Piewn2cO TONS, 
I.— On Generalised Functions of Legendre and Bessel. By the Rev. F. H. 
Jackson, H.M.S, “Irresistible.” Communicated by Dr W. Prpptn. 
(Part I. received July 27 ; Part II. August 6, 1903. Read November 16, 1903. Issued separately February 5, 1904.) 
Parr I. 
i 
In this paper some properties of the functions J,,(% ,A) , Pra(@, A), Qn(v,r) will be 
investigated. These functions are generalised forms of BrssEL’s and LEGENDRE'’S 
Functions. Two interesting expressions are obtained for the sum of the coefticients 
(2) ieee 
of x in the series P,,,(x) and ¢ pa . Throughout the paper [n] denotes = ' 
Jin(x , A) denotes the convergent series 
T=0 Mr yln+2r] 
2 fr} [0 +7} Qo () 
in which 
[nm]! represents [1][2][3]... . [x] 
(2)n ” ptl-p?t+1l-pe+1.... p™tl 
if p =1 we see that 
[nm]! reduces to m! 
(2), »» 2” 
P,,(@A) denotes 
ae [20 = 27]! T.r+2) n—27,,[n—2r] 2 
2 pEatesioo.. °° =. ° © 
which when p= 1 reduces to LEGENDRE’s function P,, 
Qin(x , +) denotes 
[1 + 2r]! [m7]! (2) nse pean Bralgl—n— a ; : (3) 
—_ “(7]! [2n + 27 + 1]! (2), 
If n be not integral then [n]! must be replaced by the function U,([n]) which is 
defined as 
1} 2 3] on ct oD Oo [x] 7 atisres 
BAe) = Lalit ee [oe eee gel! : : (4) 
TRANS. ROY, SOC, EDIN., VOL, XLI PART I. (NO. 1). 1 
< 
