Is there any approximation or solution for this integral? The independent integral for both functions is possible while combining them I couldn’t find any solution. So, please help me to solve this. $$\int_0^\infty K_0 (2\sqrt v) \exp\left(\frac{(s+v)^2}{t}\right) \Bbb dv$$
Hint: Maybe it is possible to approximate these functions before integrating, for example, the exponential function. Is there any approximate form of $\exp{(x^2+x)}$ in the form of polynomials?
Hoping that there is no typo, I think that the problem comes from $s$. But, even if $s=0$, the result is quite complicated since $$\int _0^{\infty } {K_0\left(2 \sqrt{v}\right)\, e^{\frac{v^2}{t}}} \, dv=\frac 1{2 \pi}\,\,G_{1,4}^{4,1}\left(\frac{t}{16}| \begin{array}{c} 1 \\ \frac{1}{2},\frac{1}{2},1,1 \end{array} \right)$$ where appears the Meijer G function.
May be, an idea would to expand $$e^{\frac{(s+v)^2}{t}}= e^{\frac{v^2}{t}}\sum_{n=0}^\infty a_n s^n$$ where the coefficients are defined by $$a_0=1 \qquad a_1=\frac{2v}t\qquad a_n=\frac 2{n\,t} \left(v\, a_{n-1} +a_{n-2}\right) $$to face a linear combination of integrals
$$I_n=\int _0^{\infty } {K_0\left(2 \sqrt{v}\right)\, e^{\frac{v^2}{t}}} \,v^n\, dv=\frac{2^{2 n-1}}{\pi }\,\,G_{1,4}^{4,1}\left(\frac{t}{16}| \begin{array}{c} 1 \\ \frac{n+1}{2},\frac{n+1}{2},\frac{n+2}{2},\frac{n+2}{2} \end{array} \right)$$