FW: Matrix of potentials in Legendre basis

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

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