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

2026/07/26 by Zachary P. Bradshaw
Mathematics · #math.CV #math.NT #msc:44A15

paper · pdf

19 pages

arxiv created 2026/07/26 · arxiv updated 2026/07/31

Abstract

We prove a conjecture of Bradshaw and Atale asserting a Ramanujan type master theorem for the kernel π^m/sin^m(πs), 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.

Citations

• On certain extensions of Ramanujan's Master Theorem and their applications
• Ramanujan's Master Theorem for Riemannian symmetric spaces
• The method of brackets. Part 2: examples and applications
• Definite integrals by the method of brackets. Part 1

Related

• Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems
• A note on the Ramanujan master theorem
• Functional relations for hyperbolic cosecant series
• On certain extensions of Ramanujan's Master Theorem and their applications
• Ramanujan's Master Theorem for Riemannian symmetric spaces