The modified bessel function of the second kind is the function $K_n(z)$ which is one of the solutions to the modified Bessel differential equation.
Using Wolframalpha, we get $$K_{\frac{3}{2}}(z)= \sqrt{\frac{π}{2}} \frac{e^{-z} (1 + 1/z)}{\sqrt{z}} \qquad (*)$$
I'm looking for a reference (book or article) which I can find $(*)$.
Thank you in advance
For large values of $z$, the asymptotics are given by $$\sqrt{\frac{2z}{\pi }}\, e^z\, K_n(z)=1+\frac{4 n^2-1}{8 z}+\frac{16 n^4-40 n^2+9}{128 z^2}+\frac{64 n^6-560 n^4+1036 n^2-225}{3072 z^3}+O\left(\frac{1}{z^4}\right)$$
Now, notice that $$16 n^4-40 n^2+9=16\left(n-\frac 32\right)\left(n-\frac 12\right)\left(n+\frac 12\right)\left(n+\frac 32\right)$$ $$64 n^6-560 n^4+1036 n^2-225=64\left(n-\frac 52\right)\left(n-\frac 32\right)\left(n-\frac 12\right)\left(n+\frac 12\right)\left(n+\frac 32\right)\left(n+\frac 52\right)$$ and for the particular case of $n=\frac 32$ the rhs reduces exactly to $1+\frac 1 z$. In fact, if $n=k+\frac 12$, you have simple and nice forms (as asymptotics - no $O(.)$)
$$\left( \begin{array}{cc} k & \sqrt{\frac{2z}{\pi }}\, e^z\, K_{k+\frac 12}(z) \\ 0 & 1 \\ 1 & 1+\frac{1}{z} \\ 2 & 1+\frac{3}{z}+\frac{3}{z^2} \\ 3 & 1+\frac{6}{z}+\frac{15}{z^2}+\frac{15}{z^3} \\ 4 & 1+\frac{10}{z}+\frac{45}{z^2}+\frac{105}{z^3}+\frac{105}{z^4} \\ 5 & 1+\frac{15}{z}+\frac{105}{z^2}+\frac{420}{z^3}+\frac{945}{z^4}+\frac{945}{z^5} \\ 6 & 1+\frac{21}{z}+\frac{210}{z^2}+\frac{1260}{z^3}+\frac{4725}{z^4}+\frac{10395}{z^5}+\frac{10395}{z^6} \end{array} \right)$$