Might be useful - an expression in terms of the incomplete Gamma function. The series $$ {}_1F_2(1;a,a+\frac12;-x^2)=1-\frac{x^2}{2a(2a+1)}+\frac{x^4}{2a(2a+1)(2a+2)(2a+3)}-\frac{x^6}{2a(2a+1)(2a+2)(2a+3)(2a+4)(2a+5)}+... $$ can be written as $$ \frac12\left(f(2ix)+f(-2ix)\right), $$ where $$ f(t)=1+\frac t{2a}+\frac{t^2}{2a(2a+1)}+\frac{t^3}{2a(2a+1)(2a+2)}+...=\frac{2a-1}{t^{2a-1}}e^t(\Gamma(2a-1)-\Gamma(2a-1,t)) $$

Might be useful - an expression in terms of the incomplete Gamma function. The series $$ {}_1F_2(1;a,a+\frac12;-x^2)=1-\frac{4x^2}{2a(2a+1)}+\frac{16x^4}{2a(2a+1)(2a+2)(2a+3)}-\frac{64x^6}{2a(2a+1)(2a+2)(2a+3)(2a+4)(2a+5)}+... $$ can be written as $$ \frac12\left(f(2ix)+f(-2ix)\right), $$ where $$ f(t)=1+\frac t{2a}+\frac{t^2}{2a(2a+1)}+\frac{t^3}{2a(2a+1)(2a+2)}+...=\frac{2a-1}{t^{2a-1}}e^t(\Gamma(2a-1)-\Gamma(2a-1,t)) $$