My question is on the notation of the Digamma function.
The Factorial function $n!$ (which is met in secondary school), is conceptually seminal to the Digamma function. The Factorial function is defined as: $$0!=1,\qquad n!=\frac{(n+1)!}{n+1}$$
This concept is extended with Gauss's Pi function: $$\operatorname\Pi(z)=\frac{\operatorname\Pi(z+1)}{z+1}$$ and with a simple unit offset (see here), the more familiar Euler Gamma function: $$\operatorname\Gamma(z)=\frac{\operatorname\Gamma(z+1)}z$$
The Digamma function is a further extension: $$\psi(z)=\dfrac{\operatorname\Gamma'(z)}{\Gamma(z)}$$ and can itself be extended with the Polygamma function: $$\psi^m(z)=\frac{\mathrm d^m}{\mathrm dz^m}\psi(z) %=\frac{\mathrm d^{m+1}}{\mathrm d z^{m+1}}\ln \operatorname\Gamma(z) $$
All sources I have encountered that mention the Digamma (or Polygamma) denote it as above with the greek letter 'psi' $\psi$. This befuddles and confuses me. Archaic greek's uppercase form of digamma resembles the glyph of capital gamma $\Gamma$, but with and additional horizontal cross bar, making it appear as a 'double gamma' or a latin $\mathrm F$.
The uppercase gamma, and lowercase digamma and psi characters are available in LaTeX/MathJax, where $$\Gamma,\digamma,\psi$$
produces: $$\huge\Gamma,\digamma,\psi$$
It makes sense that 'Digamma' would be chosen as a symbol that is both visually and etymologically similar to the Gamma function, because these functions are closely related. It does not make any sense to chooses the symbol $\psi$. But what really gets me, is that the name of the function is a complete mismatch to the symbol of the function.
My question(s):
If we are going to call it a 'Digamma' function why don't we use a 'Digamma' symbol?
or conversely
If we are going to use a 'psi' symbol why don't we call it a 'psi' function?
Is there some justification for the establishment or continuation of this entrenched convention? Is this merely an historical artefact of idiotic humanity (a typographical error)? Or is there more to the story?
My best guess is that it many have something to do with typographical limitations, or overloading. But it still seems to ridiculously confusing notation to learn, (let alone to teach and propagate).