A Regularised Wallis Hierarchy

Abstract

A hierarchy of regularised Wallis products is introduced by raising the reciprocal Wallis factor \[ 1-\frac1{n^2} \] to the polynomial weight $n^m$, $m=0,1,2,\ldots$. For each $m$, a minimal exponential counterterm is chosen by cancelling precisely the non-summable terms in the logarithmic expansion. This gives a convergent product $P_m$ the logarithm of which is an explicit zeta-function tail. The first non-trivial examples are \[ \prod_{n=2}^{\infty} e^{1/n} \left(1-\frac1{n^2}\right)^n = \frac{e^\gamma}{2}, \qquad \prod_{n=2}^{\infty} e\left(1-\frac1{n^2}\right)^{n^2} = \frac{\pi}{e^{3/2}}. \] The even branch has a finite closed form involving $\pi$, harmonic numbers, and odd zeta values. The odd branch reduces to finite logarithmic gamma moments, and hence to constants involving $\gamma$, logarithms, odd zeta values, and derivatives of the zeta function at positive even integers. The same subtraction rule also gives a two-factor extension involving the companion factor $1+1/n^2$. Finally, the associated $x$-dependent products factor into one-sided canonical products, giving a direct connection with Kurokawa's multiple sine functions: the even Wallis branch is obtained from odd multiple sine functions, while the odd branch appears as a symmetric companion to the even multiple sine case.