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Generalized Bernoulli polynomials revisited and some other Appell sequences. (English) Zbl 1097.33010
In this paper the authors study Appell sequences, that is, sequences of polynomials $$C_{n}(x)$$, $$n=0,1,2,\ldots$$ satisfying $$C_{n+1}^{\prime}(x)=(n+1)\,C_{n}(x)$$, in terms of dual sequences. As special cases of $$C_{n}(x)$$ the authors consider the sequences of generalized Bernoulli and Euler polynomials. In particular, the authors consider the following problem: Let $$C_{n}(x)$$, $$n=0,1,2,\ldots$$ be an Appell sequence, determine all sequences $$B_{n}(x)$$, $$n=0,1,2,\ldots$$ such that $$B_{n+1}^{\prime}(x)=(n+1)\,B_{n}(x)$$ and $$B_{n+1}(x+1)-B_{n+1}(x)=(n+1)\,C_{n}(x)$$. The case where $$C_{n}(x)$$ is the sequence of generalized Bernoulli polynomials $$B_{n}(x;k)$$, has been studied by N. E. NĂ¶rlund in the paper [Acta Math. 43, 121–196 (1920; JFM 47.0216.05)].
The authors give new results concerning the dual sequence of the generalized Bernoulli sequence. The dual sequence $$u_{n}$$ of a sequence $$P_{n}(x)$$ of monic polynomials, is a sequence of elements of the dual space $$\mathcal{P}^{\prime}$$ of the vector space $$\mathcal{P}$$ of polynomials with coefficients in $$\mathbb{C}$$, satisfying $$\langle u_{n}, P_{m}\rangle= \delta_{m,m}$$.
In the paper under review, it is shown that the canonical form $$u_{0}(k)$$ of a generalized Bernoulli sequence $$B_{n}(x;k)$$, $$k\geq 1$$, satisfies a homogeneous $$k$$th-order linear equation with polynomial coefficients and is a positive definite form. Similar results are obtained in the case where $$C_{n}(x)$$ is the sequence of monic Hermite polynomials $$\hat{H}_{n}(x)$$. The case where $$C_{n}(x)$$ is the sequence of generalized Euler polynomials $$E_{n}(x;k)$$, is also studied and its canonical form $$e_{0}(k)$$, $$k\geq 0$$ is determined.

##### MSC:
 33C65 Appell, Horn and Lauricella functions 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 11B68 Bernoulli and Euler numbers and polynomials
JFM 47.0216.05