........We arrive at the following results which provide the sine and cosine transforms of the H-function
$$int_{0}^{infty}x^{ ho}sin (ax)H_{p,q}^{m,n}igg[bx^{sigma}Bigg|{}_{(b_{q},B_{q})}^{(a_{p},A_{p})}Bigg]dx=frac{2^{ ho-1}sqrt{pi}}{a^{ ho}}H_{p+2,q}^{m,n+1}Bigg[bleft(frac{2}{a} ight)^{sigma}Bigg|{}_{(b_{q},B_{q})}^{(frac{1- ho}{2},frac{ ho}{2}),(a_{p},A_{p}),(frac{2- ho}{2},frac{ ho}{2}}Bigg]$$
where $a,alpha,sigma>0, ho,bin C,|arg b|<frac{pi alpha}{2}$,
$$Re( ho)+sigma min_{1leq jleq m}Releft(frac{b_{j}}{B_{j}} ight)>-1;Re( ho)+sigmamax_{1leq jleq n}frac{a_{j}-1}{A_{j}}<1$$
And
$$int_{0}^{infty}x^{ ho}cos (ax)H_{p,q}^{m,n}igg[bx^{sigma}Bigg|{}_{(b_{q},B_{q})}^{(a_{p},A_{p})}Bigg]dx=frac{2^{ ho-1}sqrt{pi}}{a^{ ho}}H_{p+2,q}^{m,n+1}Bigg[bleft(frac{2}{a} ight)^{sigma}Bigg|{}_{(b_{q},B_{q})}^{(frac{2- ho}{2},frac{ ho}{2}),(a_{p},A_{p}),(frac{1- ho}{2},frac{ ho}{2})}Bigg]$$
where $a,alpha,sigma>0, ho,bin C,|arg b|<frac{pi alpha}{2}$,
$$Re( ho)+sigma min_{1leq jleq m}Releft(frac{b_{j}}{B_{j}} ight)>0;Re( ho)+sigmamax_{1leq jleq n}frac{a_{j}-1}{A_{j}}<1$$
Specially,
$$E_{alpha,eta}(z)=sum_{k=0}^{infty}frac{z^{k}}{Gamma(alpha z+eta)}=H_{1,2}^{1,1}Bigg[-zBigg|_{(0,1),(1-eta,alpha)}^{(0,1)}Bigg]$$
Set $z=-a x^{2}$, we have
$$E_{alpha,eta}(-a x^{2})=H_{1,2}^{1,1}Bigg[a x^{2}Bigg|_{(0,1),(1-eta,alpha)}^{(0,1)}Bigg]$$
Thus,
$$int_{0}^{infty}cos (kx)E_{alpha,eta}(-ax^{2})dx=frac{pi}{k}H_{1,1}^{1,0}Bigg[frac{k^{2}}{a}Bigg|_{(1,2)}^{(eta,alpha)}Bigg]$$