# This Project

• What is special about special functions?
Just taking a look at their representations makes one understand that these functions make up a select set in the world of functions: their series coefficients and continued fraction elements are all known through simple explicit formulas which in addition exhibit some nice monotonicity properties.
• What is special about continued fractions?
On the one hand continued fraction representations of functions enjoy larger convergence domains than their series counterparts, while on the other hand they are equally simple to deal with. It suffices to build some understanding for them by reading part I of the handbook.
Books on special functions do mostly not contain continued fraction representations. Books on continued fractions occasionally serve up some special function as example. This handbook is the result of a systematic study of continued fraction representations of special functions. Only 10% of the continued fractions in part III can also be found in the Abramowitz and Stegun handbook or at special functions websites. The project is still ongoing and more chapters and functions will be added in the future.
• How trustworthy is the compendium of formulas?
Those formulas from the handbook which are implemented in our Maple library have been validated in two ways. They were automatically transferred from text to program to allow the detection of printing errors. They were then compared to existing implementations for the functions they represent. Any errors found in original reference material is thereby corrected.
• Which software is developed in the wake of this encyclopedic study?
Essentially three tools are made available with an accompanying web interface. All tables printed in the handbook can be tailor made interactively: one can put more formulas in the side by side comparison, choose higher or lower order approximants of the representations, or select different sample values for the arguments and parameters. All series and continued fraction representations of the elementary and special functions are preprogrammed in a downloadable Maple library, developed especially to offer the functionality required for handling (limit k-periodic) continued fractions. Easy truncation and roundoff error bounds for the series and continued fraction representations have allowed to develop a validated scalable precision (up to a few thousand digits) and multiradix (powers of 2 or 10) numeric C++ library for the evaluation of these special functions (available in the near future).

# Teams

 Handbook Software A. Cuyt F. Backeljauw S. Becuwe S. Becuwe V. Petersen M. Colman B. Verdonk A. Cuyt H. Waadeland J. Van Deun W. B. Jones

• Chapter 0
• Errata

• Tabulate
• Approximate