仅含临界零点的梅林变换：切比雪夫函数和盖根鲍尔函数
Mellin transforms with only critical zeros: Chebyshev and Gegenbauer functions
摘要：
我们考虑基于切比雪夫多项式的某些切比雪夫函数的梅林变换。我们表明，变换的多项式因子的零点都在临界线或实直线上。仅在临界线上为零的多项式被确定为某些$_3F_2(1)$超几何函数。此外，我们将这一结果推广到一个单参数族的多项式与零的临界线上。这些多项式具有函数方程$p_n(s;β)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s;β)$。然后，我们提出了推广到梅林变换的某些Gegenbauer功能。结果应感兴趣的特殊功能理论，组合数学和解析数论。
英文摘要：
We consider the Mellin transforms of certain Chebyshev functions based upon the Chebyshev polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line or on the real line. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend this result to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;β)=(-1)^{\lfloor n/2 \rfloor} p_n(1-s;β)$. We then present the generalization to the Mellin transform of certain Gegenbauer functions. The results should be of interest to special function theory, combinatorics, and analytic number theory.
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