# FW: Matrix of potentials in Legendre basis

• Subject: FW: Matrix of potentials in Legendre basis
• From: "Daniel Lozier" <lozier@nist.gov>
• Date: Mon, 12 Feb 2001 15:11:21 -0500
• Content-Transfer-Encoding: 7bit
• Content-Type: text/plain; charset="iso-8859-1"
• Importance: Normal



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.

seeing it. I was a bit surprised, but I would be even
more surprised if (3) were new.

I appreciate any information.