GENERALISED FUNCTIONS OF LEGENDRE AND BESSEL. 25 
Then 
S » Via 2r, eae = 
in Os = are [7]! (2)-(2) nar ae 
Consider the series 
DE A@A). = pe" [0 (2), he et CONS) hese eee aac mey AC) 
of which the general term is 
s..ets—sm| | |m+1l]-+-|m+s—1 2D) seas a 
1)ip feel rs | gy JP 429 (\p') . (78) 
ewe replacemmi(s 7) Ite, Jaman 2 + +» - by the infinite series which they represent 
[2m+47r] 
and collect the terms according to powers of x, then the terms containing « @ form 
the infinite series 
ne [2m+4r] s=o ee [2m-+4r] 
m+2r, 5 pee 
Nae TEN i 25s A TET STON ae J (2) Saree _petstrs (79) 
[m +7]! [7]! (2)m4r(2), 2a [mm — 1]! (2)m—a[s]! (2),[7 — s]! (2),—.[m + 7 + JM 2) Vinee 
which reduces by (73) to 
st [2m-+47r] 
1 Ameer y PI 
[2m + 47] [2 +47 — 2] -- - [442] [vr]! [r]! (2),(2), 
which is 
a M TP? 72m 
ppl Joy (a -X) . . 2 : (80) 
I” denoting the operation reversing 
iw = a ee ea } } 
8. 
It is well known that 
mn Re Ae 
Fy In + Ion + gpg Jmve + gp cgidms eB) 
Consider the series 
F Xv Pee ee : 82 
Tin(@A) ie (2), 26 ee (x?-A) + p? a ae Bea Co ete (82) 
of which the general term is 
fra, Oa oi 
P ‘@.r] diosa X): : : : - (83) 
Jin(A) denoting 
Arr ylr+2 27] 
2 GTP OA © 
Replacing Jin(X) Jiny1)(@7A) . . . in (82) by infinite series, and collecting the terms accord- 
ing to powers of x, we have for the terms involving a*"l a oroup of r+1 terms, viz., 
AMtPeryln+2r] Arty! [22+27r] 
. Meeps) ae Om Oem} 
1% ye main tan 
Seah) sp [m+ r]![7 — s]!(2),s(2)n4r(2)s 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART T. (NO. 1). 4 
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