### Applicability of mathematics: special functions

*On the Specialness of Special Functions (The Nonrandom Effusions of the Divine Mathematician).*Here's an extract:

...it seems reasonable to maintain that many special functions are special for more than simple pragmatic reasons. They are not special simply because they appear in the physicist’s, applied mathematician’s, and engineer’s toolboxes. Furthermore, special functions are not special simply because they share some deep mathematical properties. Recall this is the point of view of Truesdell and of Talman/Wigner. On their proposals, what makes some functions special is that despite “surface” differences, they are each solutions to the “F-equation” (for Truesdell), or they possess similar group representations (for Talman and Wigner). While these classificatory schemes suffice to bring some order to the effusions of the Divine Mathematician, they do not fully capture the special nature of the special functions.

From the point of view presented here, the shared mathematical features that serve to unify the special functions—the universal form of their asymptotic expansions—depend upon certain features of the world. What Truesdell, Talman and many others miss is how the world informs and determines the relevant mathematical properties that unify the diverse special functions.

As I noted, in many investigations of physical phenomena we find dominant physical features—those features that constrain or shape the phenomena. These are things like shocks and the highly intense light appearing in the neighborhood of ray theoretic caustics. They are features that are most effectively modeled by taking limits.

Limiting idealizations are most effective for examining what goes on at places where the “laws” break down—that is, at places of singularities in the governing equations of the phenomena. These “physical” singularities and their “effects”—how they dominate the observed phenomena—are themselves best investigated through asymptotic representations of the solutions to the relevant governing equations. The example of the Airy integral is a case in point. By using Stokes’ asymptotic representation we get superb representations of the nature of the diffraction relatively far (large z) from the dominating “physical” singularity—the caustic.

There's a curious tendency when it comes to discussing the applicability of mathematics to polarise one's response to the question of which of mathematics and science owes the other the most. It seems each side is only too happy to exaggerate the role of their favoured discipline, finding the benefits mathematics bestows as miraculous, or deflating mathematics to a bunch of tautologies which physics is gracious enough to give an interpretation to. In the philosophical literature of the past century, with one or two notable exceptions, the latter attitude has been prevalent. Of the two, it is the role of mathematical understanding which tends to get passed over or taken for granted.

Batterman is more sensitive to the narratives various mathematicians and physicists tell about special functions, but I wonder if there might not be a richer mathematical story which would make the still sharp dichotomy he maintains between physical and mathematical considerations less sharp. Is it not possible that the richer mathematical understanding of special functions gained since Talman (1968) could intersect with the physical considerations treated by Batterman? For instance, might the asympotic results he discusses have something to do with current ideas from the group representation understanding of special functions? And how do q-deformed special functions fit in? It's surely common to find mathematics and physics narratives creatively intertwined.

Batterman concludes:

I hope that the discussion here leads us to question the anthropocentric role of the mathematician’s appreciation for beauty (or formal analogy) as an important criterion for what arguably should be paradigm examples of mathematics’ applicability to the world; namely, the special functions.

I'd like to hear some contemporary mathematicians' aesthetic narratives about special functions.

## 2 Comments:

There's a lot one could say about special functions. For example, these days in the US pure mathematicians can get PhDs without knowing much about special functions. Undergraduate math courses are a bit spotty, so one can squeak by without learning stuff that once would have been

de rigeur. By the time grad school comes around, pure math students are busy studying the big early-20th-century theorems in real analysis, complex analysis, algebra and topology - and special functions have the reputation of being "old-fashioned" and "applied". So, I think a lot of modern mathematicians in the US meet and learn to appreciate special functions later in life, when they need them for their work.I was lucky to take physics courses as a counterweight to math, so I learned a reasonable amount about Legendre polynomials, Laguerre polynomials, Bessel functions and elliptic integrals and the like. Unfortunately, I only remember detailed information about these things when I'm working on a project that actively uses them. For example, for the last few years I've been studying elliptic curves, so I know stuff about elliptic functions and theta functions, but nothing much about Bessel functions - except what they're good for! I could relearn Bessel functions if I needed to.

But about your question! A lot of special functions arise naturally as matrix elements of important representations of important Lie groups, and Gelfand, Naimark and Vilenkin developed the theory of special functions very nicely from this viewpoint in their books. For a modern update on this viewpoint, check out the '93 paper by Etinghof and Kirillov. They stretch this viewpoint to include Clebsch-Gordan coefficients (which describe how to tensor representations) and the q-deformed version of everything.

This viewpoint is mathematically esthetic (what could be prettier than groups and their representations?) but also very hard-nosed and practical (what could be more useful than solving PDE by separation of variables with the help of symmetry assumptions?).

But, not all special functions fit into this framework in any useful way. At least, I don't think so! I could be wrong. How would the gamma function, or hypergeometric functions, fit into this viewpoint? I know Gelfand got very interested in hypergeometric functions in his later years... but I don't know his "take" on them.

Regarding hypergeometric functions, here's an extract from Kazhdan's talk at the unity of Mathematics conference for Gelfand's 90th birthday:

"Gelfand has shown that many special functions such as Bessel and Whittaker functions, Jacobi and Legendre polynomials appear as matrix coefficients of

irreducible representations. This interpretation of special functions immediately explains the functional and differential equations for these functions. It is clear now that [almost] all special functions studied in the 19-20 centuries can be interpreted as matrix coefficients or traces of representations of groups or their quantum analogs (e.g., works of Tsuchiya-Kanie, I. Frenkel-Reshetikhin on the representation theoretic interpretation of hypergeometric (respectively q-hypergeometric) functions,

works of Koornwinder, Koelink, Noumi, Rosengren, Stokman,Sugitani and others on representation theoretic interpretation of Askey-Wilson, Macdonald, and Koornwinder polynomials)."

What happens after categorification? What are the matrix coefficients or traces of representations of 2-groups?

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