# Hypergeometric Properties

 P: 607 Using Maple, I get ${{}_2F_1(1/2,b;\,3/2;\,-{x}^{2})}-{{}_2F_1(3/2,b+1;\,5/2;\,-{x}^{2})} = \sum _{k=0}^{\infty }{\frac { \left( -1 \right) ^{k}\Gamma \left( b+k \right) {x}^{2\,k}}{\Gamma \left( b \right) \Gamma \left( k+1 \right) \left( 2\,k+1 \right) }}-\sum _{k=0}^{\infty }3\,{\frac { \left( -1 \right) ^{k}\Gamma \left( 1+b+k \right) {x}^{2\,k}}{\Gamma \left( b+1 \right) \Gamma \left( k+1 \right) \left( 2\,k+3 \right) }}$ $=\sum _{k=0}^{\infty }-{\frac { \left( -1 \right) ^{k}{x}^{2\,k} \left( 4\,b+6\,k+3 \right) \Gamma \left( b+k \right) }{ \left( 2\,k+ 3 \right) \left( 2\,k+1 \right) \Gamma \left( b+1 \right) \Gamma \left( k \right) }} =1/15\,{x}^{2} \left( 4\,b+9 \right) {{}_3F_2(3/2,b+1,2/3\,b+5/2;\,7/2,2/3\,b+3/2;\,-{x}^{2})}$
 Quote by g_edgar Using Maple, I get ${{}_2F_1(1/2,b;\,3/2;\,-{x}^{2})}-{{}_2F_1(3/2,b+1;\,5/2;\,-{x}^{2})} = \sum _{k=0}^{\infty }{\frac { \left( -1 \right) ^{k}\Gamma \left( b+k \right) {x}^{2\,k}}{\Gamma \left( b \right) \Gamma \left( k+1 \right) \left( 2\,k+1 \right) }}-\sum _{k=0}^{\infty }3\,{\frac { \left( -1 \right) ^{k}\Gamma \left( 1+b+k \right) {x}^{2\,k}}{\Gamma \left( b+1 \right) \Gamma \left( k+1 \right) \left( 2\,k+3 \right) }}$ $=\sum _{k=0}^{\infty }-{\frac { \left( -1 \right) ^{k}{x}^{2\,k} \left( 4\,b+6\,k+3 \right) \Gamma \left( b+k \right) }{ \left( 2\,k+ 3 \right) \left( 2\,k+1 \right) \Gamma \left( b+1 \right) \Gamma \left( k \right) }} =1/15\,{x}^{2} \left( 4\,b+9 \right) {{}_3F_2(3/2,b+1,2/3\,b+5/2;\,7/2,2/3\,b+3/2;\,-{x}^{2})}$