论文标题
哈恩的比率 - exton $ q $ - 贝塞尔函数和$ q $ -lommel多项式
Ratios of Hahn--Exton $q$-Bessel functions and $q$-Lommel polynomials
论文作者
论文摘要
在1993年，Delest和Fédou表明，连接偏斜形状的生成功能作为比率$ j_ {ν+1}/j_ν$的hahn-- exton $ q $ -bessel函数当参数$ n为零时。他们推测，当$ν$是一个非负整数时，生成函数的系数是有理功能的，其分子和分母为$ q $，具有非负整数系数的$ q $，这是Kishore的1963年$ Q $ -Analog在Bessel函数上的作用。本文的第一个主要结果是证明了Delest和Fédou的猜想。第二个主要结果是对DeLest和Fédou的结果进行完善：具有界面对角线的连接偏斜形状的生成函数的比例为$ Q $ - lommel polyenerials，由Koelink和Swarttouw引入。还表明，比率$ j_ {ν+1}/j_ν$具有两个不同的持续分数表达式，这分别为$ r_i $的正交多项式矩的生成函数分别为生成函数，并为通常的正交多项主义的矩提供了生成功能。使用Flajolet和Viennot引起的正交多项式技术。
In 1993 Delest and Fédou showed that a generating function for connected skew shapes is given as a ratio $J_{ν+1}/J_ν$ of the Hahn--Exton $q$-Bessel functions when a parameter $ν$ is zero. They conjectured that when $ν$ is a nonnegative integer the coefficients of the generating function are rational functions whose numerator and denominator are polynomials in $q$ with nonnegative integer coefficients, which is a $q$-analog of Kishore's 1963 result on Bessel functions. The first main result of this paper is a proof of the conjecture of Delest and Fédou. The second main result is a refinement of the result of Delest and Fédou: a generating function for connected skew shapes with bounded diagonals is given as a ratio of $q$-Lommel polynomials introduced by Koelink and Swarttouw. It is also shown that the ratio $J_{ν+1}/J_ν$ has two different continued fraction expressions, which give respectively a generating function for moments of orthogonal polynomials of type $R_I$ and a generating function for moments of usual orthogonal polynomials. Orthogonal polynomial techniques due to Flajolet and Viennot are used.
