272 ON CERTAIN FORMULAS FOR DIFFERENTIATION WITH APPLICATIONS, &C. [41
where a extends from 0 to A. Hence, substituting and prefixing the summatory sign,
where X extends from 0 to oo, the formula required.
[I annex the following Note added in MS. in my copy of the Journal, and referring to the formula,
ante p. 267 ; a is written to denote X -
N.B.—It would be worth while to find the general differential coefficient of U k
from which it is easy to see that
K U k,i=l-) r K r, 0 U k-r,i
+ (-) r - e X r ,e* 2e U MJ _ 9
+ K r,r^ U k-2r,i-r-
The general term of c£ +1 U k i is
+ ( - iT, +e+1 «»+* [ - (k + i - 29 - 2 - r) U t _ r _,
which must be equal to
{ ) -”++1,0+1® u &-r-0-2,*-0-l,
therefore
A^+i, 0+i = (k + i — r — 2d — 2) K r , g+ x + i {k — r — 6) K r ^
In particular
X r +l, o -(k+i-r) K rQ =0,
K r+ i t ! -(k+i-r-2) K r> !=*(*-r)K r 0 ,
K r+l,r+l-^-r) K r>r =0,
whence
=[ft+*T
A r _ j =\r {№ + (i - r) k - £ (r -1) i} [ft + » - 2f “ 2 ,
/s, } . =[Pf,
which appears to indicate a complicated general law.
Even the verification of K x is long, thus the equation becomes
7+1 [k+i-2]»— 1 {fc 2 + (i -r -1) Tc- Jn} - (ft+i- r- 2) r {ft* + (i-r) k -\ (r-1) i} [ft + i - 2f ~ 2 = (ft - r) [ft + if,
or
r + 1 (k + i-r) {ft 2 + (i-r- 1) k - \ri) -r (k + i-r-2) {7c 2 + (i — r) k-\ (r - 1) i} = (k-r) (k + i) (k + i- 1),
which is identical, as may be most easily seen by taking first the coefficient of k 3 , and then writing k=r,
k — - i, k= - i- 1.]
