FUNDAMENTAL POWER SERIES AND THEIR DIFFERENTIAL EQUATIONS. 37 
ae oe —1 —-a-B-l1 eae 
Ce] [6 ylpren®) = [PD eh Ca) (8) ly — ehen-2-#- 
Fe] Us)yler-e-t-) = Wa lyme Tee Bla (el by— ders) (48 
fy-2-8-7] om 
[y-a- - 1) + py y -B] -F({o) [8] [y]e ae 
1 
F((-«} [8] [y- «]py-*) 
and other similar relations found by interchanging « and £. 
The expression of F([«][6][y]p"~*-*) and F([¢] [S][-y]p’"*-*-’) in the form of infinite 
F({a] [8] [y]py-*-*) 
F({e] [8] (y]py-*-*) = 
I(y—a—B)I(y) ; 
roducts analogous to is effected at once by the above relations. The 
‘ Sf Tyas) ; 
investigation of these I have given elsewhere, but note the results here as of interest in 
connection with the Fundamental Hypergeometric Equation discussed in this paper. 
Particular cases of these series are 
eye) & Felis [Pls Pe* She 
T([a+3]) [) * ON (ele 1) > BE] [e+ 1) [e+2] © 
«ag el 2,,2¢—1 3 ]°p* 
/ (2p 
~ {[e—43)+yC- FF nity t (1)! fe} (e@+1] * [2} ele era) 
ae ee: 
BESSEL’s Series :— 
Consider now a progression 
denote 
DeVoe Py by (70) 
and 
ipa eee ten OY, (= 2) 
Then the operator 
Ap, pye"D” + A{(1) — (m) — (- 2) }p,a%D" + A(n)(-12)D” = : . (45) 
operating on A, gives 
AL (77,)((m,) — (1) + {(1) = (n) — (= 2) } (m4) + (n)(- 2) Aya 
A[ (7) — (m)] [(m,) — (- mn) |A,a™) 
So that if we operate on a series y= Ane 
which is 
dey = Dillm) - (w)] [(om) - (~)]Aa™ 
(m,) =(m) and (m,) = (m+ 2) 
(m3) = (n+ 4) 
choose 
AAr gal (Mr4a) — (7)] [(7%41) — (-2)] = 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 2). 6 
