FW: Matrix of potentials in Legendre basis

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

-----Original Message-----

Sender: Richard Askey <askey@math.wisc.edu>
Subject: Re: FW: Matrix of potentials in Legendre basis


  The first problem can be solved as follows.  There is a long
paper by Polya and Szego in the early 1930s, see either Polya's
collected papers or those of Szego, in which they find an expansion
of |x-y|^t as a sum of products of ultraspherical polynomials,
I think the polynomials are C_k^(-t)(x)* same function of y and
the coefficient multiplying this term is a simple rational function.
I don't have access to either of these books now but looked yesterday.
Then the integral below, which I would put on (-1,1) for simplicity
of using known formulas, can be evaluated as a single hypergeometric
series by using the fomula for the integral of the polynomial above
times a Legendre polynomial.  This is a special case of a formula
of Gegenbauer which I have used frequently.  There are a number
of relatively simple derivations of this formula.  One of the
easiest is the one Gasper and I gave in the paper we wrote for
the meeting celebrating de Branges's proof of the Bieberbach
conjecture.  It just uses the generating function of ultraspherical
polynomials, differentiation, and orthogonality to get the result
as an integral.
  Then it is just a matter of seeing if the hypergeometric
series which comes from these calculations can be summed.  Sadov
claimed a specific formula is true, so the series can be summed.  I
would
be interested in seeing Sadov's simple evaluation of this, since mine
uses a formula of Polya and Szego which is not immediately obvious.
  Dick Askey
>
> Sender: Sergey Sadov <sadov@keldysh.ru>
> Subject: Matrix of potentials in Legendre basis
>
>
> Dear colleagues,
>
> Can anyone suggest if the following formulas
> are published elsewhere:
>
> 1. (Matrix elements of Bessel potentials in Legendre basis)
> $$
>    \int\int p_n(x) p_m(y) |x-y|^t
>    =
>    -2 \frac{(-t-1+d/2)_{n+1} \,(-t)_{d-2}}%
>            {(t+1+d/2)_{n+1}\, (t+2)_{d}}
>
> $$
>
>  Here p_n(x) are Legendre polynomials on [0,1],
>  i.e. p_n(x)=(1/n!) (d/dx)^n[ x^n(x-1)^n],
>   d=m-n,
>  assumptions: d=2,4,6,...; Re t>-1.
>
> 2. (Matrix elements of logarithmic potential in Legendre basis)
> $$
>    \int\int p_n(x) p_m(y) \ln |x-y|
>    =
>   \frac{2}%
>        {(m+n)(m+n+2)(m-n+1)(m-n-1)}
> $$
>  where (m-n) is even.
>
> 3. (Some hypergeometric summation formula)
> $$
>    _3 F_2(a, 1-a, (b+c-1)/2;
>           b,c;
>              1)=
>    2^{2-b-c} \pi
>    \frac{\Gamma(b)\Gamma(c)}%
>         {\Gamma((b+a)/2) \Gamma((b+a')/2)
>          \Gamma((c+a)/2) \Gamma((c+a')/2)}
> $$
>  Here a'=1-a.
>  The series is not balanced, well-poised or terminating.
>
> Comment.
> I found formula 2 experimentally in 1998 and used
> it in an algorithm for solving integral equations of diffraction
> theory on polygonal contours, but I couldn't
> prove it or find in literature. A proof was communicated
> to me by Prof. M.Rahman (Ottawa) in Feb.1999. It was
> based on Din-Askey integral formula involving Legendre
> polynomials and the Legendre function of 2nd kind.
> Formulas 1 and 3 are my recent findings. (3) in case
> of terminating series (integer a) is a by-product
> of a relatively simple proof of (1).
> (2) follows from (1) by taking derivative  at t=0.
>
> I asked few people about (2) but no one could recollect
> seeing it. I was a bit surprised, but I would be even
> more surprised if (3) were new.
>
> I appreciate any information.
>
> Sergey Sadov
>
> email: sadov@keldysh.ru
>
>
>
>