Products of Legendre Functions. 779 



or finally, if in > n + 1, 



" 2 U !/ ' 2n + l ! 2m-2n-2 ! | m ~ n - ] 



1 2??i . 2n + 2 . 2m - 2m - 1 . 2m — 2n - 5 p 

 + 2 " 2m-1.2n + 3.2m-2n-2.2m-2»^T " ,! - n ~ 3 + 



1.3 2m .2m -2 2n + 2 . 2-n + 4 

 •2.4-' 2m -1.2m -3 ' 2rc + 3.2?? + 5 



2m- 2>z - 1 . 2m- -2w-3 2m ~2n— 9 



2m -2n— 2 . 2?n— 2n-4 2m— 2/2-1 



-£?n -B-5T"" 



If m==fi.+ l, the expression is known to be constant. 

 If m = n, it is zero. It is constant again i£ m = n — 1, 

 and if n>m+l the factorials clearly only need re-writing. 

 If expressed as Gamma functions, the same theorem is true 

 throughout. 



Series for Q 2m P„. 

 We know that 



<Juw =-22; g- g+Sga -^ = Q 2 „ lV) p. m . 



Thus, multiplying by /4, and using a recurrence formula, 



2 V^r, - 2, 2m _ 2r -l . 2m + 2r+2 



r 2r + 2. P 2 , +2 + 2r-f-l.P 2 ,- j 

 X { ~ 4r + 3 { 



= Po s - j 2r + l 



2m— 1.2m + 2 + I 2m-2r— 1 .2m + 2r + 2 



, 2r 1 



+ 2ro-2r + 1.2m + 2rJ 2n 



which, on reduction, becomes 



— 2 H2m±l 



_ y™ 4r+1.2»i — 2r 2m + 2r4-l 



° 2m - 2r- 1 . 2m + 2r + 2 .2m- 2r + l .2m + 2r 2r ' 



the term P falling naturally into the series. 



