The Wikipedia article about "Conjugate Prior" has a table containing information about Likelihood Distributions with their Conjugate Priors.

In the "Continuous Likelihood" table, the last entry shows a Beta Likelihood and a formula for a Conjugate Prior, (with 3 hyperparameters: p, q, k).

Can anybody give pointers to literature on this?

I did check the key reference given in Wikipedia: "A Compendium of Conjugate Priors" by Daniel Fink. Couldn´t find it there or anywhere else.

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    $\begingroup$ This question would be clearer if you edited to include the probability distribution for which you would like to have a reference. $\endgroup$
    – Sycorax
    Jul 9, 2022 at 0:52
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    $\begingroup$ The Wikipedia conjugate prior for $(\alpha,\beta)$ of a Beta distribution namely a density $\propto \dfrac{\Gamma(\alpha+\beta)^k \, p^\alpha \, q^\beta}{\Gamma(\alpha)^k\,\Gamma(\beta)^k}$ is fairly clearly correct as each new observation multiplies this by the likelihood $\dfrac{\Gamma(\alpha+\beta) \, x^\alpha \, (1-x)^\beta}{\Gamma(\alpha)\,\Gamma(\beta)}$ i.e. adds $1$ to $k$, multiplies $p$ by $x$ and multiplies $q$ by $1-x$. $\endgroup$
    – Henry
    Jul 9, 2022 at 1:27
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    $\begingroup$ Hi: Here's another thread that has some information. stats.stackexchange.com/questions/67443/… $\endgroup$
    – mlofton
    Jul 9, 2022 at 3:13