12)

II1 + 55" = 1 4 £yx 4 £0x® 4 (Bytg + £5)2% 4 (Eybg + 0)5% 4 Loty +tytq4tg)ePoee,

As a special case, in II[1+x"] each t = 1, and

TI[1 +x™) =1 + x4 2% + 2x® 4 2x* 4 3a® 4 4x% + *veees
As anotber case, in II[1 -x®) each ¢t = -1, and
I[1=aP) =1 ~x=x24 %427 coceee

The product of the type II[1 + £,4**] can be expanded by
changing x to x® in the preceding,

Writing out the product II[1 + topeq x*3] = :
" 4tx 4040+ I1+ 04C+2ax® 40+ 1140404040+ #52° 4 *+7) o+
we see that the only #'s which are not zero are of the form fiojongses
where 4, j, k, **° are only zero or one. Using (12)

TT[1 4 tonma x ¥ = 14 tax + tex” 4 tytex® 4 53° 4 atex® 4 tox" 4 (Egtydtyt,)a®
When all the ¢'s are equal, the preceding expensions are
TI[1 % 8a™ = 1% txg 6x® 4 (822 £)x® 4 (823 )2 + (224 t)a® 4 oove g
TH1 2 ¢x®™) =1 2 ta® 4 tx* 4 (124 2)2€ 4 (82 3 #)2® 4 *o0ee
TH1 2 ¢x®3) = 1 4 tx 4 6x® 4 £%2% 1 25 4 t%%% 4 £x7 4 2508 4 evvee

Now let II[1 + gux" 4 rax®®] be expanded into series. This

product is of the form II[1 + ap® + b,;xB + e, 4 *0*¢ 1 where a, = 0
~33=