关于余割幂的拉马努金型主定理猜想的证明
Proof of a Conjectured Ramanujan Type Master Theorem for Powers of the Cosecant
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中文总结 AI 辅助
该研究证明了Bradshaw和Atale关于余割幂的拉马努金型主定理猜想，通过归纳微分递推建立洛朗级数恒等式完成证明，所得定理在更弱增长假设下成立，还得到余割幂积分表示等应用。
中文摘要 AI 辅助
我们证明了Bradshaw和Atale的一个猜想，该猜想断言存在关于核函数π^m/sin^m(πs)的拉马努金型主定理，其在非正整数处的极点阶数为m。这些极点处的留数数据由Airault研究的一族多项式整理，其系数为第一类中心阶乘数。证明将该猜想归结为csc^m的洛朗级数恒等式，该恒等式通过初等微分递推归纳得到，且所得定理在比Hardy的条件更弱的增长假设下成立。作为应用，我们得到了余割幂的积分表示、柯西核(1+x)^{-1}的Mellin卷积幂的闭式，以及Airault多项式满足的一族微分恒等式。
英文摘要
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.
