Proof of a Conjectured Ramanujan Type Master Theorem for Powers of the Cosecant

Abstract

We prove a conjecture of Bradshaw and Atale asserting a Ramanujan type master theorem for the kernel ${\pi }^m/{\sin }^m(\pi s)$, whose poles at the non-positive integers have order $m$. The residue data at these poles are organized by a family of polynomials studied by Airault with coefficients the central factorial numbers of the first kind. The proof reduces the conjecture to a Laurent series identity for ${\csc }^m$, established by induction from an elementary differential recursion, and the resulting theorem holds under a growth hypothesis strictly weaker than Hardy's. As applications, we obtain integral representations for powers of the cosecant, a closed form for the Mellin convolution powers of the Cauchy kernel $(1+x{)}^{-1}$, and a family of differential identities satisfied by the Airault polynomials.

19 pages