Request for Information for DLMF Project

["Daniel Lozier" <lozier@nist.gov>]

Editor's Note:  The Web site for the DLMF (Digital Library of
Mathematical Functions) is http://dlmf.nist.gov.  For a long time the
information there was very limited but that is beginning to change.
Authors are writing chapters, and as these take shape details are being
posted at the Web site.

The DLMF will include selected typical applications of the various
special functions, especially in physics.  The following
message from Dick Askey, a DLMF author as well as a DLMF Associate
Editor, asks the readers of this list to help by providing examples.
Please send them to opsftalk@nist.gov.  For information about the
OPSF-Talk List Service and access to the OPSF-Talk Archive, see
http://math.nist.gov/opsf.

I am general editor for the DLMF project.  The other primary editors
are Frank Olver (mathematics), Charles Clark (scientific applications),
and Ron Boisvert (information technology).  The associate editors are
named at the Web site.

Dan Lozier

-----Original Message-----
From: Richard Askey [mailto:askey@math.wisc.edu]
Sent: Sunday, March 04, 2001 12:59 PM
To: Daniel Lozier
Cc: Richard Askey; Frank Olver
Subject: Request for information


  As many of you know, NIST is doing a revision of the Handbook of
Mathematical Functions, now titled "The digital library of mathematical
functions".  In addition to including important facts about many
special functions, there is a need to include references to uses of
them.  Each of us knows some, but collectively we know much more.
It would be useful to the authors of the individual chapters if
people sent examples of important uses of the usual special functions
and their properties.
  Let me give an example of how an old handbook was used to suggest
an interesting property of Legendre polynomials, and then mention
a use of the inequalities which were suggested by one graph in
Jahnke and Emde.
  When Legendre polynomials are graphed on (-1,1) a number of things
are suggested.  First, all the zeros lie in this interval.  Second,
the polynomials are bounded by 1 in absolute value on this interval.
Third, if the absolute values of the successive extrema are considered,
they decrease when going from 1 to 0.  All of these facts had been
observed and proved more than 100 years ago.  About 55 years ago,
John Todd looked carefully at a graph consisting of P_n(x) on (0,1)
for n=1,2,...7.  He noticed that the first minimum value of each
of these functions when you look back from 1 seems to be an
increasing function of n, the first maximum seems to be decreasing,
and so on.  These extrema have limits of the extrema of the Bessel
function J_0(x).  In a few years a proof was found by Gabor Szego,
extended to ultraspherical polynomials suitably normalized by
Otto Szasz, and a similar result was found for Hermite polynomials
when suitably normalized.  For many years these inequalities sat there
as beautiful results but without serious use.
   The result for Legendre polynomials was rediscovered by Cornille
and Martin in some work on pi-pi scattering.  They also found a
dfferent weighted result for ultraspherical polynomials.
Between the publication of their first paper and their second, someone
told them of Szego's work.  References are given in my SIAM regional
conference lectures.  It would have helped Cornille and Martin if
this result had be included in a book they looked at.  As we all know,
results which are not eventually published in a book are very likely
to get lost.  Also, users of special functions will find it helpful
to know the general areas in which these functions are used.  Finally,
mathematicians need to know about the uses of the functions since
the users frequently need something just a little bit different than
what mathematicians have done.
   There is another graph in Jahnke and Emde which was not looked at
for even longer.  This is a graph of Q_n(x) on (0,1) for n=1,2,3,4,5.
A similar monotocity result is suggested by this graph, and has been
proven.  I don't know applications of it, but strongly suspect there
will be some.
   I am writing the chapters on gamma and beta functions and on
higher hypergeometric series.  Any references which can be sent about
uses of these functions would be helpful.  I am sure that the other
authors would also find it useful to hear about applications of the
functions they are writing about.  To avoid duplication, it would
be best to send them to the list this is being sent to, and Dan Lozier
can forward them the author of the chapter if necessary.
   Thanks in advance for help.
Dick Askey