Summation theorems for \({_p}F_{q+1}[(\alpha_p);g\pm m, (\beta_q); z]\) via Mellin-Barnes type contour integral and its applications (Q345224)

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