THEOREMS RELATING TO A GENERALIZATION OF BESSEL'S FUNCTION. 
E, ( - 2) Lin(2) 
= E, (~ #)Iin(2) 
fy yy Pmt3] go [2045] 
peo eo epee 6 ; 
Tin(#) = 0" S puy(2a0) 
Ti (X) = 1" Ieny( 4) 
_ the conditions for convergence being as follows : 
Case i |p| >1 E,(x) and I,,,(z) are absolutely convergent for all values of a. 
E,(x) and J[,,(x) are absolutely convergent if a< P a 
; zs 
Case li | ol 1 E,(«) and J,,,(w) are absolutely convergent for all values of «. 
PD 
E,(z) and I,,(x) are absolutely convergent if «< i s 
The series (27) is convergent for all values of p. 
It is easily deduced that 
J iny(e) = E, (ter) E,( — 020) Aem(a) - 
Sen") = FE, (ta) E,( — ta) Sin) 
403 
(27) 
From these relations some interesting expressions for various elliptic functions may be 
found. 
3. 
RELATIONS WITH ELLipric FUNCTIONS. 
By means of equation (29) we are able to write 
Im(@t)Im(@t™) _ p (, 2G et) EB, ( — it) 
Taaaae ays Re ( es (tect yes (-—tt) . 
Replacing x by u, (u=7x,/p/(p—1)), we obtain by means of result (11) 
Atn(vt) al, (ut!) oo { : "i . 
c EN = 242 2m—] 1 a Qe 2,.9m—1 
Jin Ut) J pny(ut) Ne ( COP. )( x P ) 
Using result (12), the right side of this equation may be written 
tt? p22 
Ge) end 
oT) aI 
or 
] 90242 1 2 t—2 
2% a Ee) E x _— 
eile a2) 1p 
(30) 
(31) 
(32) 
(33) 
