What is a special function?

When I read some issues here I see from time to time incorrect references to the field special functions, it might e.g. be a discussion around Dirac's $\delta$-function which is tagged (special-functions) or a discussion around some function reminding of the Weierstrass no where differentiable continuous function. These examples make me think - what would classify a special function?

A vague bad definition could be "A function is a special function if it has some resemblance to some Hypergeometric function" or "A function is a special function if it fits into the Bateman manuscript project.

To me the Gamma function and the Zeta function are definitively special functions.

Also, I have worked on Legendre functions $P_\lambda$ and $Q_\lambda$ of the first and second kind , which I would call special functions, but not individually however (except perhaps $P_{-1/2}$).

I would not say that elementary functions (such as trigonometric functions and the exponential function) are special functions - but I am not totally convinced about this..

I do not agree with Wikipedia, it says: "Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.

There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. In particular, elementary functions are also considered as special functions."

Also, looking at the Wikipedia list (linked above) the indicator function, step functions, the absolute value function and the sign are special functions -- this sounds very wrong to me.

So what is a special function and what should be under the (special-function) tag?

The term "special function" has a historical connection with solutions of ordinary differential equations, often second order differential equations that arise from a separation of variables treatment of second order linear partial differential equations with constant coefficients. This connection is mentioned briefly in the Wikipedia article.

The Sage documentation of special functions, for example, lists the ODE satisfied by the function together with remarks about boundary conditions. Stephen Wolfram gave a nice historical review in his talk The History and Future of Special Functions, in which he says, "Most continuous special functions are in effect defined implicitly, usually from differential equations." Naturally he is not shy about what he perceives as Mathematica's contribution to this history, developing automated searches for accurate numerical methods that proceed directly from the underlying definition by differential equations.

He is not alone in exploring the automatic generation of algorithms for special functions using symbolic computation. Focusing on special functions that are the solutions of linear ordinary differential equations with polynomial coefficients, Meunier and Salvy (2003) describe the design of their website The Encyclopedia of Special Functions (ESF).

The elementary transcendental functions fit easily into this scheme as special special functions, since they solve differential equations too. For example, the exponential function solves $y' = y$ and sine and cosine, with different boundary conditions, solve the constant coefficient ODE $y'' = -y$.

I (physicist) associate the term mostly with this huge family of often indexed functions which happen to bear some magical relations to each other. My handwavy explaination why these things exists goes as follows:

In physics, we're dealing with the dynamics of certain degrees of freedom. These (the dynamics, given by differential equations) often employ smooth symmetries, that is we're dealing with Lie groups, which are also manifold in themselfs. Take e.g. the Laplacian $\Delta=\nabla\cdot\nabla$ and the associated symmetries $R$ acting as $\nabla\to R\nabla$ in such a way that that $R\nabla\cdot R\nabla=\nabla\cdot\nabla$. Now in case one is dealing with a "rotation" in the broadest sense of the word, the $R$'s often form compact manifold, where we can savagely define things like integration on the group, and these symmetry groups also permit pretty unitary matrix representations. That is there are necessarily matrices $U$ with

$$\sum_kU_{kn}^*U_{km}=\delta_{mn},$$

and well, the matrix coefficients $U_{kn}$ must be some complex functions. You see the direct relation to special functions if you take the abstract Lie group theory and actually sit down and write down the matrices in some base. E.g. for the rotation group matrices $D$, you find

$$D^j_{m'm}(\alpha,\beta,\gamma)= e^{-im'\alpha } [(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\cdot$$ $$\cdot\sum\limits_s \left[\frac{(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!} \right.\cdot$$ $$ \left. \cdot \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2s}\left(\sin\frac{\beta}{2}\right)^{m'-m+2s} \right] e^{-i m\gamma}.$$

Very sweet, right? Now here you have the Legendre Polynomials $P_\ell^m$

$$ D^{\ell}_{m 0}(\alpha,\beta,0) = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^{m*} (\beta, \alpha ) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\beta} ) \, e^{-i m \alpha } $$

so that you have the special relation relating the functions

$$ \int_0^{2\pi} d\alpha \int_0^\pi \sin \beta d\beta \int_0^{2\pi} d\gamma \,\, D^{j'}_{m'k'}(\alpha,\beta,\gamma)^\ast D^j_{mk}(\alpha,\beta,\gamma) = \frac{8\pi^2}{2j+1} \delta_{m'm}\delta_{k'k}\delta_{j'j},$$

which I read as $U^* U=1$.

For there are representable symmetries in the geometric structure of spaces, there must be functions which have some magical interrelating properties.