Motivated by the problem of determining the values of $$\alpha >0$$ α > 0 for which $$f_\alpha (x)=\mathrm{e}^\alpha - (1+1/x)^{\alpha x},\ x>0$$ f α ( x ) = e α - ( 1 + 1 / x ) α x , x > 0 , is a completely monotonic function, we combine Fourier analysis with complex analysis to find a family $$\varphi _\alpha $$ φ α , $$\alpha >0$$ α > 0 , of entire functions such that $$f_\alpha (x) =\int _0^\infty \mathrm{e}^{-sx}\varphi _\alpha (s)\,\mathrm{d}s, \ x>0.$$ f α ( x ) = ∫ 0 ∞ e - s x φ α ( s ) d s , x > 0 . We show that each function $$\varphi _\alpha $$ φ α has an expansion in power series, whose coefficients are determined in terms of Bell polynomials. This expansion leads to several properties of the functions $$\varphi _\alpha $$ φ α , which turn out to be related to the well-known Bessel function $$J_1$$ J 1 and the Lambert W function. On the other hand, by numerically evaluating the series expansion, we are able to show the behavior of $$\varphi _\alpha $$ φ α as $$\alpha $$ α increases from 0 to $$\infty $$ ∞ and to obtain a very precise approximation of the largest $$\alpha >0$$ α > 0 such that $$\varphi _\alpha (s)\ge 0,\, s>0$$ φ α ( s ) ≥ 0 , s > 0 , or equivalently, such that $$f_\alpha $$ f α is completely monotonic.

中文翻译：

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连接贝塞尔函数 $$J_1$$J1 和兰伯特 W 函数的完整函数族

由确定 $$\alpha >0$$ α > 0 的值的问题驱动，其中 $$f_\alpha (x)=\mathrm{e}^\alpha - (1+1/x)^{\ alpha x},\ x>0$$ f α ( x ) = e α - ( 1 + 1 / x ) α x , x > 0 ，是一个完全单调的函数，我们结合傅里叶分析和复分析找到一个族$$\varphi _\alpha $$ φ α , $$\alpha >0$$ α > 0 ，整个函数使得 $$f_\alpha (x) =\int _0^\infty \mathrm{e}^ {-sx}\varphi _\alpha (s)\,\mathrm{d}s, \ x>0.$$ f α ( x ) = ∫ 0 ∞ e - sx φ α ( s ) ds , x > 0 . 我们表明每个函数 $$\varphi _\alpha $$ φ α 都有幂级数的展开，其系数是根据贝尔多项式确定的。这种展开导致函数 $$\varphi _\alpha $$ φ α 的几个性质，结果证明这与著名的贝塞尔函数 $$J_1$$J 1 和兰伯特 W 函数有关。另一方面，通过对级数展开的数值评估，我们能够展示 $$\varphi _\alpha $$ φ α 当 $$\alpha $$ α 从 0 增加到 $$\infty $$ ∞ 时的行为并获得最大 $$\alpha >0$$ α > 0 的非常精确的近似值，使得 $$\varphi _\alpha (s)\ge 0,\, s>0$$ φ α ( s ) ≥ 0 ， s > 0 ，或等效地，使得 $$f_\alpha $$ f α 是完全单调的。