MATHEMAIICAL AX A LYSIS OF RASDOM NOISE 151 



\\'hen R (z) > we have the asymptotic expansions 



i){c — a) 



r ( ^ r(c)g-' fi . (1 - «) 



\z 



, (1 - a){2 - a){c - a)(c - a + 1) "1 



2!s- i- •••J 



P / N r(<;) r, , a(l + a 



iFi(a; c; -s) ~ — — ^^ 1 + -^ — - 



r(c — a)2" |_ l!z 



-c) 



(4B-3) 



, a(a + 1)(1 + a - c){2 + a - c) . 



212- 



] 



Many of the hypergeometric functions encountered may be expressed in 

 terms of Bessel functions of the first kind for imaginary argument. The 

 connection may be made by means of the relation^^ 



iFi ^. + ^ 2. + 1; z^ = f'r{u + l)z~V"lJ^ (4B-4) 

 together with the recurrence relations 



For example, the first recurrence relation is obtained from line 1 as follows 

 aF{a + 1; c; 2) + (a — c)F{a — 1; c; s) 



+ (c - 2a - z)F{a; c; g)= (4B-5) 



These six relations between the contiguous i^'i functions are analogous to 

 the 15 relations, given by Gauss, between the contiguous 2F1 hypergeometric 

 functions and may be derived from these by using 



{a, b; c- fj 



iFi{a; c; z) = Limit 2FA a, b; c; -) (4B-61 



A recurrence relation involving two i/'\'s of the type (4B-4) may be ob- 

 tained by replacing a by a + 1 in the relation given by row four of the table 



" G. N. Watson, "Theory of Bessel Functions" (Cambridge, 1922j, p. 191. 



