双曲余割级数的函数关系
Functional relations for hyperbolic cosecant series
摘要：
研究了整数$m$和复数$ϕ$的函数级数$\sum_{n=1}^\infty ϕ^{2m+2} \text{cosch}^{2m+2}(nϕ/2)$和相似级数.这个双曲级数与兰伯特级数线性相关。已知兰伯特级数满足定义拉马努金多项式的函数方程。利用留数定理（求和定理），求出了该双曲级数所满足的函数方程。该函数方程确定一类多项式，可以看作是一个推广的拉马努金多项式。这些多项式与双曲级数在原点的渐近展开式一致，并且对于$ϕ=\pm 2πi$它们都为零。我们还得到了调和数与普通和广义伯努利多项式之间的几个恒等式。
英文摘要：
We study the function series $\sum_{n=1}^\infty ϕ^{2m+2} \text{cosch}^{2m+2}(nϕ/2)$, and similar series, for integers $m$ and complex $ϕ$. This hyperbolic series is linearly related to the Lambert series. The Lambert series is known to satisfy a functional equation which defines the Ramanujan polynomials. By using residue theorem (summation theorem) we find the functional equation satisfied by this hyperbolic series. The functional equation identifies a class of polynomials which can be seen as a generalization of the Ramanujan polynomials. These polynomials coincide with the asymptotic expansion of the hyperbolic series at the origin and they all vanish for $ϕ=\pm 2πi$. We furthermore derive several identities between Harmonic numbers and ordinary and generalized Bernoulli polynomials.
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