118 THEOREMS RELATING TO A GENERALISATION OF THE BESSEL-FUNCTION. 
If, however, p> 1, 
n=0 (at+pb)\(a+p*b) . . . (a+ p-**b) 
pel : ; 
is the effective representative of the product (a+b)(a+p*b) ... . to n factors. This 
corresponds to the change of n! in the Bessel coetticients into P(m+1) in the case of 
Bessel-Functions. ‘The series expansions of the products given above may be found in | 
Proc. L.M.S., series 2, vol. i. pp. 68-88. The theorem analogous to NEUMANN’S 
theorem 
Jy(a2 + b? + 2ab cos 6) = JF(a)T (2) + 2>°( —1)'J,(a)J (0) cos 86 ; : (€) 
I have investigated in a paper (Proc. L.M.S.). The function H, being used in a 
manner similar to the use of the exponential (pp. 25, 26, 27, Gray and Matthew's 
Treatise on Bessel- Functions), gives us a rather complicated extension of (é). 
