关于某些β分布的广义Cauchy-斯蒂尔切斯变换
On generalized Cauchy-Stieltjes transforms of some Beta distributions
摘要：
我们将某些特殊β分布（正交多项式的超球型母函数）的广义Cauchy-斯蒂尔切斯变换表示为某种测度的幂Cauchy-斯蒂尔切斯变换.对于合适的功率参数值，后一个测量结果是一个概率测度和它的密度是使用马尔可夫变换写下来。被丢弃的值给出了一个否定的答案变形的自由概率，除非对功率参数的限制。得到了一种特殊的对称分布，它介于Wigner分布和反正弦分布之间.它的时刻表示通过一个终止超几何级数Catalan和移位Catalan数之间的插值。对于功率参数的小值，计算自由累积量。讨论了无限对称群的变形表示论和变形Bozejko卷积的有趣的opne问题。
英文摘要：
We express generalized Cauchy-Stieltjes transforms of some particular Beta distributions (of ultraspherical type generating functions for orthogonal polynomials) as a powered Cauchy-Stieltjes transform of some measure. For suitable values of the power parameter, the latter measure turns out to be a probability measure and its density is written down using Markov transforms. The discarded values give a negative answer to a deformed free probability unless a restriction on the power parameter is made. A particular symmetric distribution interpolating between Wigner and arcsine distributions is obtained. Its moments are expressed through a terminating hypergeometric series interpolating between Catalan and shifed Catalan numbers. for small values of the power parameter, the free cumulants are computed. Interesting opne problems related to a deformed representation theory of the infinite symmetric group and to a deformed Bozejko's convolution are discussed.
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