The double square root, Jacobi polynomials and Ramanujan's master theorem (Q5937197)

Let NEWLINE\[NEWLINEN_{0,4}(a;m): =\int^\infty_0 {dx\over (x^4+2a x^2+1)^{m +1}}, \;a>-1,\;m=1,2, \dotsNEWLINE\]NEWLINE and define NEWLINE\[NEWLINEP_m(a): ={1\over\pi} 2^{m+3/2} (a+ 1)^{m+1/2} N_{0,4}(a;m).NEWLINE\]NEWLINE The authors prove that \(P_m(a)\) is a polynomial in \(a\) given by NEWLINE\[NEWLINEP_m(a)= 2^{-2m} \sum^m_{k=0}2^k{2m-2k \choose m-k} {m+k \choose m}(a+1)^k.NEWLINE\]NEWLINE The proof is based on the Taylor expansion of the double square root and Ramanujan's Master Theorem.