AI 中文总结
研究将反正弦幂展开相关系数族放入双曲反正弦核，通过有限傅里叶投影和梅林变形等方法，得到含特定参数的恒等式及相关结果，还记录了核间卷积等细节，指出直接四次核会导致不同的\({}_4F_3\)族。
AI 中文摘要
反正弦幂的经典展开包含中心二项式系数和有限重复调和和。我们将奇平方和普通平方系数族放入两个双曲反正弦核中，并将这些核用作生成函数，在特化之前可以对其进行有限傅里叶投影和梅林变形。二次投影提取四次子序列，并给出涉及\(\binom{4r}{2r}\)、\(\pi\)和\(L = \log(1 + \sqrt{2})\)的恒等式。相同的投影允许加速内部形式，并且在梅林变形之后，在\((\sqrt{2}-1)^2\)处有分母幂和对数伴随多对数。本文还记录了两个核之间的平方律卷积、周期权重滤波器、负平方参数处的有限谱截断，以及分支选择、边界收敛和逐项梅林运算所需的解析细节。最后的比较表明，直接四次核导致不同的\({}_4F_3\)族。
英文摘要
Classical expansions of powers of the inverse sine contain central-binomial coefficients and finite repeated harmonic sums. We place the odd-square and ordinary-square coefficient families into two hyperbolic arcsine kernels and use these kernels as generating functions on which finite Fourier projection and Mellin deformation can be carried out before specialization. The quadratic projection extracts quartic subsequences and gives identities involving \(\binom{4r}{2r}\), \(π\), and \(L=\log(1+\sqrt2)\). The same projection admits accelerated interior forms and, after Mellin deformation, denominator-power and logarithmic companions with polylogarithms at \((\sqrt2-1)^2\). The paper also records the square-law convolution between the two kernels, periodic-weight filters, finite spectral truncations at negative square parameters, and the analytic details needed for branch choices, boundary convergence, and termwise Mellin operations. A final comparison shows that direct quartic kernels lead to a different \({}_4F_3\) family.
Comments28 pages
