1917-18.] Reduction-formula for Functions of Second Kind. 67 
For v = i, we have ordinary Legendrian functions; the first formula 
becomes 
Q«(*) = - + /+’ + Lh+O- 
1 — Z z z( 1 — Z L ) Z 
0+1 
As Q x {z) is \z log 
0-1 
:(1 — z 2 ) 
— 1, this can be written 
Q-W = + jp n (z) log , 
which is the formula we spoke of at the beginning of this paper. 
Professor Whittaker having suggested that the function B n might be 
put under a determinantal form, I have found for B n v the following ex- 
pression as a determinant with (n— 1) rows: 
( 2 v+l )z + ( 2 v+l) 0 0 0 . .. 
2 ( 2 v + 4 )z 2v + 2 0 0 ... 
0 3 (2 v + Q)z 2i/ + 3 0... 
0 0 4 (2j/ + 8)2 2v + 4 
Bn v = 
1 
2v(2v+ 1 ) . . . (2v + n— 1 ) 
This expression includes as a particular case the determinantal form for 
the function B n connected with the Legendrian polynomials. 
§ 5. Laguerre’s Polynomials. 
The polynomial L n of Laguerre * is defined by 
L “( z ) =e ~^( zV ) 
and satisfies the differential equation 
zLn(z) + (z+ 1 )L n '(z) - nL n (z) = 0 . 
The function of the second kind connected with this polynomial is 
«->£iirar 
We do not develop this case, which is extremely simple. A recurrence- 
formula can readily be obtained between the B’s and L’s ; it may be proved 
that Ln(0) — Til, and B»(0) = — ; , ; and we obtain 
n\ 
[n !] 2 M W (^) = e- z Ii n -i(z) + L n (z)M 0 (z) 
where II n _ 1 (0) is the polynomial 
i[L n (z) - (n\yB n (z)] ; 
* Laguerre, (Euvres , 1 , p. 428. Appell and Lambert, op. cit ., p. 236. These polynomials 
can be expressed in terms of the confluent hypergeometric function m . 
