The double square root, Jacobi polynomials and Ramanujanʹs Master Theorem

LetN0,4(a;m)≔∫0∞dx(x4+2ax2+1)m+1, a>−1, m∈Nand definePm(a)≔1π2m+3/2(a+1)m+1/2N0,4(a;m).We prove that Pm(a) is a polynomial in a given byPm(a)=2−2m∑k=0m2k2m−2km−km+km(a+1)k.The proof is based on the Taylor expansion of the double square root and Ramanujanʹs Master Theorem.