276 
2(-~lyert n= © 008 2ant aje n=» sin Zant 
gek(t) = Rm Ja Ad gertilf) = (272 pg nt 
These series, satisfying the relation 
gt (€) = gm (6) 
represent, with the exception of g, (f), continuous periodic functions of 7. 
If O0<t<1 is, they are equal to simple polynomials in ¢, which 
are in fact the derivatives of the polynomials of BERNOULLI. Otherwise 
stated, for O<ct<c1 the functions ¢,, (¢) are determined as coefficients 
of expansion ; we have 
wert m= 0 
' dd gam (bet, 
pret es 8 
and from this it ensues, always supposing 0 << 1, 
t B, de ; he Ee fi 
= fu} Bek 5 q,(t) —-. —}4.—+—+t, etc. 
ADN HO KE ae, MN a) | 3) 
It is to be seen that every polynomial in ¢ is to be expressed 
in g-funetions for O<¢< 1. This holds true in particular for the 
product of two g-functions and such a product is reduced in the 
following way. 
From the identical equation 
val: yely ay (type oT. | etl ‘ 
BSA seal nh ader ted, ees De 1) 
aa peten @ 1 RAAR : 
aT . ss Ter +] 
ety EEA gy 
hein: 
ied Eg (te JEE = g,(0y"| = 
n= =| 
M= 1—co yen -1 2n—l 
=|! + 2 g (t)e+y)n | | ae M= 1, (0) jay. ay 
renee te | nl, Hy 
If on both sides of this equation the coefficient of zv” yk is taken 
the product git) g(t) will be found expressed linearly by g-functions 
by equating the results. 
In this way we obtain 
7,09, OAD, 94,0 HAHA), HAHA) 91, 209M 
FAHAS), (A! md RE (07, (0) 
a AD 4 (/ T bly 1, ae (B, (0) 
hk 
If 4+ is odd the last term on the second side may contain 
gt, if, however, h +k is even, half of the last term must be taken 
and the latter will therefore be 
