The Laplace transform of the Elephant Random Walk grows as an explicit φ(a;x)^n times a prefactor that is 1 for a<0 and 2(1-q)/(a+1) or 2q/(a+1) for a>0.
Elephants Can Always Remember: Exact Long-Range Memory Effects in a Non-Markovian Random Walk
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Asymptotics for the Laplace transform of the Elephant Random Walk via Schwarz-Christoffel mappings
The Laplace transform of the Elephant Random Walk grows as an explicit φ(a;x)^n times a prefactor that is 1 for a<0 and 2(1-q)/(a+1) or 2q/(a+1) for a>0.