112 BELL SYSTEM TECHNICAL JOURNAL 



The series expansions required here could also be obtained from the appropri- 

 ate series for Bessel functions. It will be noted, however, that the typical 

 modulation coefficient can be expressed in terms of either a single iFi function 

 or several Bessel functions, so that manipulations must be performed on the 

 series for the latter to give the final result. The Bessel functions on the 

 other hand are more convenient for numerical computations because of the 

 excellent tables available. 



Reduction formulas for certain other h}^ergeometric functions are needed 

 in evaluating the higher order products. They are: 



iFi(3/2; 1; -2) = e-'"[{l - z)h{z/2) + h{z/2)] (66) 



iF^{Z/2- 2; -2) = e-'"[h{z/2) - Uz/2)] (67) 



iFi(5/2; 4; -2) = \ e-'" \^^ + 1)71(2/2) - 7o(z/2)] (68) 



Derivation of these is facilitated by the use of the easily demonstrated 

 relations: 



iFi(a; 1; -2) = -^ [2iF:(a; 2; -2)] (69) 



az 



2ziFi(a; 2; -2) = ^ [z\Fiia; 3; -2)] (70) 



az 



iFa(3/2; 3; - 2) - :F:(3/2; 2; -2) = | iFx(5/2; 4; -z) (71) 



APPENDIX III 



HIGHER ORDER PRODUCTS 



The methods described in Section II may be applied to calculate the gen- 

 eral expression for the general modulation coefficient. The result is for the 

 amplitude of the term cos mpot cos pnj cos pnJ • • • cos pnu^' 



OT+Af 



^ (-) ^ ^ Pn.Pn, ■" Pn^ ^ / ^ + M - l \ {W sT" 



-KiWnny^-'^i^'ml \ 2 ) {Wn) (72) 



^ ^ (m + M - \ ,. -W\ 



X.F:(^ -2 '" + ^'177; 



The coefficient of the term cos {mpa ± ^„i ± Z?,,, ± ... pnj^) ^ is amM divided 

 by 2"~^ em • The number of terms of a particular t>TDe falling in a particular 

 frequency interval can be calculated by a method previously described by 



