Efficiently extendable hash function?

I'm wondering whether there exist any good hash functions with the following property: Assume that $x$ is some string over some alphabet $A$, then given $H(x)$ we can compute in $O(1)$ time both $H(ax)$ and $H(xa)$ for any letter $a\in A$. In practice one can assume that $A$ is for example the set of $8$-bit integers.

In other words, a hash function for strings that can quickly be extended in both directions. I'm only interested in hash functions that actually distribute the data well and which are very fast to compute in practice.

Use any rolling hash, e.g., the Rabin-karp rolling hash, Buzhash, or CRC. See also the following resources:

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additive hash function

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Can ropes (AVL trees) be interned?

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https://crypto.stackexchange.com/q/17935/351

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https://crypto.stackexchange.com/q/11420/351

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the answers to https://crypto.stackexchange.com/q/1558/351

StackExchange Computer Science Q#118265, answer score: 2