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IJ.—Certain Fundamental Power Series and their Differential Equations. By the 
Rev. F. H, Jackson, H.M.S. “ Irresistible.” Communicated by Dr W. Prpptig. 
(MS. received December 7, 1903. Read January 4, 1904. Issued separately February 16, 1904.) 
The series which will be discussed in this paper are of the general type 
Awl + A+ oo, Beier gree PTs ane) 
Consider a sequence of elements 1, P2,3,.......--. then (m,) will denote the sum 
of the first m, elements of the sequence. The simplest connection between the terms 
of the sequence is equality 
Py =Po=P3 = bo Oo Oo b SJ oO 9 
The series expansions of the ordinary functions of analysis and the series which are 
solutions of ordinary linear differential equations belong to this simplest type. Another 
more general type of series is formed when the elements p,,p.,p,.... are in 
geometrical progression as @,ap,ap,..... In this case the index (n) denotes 
Gap Oper ..... +ap"", which is a” = the limitation of n to positive 
Jr 
integral values (where the sequence is geometrical) may be removed, as will be seen in 
the particular discussion of the functions J (A), Pin(AL), QA), F(Le] [6] [yPx). If p 
be made unity, the geometrical progression becomes a progression of equal elements 
and the properties of the general functions reduce to analogous properties of the simpler 
functions F(a8ya) ; P(x); Q,(x), ete. HKuLER’s expansion 
+90 
(1 —«)(1 — 2?)(1 — 2%) ad inf. = >\(-1)"aie™ 
(je <= Il) 
and GAUSS’S series 
(1—2*)(1-a*)\(1-2°)... ad MS as 
(fa) =23)(S eee adi. — 
(Ga <2 1). 
are particular cases of the general series (1) for the sequences 1,1,3,2,5... 
20 eee and 1, 2,3,4, ... , respectively. 
The fundamental Hypergeometric Series is 
Pa] 4 Witatg ee ee Gs 41) Oy (Cantal) Neat cals pap stele a(a,+(1)) OSC pee 9 
(0%: Ces ere Bi. ” (1)(2)8,(B, + (@ — )) cea BAB. +(2)- (1) @) 
In the case when there are only three elements «6 y in the Hypergeometric Series and 
the sequence of p.s is 1,3,5,....,2n+1,.. . the following series are interesting 
cases : . 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 2). 5 
