The communication of scientific knowledge is traditionally carried out via written media. Such documents are essentially fixed, even though some of them are now and then brought up to date in successive editions. In contrast, the web has for a short while allowed a rapid and reactive diffusion of discoveries, as well as an interactive access to various forms of or views to the same electronic content. In this perspective, encyclopedias and handbooks specialised to such or such field of mathematics, including the On-Line Encyclopedia of Integer Sequences (“EIS”), the Encyclopedia of Combinatorial Structures (“ECS”), the Encyclopedia of Special Functions (“ESF”), and the soon-to-be Digital Library of Mathematical Functions (see already its Chapter on the Airy function), are first, ambitious realisations. The oldest of those have existed for over a decade already.
Encyclopedias like the EIS, ECS, and ESF make essential use of symbolic calculations performed by computer-algebraic systems, that is, calculations with terms representing algebraic quantities like polynomials, matrices, differential equations, etc, constructed over integers and symbols. Specifically, the three encyclopedias above are based on software written in the general-purpose computer-algebra system Maple, but other systems could be used in the future for other encyclopedias, whether general-purpose ones like Axiom, Derive, Magma, Mathemagix, Mathematica, Macsyma, Reduce, Sage, etc, or specialised ones, dedicated to various branches of mathematics. In particular, other scientific communities with different habits, for instance in applied mathematics or mathematical physics, could conceivably prefer another system.
In this context of symbolically-generated mathematical knowledge, it is just natural to want to further exploit the ability of online computations. For instance, the ECS identifies a combinatorial structure selected by the user through its first terms. The reply by the system consists in a collection of data statically attached to the sequence and merely retrieved from a database: name, combinatorial interpretation, references, etc. A natural wish of the user would be, via new requests, to obtain an arbitrary number of terms of the sequence, or to draw an instance of the structure randomly. This new, dynamic information would be calculated at the time of the request by a computer-algebra server. For its part, the ESF provides for a special function given by the user a large number of formulas satisfied by the function, and several evaluation graphics. In this output, such or such series expansion of fixed order, for example, should be able to be extended at will, based on a user-supplied parameter. Similarly, graphics should be able to be replotted interactively with respect to different ranges of the variable.
Another goal of the project is to provide the readers with certificates of the symbolic calculations, and more generally with a displayed proof of the symbolic calculation. Since, in the existing ESF, symbolic objects that are made into displayed documents are computed using an externally-compiled symbolic library, the wanted self-documentation implies revising the existing symbolic code so as to embed it within the code. An advantage of this model that shares much with literate programming is that it permits the extraction of independent symbolic libraries from the implementation of an encyclopedia.