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Positive intertwiners for Bessel functions of type B
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-01-21 , DOI: 10.1090/proc/15312 Margit Rösler , Michael Voit
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-01-21 , DOI: 10.1090/proc/15312 Margit Rösler , Michael Voit
Abstract:Let 
denote Dunkl's intertwining operator for the root sytem 
with multiplicity 
with 
. It was recently shown that the positivity of the operator 
which intertwines the Dunkl operators associated with 
and 
implies that 
. This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: for 
and 
, the operator 
is positive when restricted to functions which are invariant under the Weyl group, and there is an associated positive Sonine formula for the Bessel functions of type 
. Moreover, the same positivity results hold for arbitrary 
and 
The proof is based on a formula of Baker and Forrester on connection coefficients between multivariate Laguerre polynomials and an approximation of Bessel functions by Laguerre polynomials.
中文翻译：
B型贝塞尔函数的正互缠
摘要：设表示Dunkl的交织运营商根系统正与多样性与。最近证明，与Dunkl算子交织在一起的算子的正性并暗示。这也是在相关的贝塞尔函数之间存在正Sonine公式的必要条件。在本文中，我们给出了两个部分相反的正结果：对于和，当运算符限于Weyl组下不变的函数时，它是正的；对于类型Bessel函数，存在一个相关的正Sonine公式。而且，相同的阳性结果适用于任意和 该证明基于Baker和Forrester的公式，该公式基于多元Laguerre多项式之间的连接系数以及Laguerre多项式对Bessel函数的逼近。
更新日期：2021-02-08
denote Dunkl's intertwining operator for the root sytem with multiplicity with . It was recently shown that the positivity of the operator which intertwines the Dunkl operators associated with and implies that . This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: for and , the operator is positive when restricted to functions which are invariant under the Weyl group, and there is an associated positive Sonine formula for the Bessel functions of type . Moreover, the same positivity results hold for arbitrary and The proof is based on a formula of Baker and Forrester on connection coefficients between multivariate Laguerre polynomials and an approximation of Bessel functions by Laguerre polynomials. 中文翻译：
B型贝塞尔函数的正互缠
摘要：设
表示Dunkl的交织运营商根系统正与多样性与。最近证明，与Dunkl算子交织在一起的算子的正性并暗示。这也是在相关的贝塞尔函数之间存在正Sonine公式的必要条件。在本文中，我们给出了两个部分相反的正结果：对于和，当运算符限于Weyl组下不变的函数时，它是正的；对于类型Bessel函数，存在一个相关的正Sonine公式。而且，相同的阳性结果适用于任意和 该证明基于Baker和Forrester的公式，该公式基于多元Laguerre多项式之间的连接系数以及Laguerre多项式对Bessel函数的逼近。 
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