THEOREMS RELATING TO A GENERALIZATION OF BESSEL’S FUNCTION. 407 
5. 
In this section of the paper I propose to state briefly some results which may 
be deduced by means of (53), (28), (29), 
E, (ia) E,( — 1) | 
2 ot 
= 1 - ed 2 w — 1 aa) ree 
il Pp) ae p A il er ?) | 
= Tio Pe)Iin(e) + 2pTuy( px) I(x) + WT y(pe)Jp(x)+...... 
Indicating the nature of the base of each function by an index, we write 
Pp x ; go a fe te 
Tul a vl Gar a =") a a> [m] ‘Tom 7 hee + =) 
whence by (29) 
Arete asi aeTt =) ee > Jom * Som = n= BS ~ + )B( = ois ct 
by (29) ga) eat) er) et) 
by (4) and 12) i‘ B,(# ) F(z aes mi vl =) 
From (19) we find 
(= a — 2p) { lay + ee aN a i) j 
ete 
i 1,72 
noge" Ti Zoy + 2 yal Spans 
l-p p-1 ( m2 ) px? ~ ee ee 
pay oak \y (P \+2 = 1)", \I (=. ) 
tag Dey aes Sia is oli, mT —p 
B, (as (1402) 
je fe “6 2 
is =To( hol = = 2>°( =p el 5 AP) 
1 : 
=osea I+ 2cos 26Iy;+... t { Lin — 2p cos 260+ . . : 
2 8 
{ Ju f =| Spon p Tapa <a ok 
ale 
dof = [ayo +2 Hep 2 tp NS rind: + + + 
é. ; A |fat 2 | 47 2 
E, (ix)E,( — iz) = {Ju} the} eee +p | Sea} 2 eae 
: Sg? voile] fy 1? 
Ki meee) ERK U0) = { Aro) ff Mp} 1 In| See vst rs +p" "12 1 Ae } Arties Greco Manne 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 17). 
=)+ + (= 1 p"(2" + 2"), (#)} ie 
60 
(59) 
(60) 
(62) 
(63) 
(64) 
(65) 
(66) 
(67) 
(68) 
(69) 
(70) 
