Summation theorems for \({_{p+1}}F_{q}[a\pm m, (\alpha_p); (\beta_q); z]\) via Mellin-Barnes type contour integral and its applications (Q527351)

Summary: In this paper, we find summation theorems for generalized hypergeometric functions \({_{p+1}}F_{q}[a\pm m, (\alpha_p); (\beta_q);\;z]\) by means of Mellin-Barnes type contour integral representation. Some closed forms (believed to be new) of summation theorems for Clausen hypergeometric functions \(_3F_2[a+m,b,c;a,1+b-c\pm p; -1] \), \(_3F_2[a+m,b,c;a,b-c\pm p; -1] \), \(_3F_2[a+m,b,c;a, (1+b+c\pm p)/2; 1/2] \), \(_3F_2[a+m,b,c;a, (b+c\pm p)/2; 1/2] \), \(_3F_2[a+m,b,1-b \pm p;a,h; 1/2] \), \(_3F_2[a+m,b,-b \pm p;a,h; 1/2] \), \(_3F_2[a+m,b,c;a,c+1; 1] \) and Gauss hypergeometric functions \(_2F_1[a+m,b;1+a-b\pm p; -1] \), \(_2F_1[a+m,b;a-b\pm p; -1] \), \(_2F_1[a+m,b;(1+a+b\pm p)/2; 1/2] \), \(_2F_1[a+m,b;(a+b\pm p)/2; 1/2] \), \(_2F_1[a+m,1-a\pm p;g; 1/2] \), \(_2F_1[a+m,-a\pm p;g; 1/2] \) with suitable convergence conditions, are also obtained.