Generalized Watson's Summation Formula For ...

. A summation formula is given for 3 F2(a; b; c; (a + b + i + 1)=2; 2c + j; 1) with fixed j and arbitrary i (i; j 2 Z). This result generalizes the classical Watson's theorem which deals with the case i = j = 0. Extensions to the cases of 3F2(a; 1 + i + j \Gamma a; c; e; 1 + i + 2c \Gamma e; 1), and 3F2(a; b; c; 1 + i + a \Gamma b; 1 + i + j + a \Gamma c; 1) are given. Notice that the case i = j = 0 corresponds to the classical theorems due to Whipple and Dixon, respectively. J. Comput. Appl. Math. 86 (1997), 375--386 1. Introduction The classical Watson's summation theorem may be written as 3 F 2 ` a; b; c d; 2c fi fi fi fi 1 ' = 2 a+b\Gamma2 \Gamma( 1 2 a)\Gamma( 1 2 b)\Gamma(d)\Gamma(c \Gamma d + 1)\Gamma(c + 1 2 ) \Gamma( 1 2 )\Gamma(a)\Gamma(b)\Gamma(c \Gamma 1 2 a + 1 2 )\Gamma(c \Gamma 1 2 b + 1 2 ) (1.1) (d = (a + b + 1)=2); provided !(2c \Gamma a \Gamma b) ? \Gamma1 (see, e. g., [3, Section 4.4], or [9, Section 5.2.4], where the duplication formula fo..