General form of dilations and translations of wavelet function In some papers, the dilations and translations of a wavelet function is written as follows:

$$\psi_{j,k}=\frac {1}{\sqrt {2^j}} \psi\left({2^{-j}t-k}\right),\quad\text{where $j$ and $k$ are integers}$$

However, in some literature, the above formula is written in a different way:

$$\psi_{j,k}=\ {\sqrt {2^j}} \psi\left({2^{j}x-k}\right), \quad\text{where $j$ and $k$ are integers}$$

Are these two formula equivalent?
Yes, they are equivalent, since $j\in\mathbb{Z}$. Note that your first definition becomes equal to the second when $j$ is substituded by $-j$. It is just a convention, if you consider a positive $j$ to be a up- or downscaling. 

So, when writing a paper or reading a book, it is always important to understand the definition of the transform used.