Generating Special Functions.

Question

I find it quite intriguing that I can generate a lot of special functions, my questions are as follows:

Is there any limit on the number of special functions?
What current literature in applied and rigorous Special Functions can you recommend me?
I looked at the integral $$\int \frac{log x}{Ei(x)}$$ in mathematica, and there doesn't seem to be a lot of information regarding this function, does it have any applications in analysis or mathematical physics?

Thanks in advance.

1. There is not a "special function commission", so anyone can make up their own. 2. Start with Whittaker and Watson 3. If you looked at it, it must have had an application. — Igor Rivin, Commented Oct 5, 2011 at 11:20
As usual, you might start with reading the Wikipedia article en.wikipedia.org/wiki/Special_functions . In this case, the Springer EoM article has a bunch of references on various approaches too eom.springer.de/s/s086280.htm — j.c., Commented Oct 5, 2011 at 12:45
Unless you actually meant, if one of the functions that you are looking at can be written using standard "special" functions such as Hypergeometrics etc... — Suvrit, Commented Oct 5, 2011 at 13:27
jc's link has changed to encyclopediaofmath.org/index.php?title=Special_functions — George Lowther, Commented Dec 2, 2011 at 2:43

Jacques Carette · Accepted Answer · 2011-10-05 12:37:04Z

There is no limit on the number of functions you can explicitly name. But a function does not get to be called 'special' just if someone can name it! It has to recur a sufficient number of times, in different applications, to acquire that 'special' moniker. The addition of new special functions to various compendiums like Abramowitz and Stegun has slowed down dramatically over the years. Even its electronic companion (easier to update!), the DLMF, has few new additions. My favourite 'new' special functions are definitely the Heun Functions as well as the Lambert W function.
[I will come back later and edit this part of my answer, as I don't have the references handy right now]. There is some solid literature that shows that a lot of special functions arise quite naturally as solutions of (initial value problems for) linear ODEs with polynomial coefficients. Most of the named functions in fact correspond (exactly!) to a particular placement of singularities, each with specific local behaviour. There are decent explanations for most of the special functions (i.e. they are not 'randomly named' functions at all, but would have inevitably have been discovered).
Very unlikely. Just because you can name it doesn't mean that it shows up anywhere.

many special functions are just solutions of homogeneous ODE equations with analytic coefficients, not initial value problems. For example, Hypergeometric, Hankel and Bessel. — user36539, Commented Sep 15, 2013 at 21:59

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