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I've heard of Euler Gamma function: $\Gamma(x)$, and Euler's beta function: $\text{B}(x,y)$. Did Euler have an alpha function?

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If Euler had named the functions himself, perhaps one might expect him to start with alpha and then go on to beta and gamma. But it is not Euler who is responsible for these functions' names. From Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics:

BETA and GAMMA FUNCTIONS. These terms derive from the symbols $\text B$ and $\Gamma$ used to denote the functions that Adrien Marie Legendre (1752-1833) called the Eulerian integral of the first kind and second kind. Legendre introduced the symbol $\Gamma$ and Binet introduced the symbol $\text B$.

The linked page "Earliest Uses of Function Symbols" has some amusing speculation(?) on why those letters were chosen.

In any case, there is no reason to expect that someone took another function of Euler's and named it alpha just to complete the pattern. Although such things have happened in theoretical physics...

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