A class of hypergeomtric identities over the roots of unity : known? [repost]
galathaea
article research has still failed to arrive at any references. So I thought
may be a repost might elicit responses from some people who might know where
to look better than I do ;).
This is a repost of Message-ID:
b22ffac3.03052...@posting.google.com
Several years ago I was playing around with hypergeometric equations when I
noticed a beautiful class of identities that were extremely easy to arrive
at. In fact, to me they looked like the type of identity that would
probably be regularly rediscovered by undergrads. However, I have yet to
see the results in any book, and my (not too detailed) search of the
literature failed to find them as well. So I was curious if the brilliant
minds here had seens this class of identities discussed anywhere and could
give me a reference.
The basic steps for proving the class of identities goes like this:
Let (a; n) be the rising factorial (= a (a + 1) ... (a + n - 1))
The addition theorem:
(a; x + y) = (a; x) (a + x; y)
The multiplication theorem:
(a; x y) = x ^ (x y) Product(0 <= j < n; ((a + j) / x; y))
So now, look at the (m, n) multisection of the hypergeometric series for
pFq({aj}(0 < j <= p), {bk}(0 < k <=q); x). The multisection begins with a
sum over i such that i = m (mod n) -- thats class equivalence, of course, as
I can't figure out a three line equivalence on this display.
The first step is to transform the sum to one over i = 0 (mod n) by using
the addition formula and extracting the coefficient m from the power of x.
This gives:
(m, n) multisection of pFq =
[((a1; m) (a2; m) ... (ap; m)) / ((b1; m) (b2; m) ... (bq; m))] x^m / (1; m)
Sum((i = 0 (mod n)); [((a1 + m; i) ... (ap + m; i)) / ((b1 + m; i) ... (bq
+ m; i))] x^i / (1 + m; i))
Lets call the term in front of the sum alpha for simplicity in notation.
Now one can transform the sum into a sum over all nonnegative i and apply
the multiplication formula. This gives
= alpha n^(n i (p - q - 1))
Sum((0 <= i); Product((0 <= h < n); [(((a1 + m + h) / n; i) ... ((ap + m +
h) / n; i)) / (((b1 + m + h) / n; i) ... ((bq + m + h) / n; i) ((1 + m + h)
/ n); i))] (x^n)^i))
After that, you just insert a (1; i) into the numerator and denominator (not
really necessary since one of the (1 + m + h) / n will equal 1, but it keeps
it good looking) and you have hypergeometric form again for a (n p + 1)F(n(q
+ 1)) function.
But the multisection also has the standard form as 1/n times the sum of the
original hypergeometric equation at roots of unity. So one has an identity!
I spent a summer working out a bunch of consequences of this class of
identities. The identical steps, of course, also lead to a class of basic
hypergeometric (q-) identities, and when one can evaluate the function at
some of the roots, the identities simplify. There are all sorts of nice
differential equation identites that arise, as well as integral
relationships, relationships on the groups of various representation spaces,
and basically the study of these identities really helped me alot in
grasping a good hold of the various realms of mathematics which intersect
the study of normal and basic hypergeometric equations. It was a great
learning experience. However, I have always been curious why such an
exposition isn't found in any of my books. Granted, my library is somewhat
limited, but even library study and a little article research has failed to
turn anything up, even though I am sure it is something Gauss would have
known (he did popularize the multiplication formula) and really it uses
nothing beyond the tools of a regular undergrad. So I would appreciate
dearly if anyone can give me any references where this class of identities
might be found.
Thank you all for your time!
--
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galathaea: prankster, fablist, magician, liar
Carl Devore
> I posted this last May, but I didn't receive any responses, and furthur
> article research has still failed to arrive at any references. So I thought
> may be a repost might elicit responses from some people who might know where
> to look better than I do ;).
Two books that spring to mind that may be helpful to you are
_A=B_ by Marko Petkovsek, Herbert Wilf, and Doron Zeilberger (AK
Peters, 1996)
and
_Concrete Mathematics: A Foundation for Computer Science_ by Ronald
Graham, Donald Knuth, and Oren Patashnik (2nd ed., Addison Wesley,
1994)
Axel Vogt
at http://www.mathematik.uni-kassel.de/~koepf/hyper.html
(i am missing time to read/study the book and it contains
a huge bibliography to be used for Goggle). The emphasis
is on implementation of algorithms, but have a look.
On Koepf's page you will find additional articles, some
are overviews, one is [41] Computer algebra algorithms
for orthogonal polynomials and special functions =
http://www.mathematik.uni-kassel.de/~koepf/koepf_minicourse.pdf
galathaea
I appreciate the response, as I am always looking for more references to
check out. I actually have read many books on hypergeometric functions and
their basic counterparts, and regularly enjoy working on identities. The
A=B book is by far one of my favorites (along with the classics like
Andrews, Askey, and Roy, etc.). The second book I read a while back, but
that was before I had encountered this little gem, and if my memory serves
me, it was concerned more with the actual computational aspects of things
like special functions and only presented transformations to assist in
translating the calculation to a tractable form (I may be wrong, and will
grab the book and flip through it next time I head over to the library).
However, my question was more on whether there were any known references to
this particular class of identities, since it seems to me to be such a
trivial derivation, yet I have not seen it in print. It also appears to be
such a useful teaching example, since it provides a nice class of identities
on which to illustrate various techniques (such as writing the identities
out in integral form and simplifying). And maybe, I was thinking, if I
couldn't find a reference (and I've been searching now for a while), it
might be something publishable, which would do wonders for my ego...
But again, I thank you very much for your response, and if you have any
other references, I will do my best to check them out.
galathaea
: Besides the books mentioned by Carl Devore you may look
Thank you for your interesting suggestions. The latter document does not
appear to include the class of hypergeometric and basic hypergeometric
identities in my post, but is a very nice reference to have. The former
book appears to be a great guide to the algorithmics of proving identities,
much in the spirit of A=B, so even if it doesn't include the actual
identity, I think it is probably a must for my library. I will certainly
check that one out!