AN EXPANSION FOR LAPLACIAN INTEGRALS 



571 



Au+c-l-k)t 



From these equations we deri\e 

 <i>u,{x)(dxldt) = e-'{e' - \y-^ 



5 = c — 1 + m. 

 Therefore (see any work on finite dififerences), 



Ac-i+m = A<=-i(w)<^-i+'" = (- l)'"A<^-i(- IV - c -{■ 1)'-^+'". 



Now 



we have 

 or 



= {w -^ c - l)A'-'{uy-'+^ + {c - l)A'-2(7£;)<=-2+"'+i. 

 Increasing c by 1, decreasing m by 1 and setting 

 Ac-i+m = (c - l)lA{m, c - 1), 



A{m, c) — A{m, c — 1) = (w + c)A{m — 1, c) 



AcA(m, c — 1) = {w + c)A{m — 1, c) 

 where the subscript c implies that Ac operates on c. Assume that 



Mm, c - i) = tQ(n,, k) d\Y 

 where the coefficients Q(jn, k) are functions of -tv; then 



AMm+l. c-l)= toini. k){ti'+c) ('J'_^^'^ 



\m-\~k-{-l/ \m-\-k 



= L[(w + ^)(2(m, k-\) + {w^k)Q{m, k)'][^J_^^y 



from which, by finite integration with reference to c, we obtain 



'"+1 fr—\A-ni-\-\ 



A{m + \,c-\)=j:[_{m + k)Q{m,k-\) + {w-^k)Q{m,k)-][^ „,+ l+I 



so that 



Q{m + 1,0) = ivQim, 0), 



Q{m + \,k) = (m + k)Q{m, k - \) + {iv + k)Q{ni, k), 



Q < k < m + \, 

 Q{m + 1, w + 1) = {2m + l)Q{m, m). 



Since 



c - \ 



i = 

 t=0 



^(0, c - 1: 



1 = 







