Abstract In this paper, we present two characterizations of the sequences of Kummer hypergeometric polynomials B a , b , n ( x ) and Kummer hypergeometric polynomials of the second kind K a , b , n ( x ) , which are respectively defined by the exponential generating functions: e x t M ( a , a + b ; t ) = ∑ n = 0 ∞ B a , b , n ( x ) t n n ! and e x t U ( a , a + b ; t ) = ∑ n = 0 ∞ K a , b , n ( x ) t n n ! with M ( a , b ; t ) = ∑ n = 0 ∞ ( a ) n ( b ) n t n n ! , where U ( a , a + b ; t ) is the Kummer hypergeometric function of the second kind. First we construct Gauss–Weierstrass-type convolution operators T w a , b with a well-chosen kernel (density) function for each sequence of Kummer hypergeometric polynomials and for Kummer hypergeometric polynomials of the second kind. Then we characterize Kummer hypergeometric polynomials as the only Appell polynomials having a weighted-integral mean equal to zero. Our approach is inspired by the Gauss–Weierstrass convolution transform for Hermite polynomials and the Kummer integral representation for confluent hypergeometric functions.

中文翻译：

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Kummer 超几何伯努利多项式的表征和应用

摘要 在本文中，我们给出了 Kummer 超几何多项式 B a , b , n ( x ) 和第二类 Kummer 超几何多项式 K a , b , n ( x ) 的序列的两个表征，它们分别由指数定义生成函数： ext M ( a , a + b ; t ) = ∑ n = 0 ∞ B a , b , n ( x ) tnn ！和 ext U ( a , a + b ; t ) = ∑ n = 0 ∞ K a , b , n ( x ) tnn ！M ( a , b ; t ) = ∑ n = 0 ∞ ( a ) n ( b ) ntnn ！，其中 U ( a , a + b ; t ) 是第二类 Kummer 超几何函数。首先，我们为每个 Kummer 超几何多项式序列和第二类 Kummer 超几何多项式构造了具有精心选择的核（密度）函数的 Gauss-Weierstrass 型卷积算子 T wa , b 。然后我们将 Kummer 超几何多项式描述为唯一具有等于 0 的加权积分均值的 Appell 多项式。我们的方法受到 Hermite 多项式的 Gauss-Weierstrass 卷积变换和汇合超几何函数的 Kummer 积​​分表示的启发。