276 THE REV. F. H. JACKSON 



§ 2. Special ^-Difference Equations. 



Three linear differential equations whose solutions enter into many interesting 

 relations with one another are 



x*y" + xy' + (x 2 -'n*)y = 0, . . . , . (79) 



vty"" + 6.<V " + (7 - 2m 2 - 2n 2 )xV + ( 1 - 2m 2 - 2n 2 )xy l + (x 2 - (m 2 - n 2 f)y = , . (80) 



xY" + 6*V" + (7 - 2??i 2 - 2n 2 + 4x 2 )xV ' + ( 1 - 2m 2 - 2w 2 )x// 1 - (to 2 - rc 2 ) 2 y = . . (81) 



By means of (74) it is not difficult to construct the corresponding A-equations : 

 among the solutions of — 



(79) is Bessel's Function J„(x) , 



of (80) is J,„ . n (x) = £ ( - I ) 2 r(?n +n + r+ ^ T —— i - i)T{n + r+l)T{j . + l )2 ,„ +M+2 ,. , 



f«m TM WWV T{m + n + 2r+\) x m+n+tr 



ox (M) ' ) >" { - t > x ' ) "\ x >- 2sT(m+7i + r+l)T(m + r + l)T(n + r+\) T(r + l)2" , + n+ip ' 



The analogous A -equations are 



r/x 2 A 2 ?/ + (l -[«]-[- w])xAy + (x 2 -[ra][ -«])y = 0, . . . (82) 



7 G x 4 A 4 // + r/([l] + [2] + [3] + [to + n] + [to - re] + [re - m] + [ - m - n])x s A?y 



+ cx 2 A 2 y + bxAy + x 2 y + ay = 0, (83) 



</ 6 x 4 A 4 */ + <f([l] + [2] + [3] + [m + n] + [to - re] + [re - m] + [ - m - n].c s A 3 y 



+ (ex 2 + [2] 2 x 4 )A 2 y + bx\y + ay = , (84) 



in which a = \_m + n][?n - n][n - m][ - m - re] , 



b — [to + 7i+ 1][to -7i + l][re - to + 1][ - m - n + 1] - [to + ?j][to - re][re - to][ - w - »] , 



[?;* + re + 2] [ - m - ?2 + 21 r ,-, r , n , T?" + «1 • • • • r - m - rel 



c = L J „T^ J-[m + n+l] .... [-TO-n+l] + f/ J rg-., L — ', 



which, when g , = l, reduce to the differential equations (79), (80), (81) respectively. 

 We solve these A-equations by substituting a series ^c s x a+2s and, in the case of 

 equation (82), obtain an indicial equation 



[a + n][a - re] = , 

 and an indicial function 



y =6 '° { *" ' [a + re+2][a-re + 2] + } ' 



The principal roots of the indicial equation are +n , — n. From these we obtain two 

 principal solutions, which are not distinct when n = ; also in case n is an integer, one 

 of the solutions becomes formally infinite. By a suitable choice of the arbitrary 

 constant we write down the solution corresponding to the root n, as 



'-SK-' V+^wi "^^ <85) 



{2?*}! = [2][4][6] . . . [2r] in case r is a positive integer. The form of {2n}\ 

 when n is not integral, I have discussed in Proc. L.M.S., ser. 2, vol. ii. p. 195. The 

 cases mentioned above, when one solution becomes formally infinite, or when the two 

 solutions are not distinct, may be treated in the manner usual in finding solutions for 

 differentia] equations in similar cases. The following is an example : — 



