If we consider a simple Random Walk on the positive integers (discrete Markov chain), with symmetric transition probabilities. We start at time $0$ at the integer $i_0 = m$ and at each time step $P(i_{t+1}=i_{t}+1) = P(i_{t+1}=i_{t}-1) = .5$. The First Passage Time Density (FPTD) is the probability that we first reach the integer $1$ at time t. I am looking for a close form formula for any $t$ of the First passage time density of $i_t$ in the integer $1$ (the density of the hitting time of integer $1$). Note that there are no boundary or reflection possible for big integers. I am specifically interested in the case when $m\neq 1$ (not the first return problem). Does a close from formula exists for First passage time density in this case or maybe even an upper bound on it in order to get the rate of it?

In Wikipedia I see a result for continuous time that makes the Levy distribution intervene. I am looking for a similar result in discrete time and discrete Markov chain with no boundary. https://en.wikipedia.org/wiki/First-hitting-time_model

Thanks for any help!